SOCR - User contributions [en]

SOCR Courses 2011 2012 Stat13 1 Lab6

2012-03-01T15:52:22Z

SeanWang: /* Questions */

== [[SOCR_Courses_2011_2012_Stat13_1 | Stats 13.1]] - Laboratory Activity 6==

===Central Limit Theorem (CLT) Activity===

This activity represents a very general demonstration of the the [http://en.wikipedia.org/wiki/Central_limit_theorem Central Limit Theorem (CLT)]. The activity is based on the [http://socr.ucla.edu/htmls/SOCR_Experiments.html SOCR Sampling Distribution CLT Experiment]. This experiment builds upon a [http://www.ruf.rice.edu/~lane/stat_sim/sampling_dist/ RVLS CLT applet] by extending the applet functionality and providing the capability of sampling from any [[About_pages_for_SOCR_Distributions | SOCR Distribution]].
----

==Goals==
The aims of this activity are to:
* Provide intuitive notion of sampling from any process with a well-defined distribution.
* Motivate and facilitate learning of the [http://en.wikipedia.org/wiki/Central_limit_theorem central limit theorem].
* Empirically validate that sample-averages of random observations (most processes) follow approximately [http://en.wikipedia.org/wiki/Normal_distribution normal distribution].
* Empirically demonstrate that the ''sample-average'' is special and other [http://en.wikipedia.org/wiki/Point_estimator sample statistics] (e.g., median, variance, range, etc.) generally do not have distributions that are normal.
* Illustrate that the expectation of the sample-average equals the population mean (and the sample-average is typically a good measure of centrality for a population/process).
* Show that the variation of the sampling distribution of the mean rapidly decreases as the sample size increases (<math> ~1\over{\sqrt{n}}</math>).
* Reinforce the concepts of a native distribution, sample, sample distribution, sampling distribution, parameter estimator and data-driven numerical parameter estimate.

==The SOCR CLT Experiment==
To start the this Experiment, go to [http://www.socr.ucla.edu/htmls/SOCR_Experiments.html SOCR Experiments] and select the SOCR Sampling Distribution CLT Experiment from the drop-down list of experiments in the left panel. The image below shows the interface to this experiment. Notice the main control widgets on this image (boxed in blue and pointed to by arrows). The generic control buttons on the top allow you to do one or multiple steps/runs, stop and reset this experiment. The two tabs in the main frame provide graphical access to the results of the experiment (Histograms and Summaries) or the Distribution selection panel (Distributions). Remember that choosing sample-sizes <= 16 will animate the samples (second graphing row), whereas larger sample-sizes (N>20) will only show the updates of the sampling distributions (bottom two graphing rows).

<center>[[Image:SOCR_Activities_General_CLT_Dinov_012207_Fig1.jpg|400px]]</center>

===Experiment 1===
Expand your Experiment panel (right panel) by clicking/dragging the vertical split-pane bar. Choose the two sample sizes for the two statistics to be 10. Press the '''step'''-button a few of times (2-5) to see the experiment run several times. Notice how data is being sampled from the native population (the distribution of the process on the top). For each step, the process of sampling 2 samples of 10 observations will generate 2 sample statistics of the 2 parameters of interest (these are defaulted to ''mean'' and ''variance''). At each step, you can see the plots of all sample values, as well as the computed sample statistics for each parameter. The sample values are shown on the second row graph, below the distribution of the process, and the two sample statistics are plotted on the bottom two rows. If we run this experiment many times, the bottom two graphs/histograms become good approximations to the corresponding sampling distributions. If we did this infinitely many times these two graphs become the sampling distributions of the chosen sample statistics (as the observations/measurements are independent within each sample and between samples). Finally, press the '''Refresh Stats Table''' button on the top to see the sample summary statistics for the native population distribution (row 1), last sample (row 2) and the two sampling distributions, in this case ''mean'' and ''variance'' (rows 3 and 4).

<center>[[Image:SOCR_Activities_General_CLT_Dinov_012207_Fig2.jpg|400px]]</center>

===Experiment 2===
For this experiment we'll look at the mean, standard deviation, skewness and kurtosis of the sample-average and the sample-variance (these two sample data-driven statistical estimates). Choose sample-sizes of 50, for both estimates (mean and variance). Select the '''Fit Normal Curve''' check-boxes for both sample distributions. '''Run''' through the experiment 10 times (by clicking the Run button and selecting Stop 10) and then click '''Refresh Stats Table''' button on the top to see the sample summary statistics. '''Run''' again 100 times by selecting Stop 100 instead of Stop 10. Try to understand and relate these sample-distribution statistics to their analogues from the native population (on the top row). For example, the mean of the multiple sample-averages is about the same as the mean of the native population, but the standard deviation of the sampling distribution of the average is about <math>\sigma\over{\sqrt{n}}</math>, where <math>\sigma</math> is the standard deviation of the original native process/distribution.

Question 1: Verify that the standard deviation of the sampling distribution of the average is about <math>\sigma\over{\sqrt{n}}</math> after running the experiment 100 times, where <math>\sigma</math> is the standard deviation of the original native process/distribution.

<center>[[Image:SOCR_Activities_General_CLT_Dinov_012207_Fig3.jpg|400px]]</center>

===Experiment 3===
Now let's select any of the [[About_pages_for_SOCR_Distributions | SOCR Distributions]], sample from it repeatedly and see if the central limit theorem is valid for the process we have selected. Try Poisson, Beta, Gamma, Cauchy and other continuous or discrete distributions. Choose a sample size 50 or bigger. Are our empirical results in agreement with the CLT? Go to the '''Distributions''' tab on the top of the graphing panel. Reset the experiments panel (button on the top). Select a distribution from the drop-down list of distributions in this list. Choose appropriate parameters for your distribution, if any, and click the '''Sample from this Current Distribution''' button to send this distribution to the graphing panel in the '''Histograms and Summaries''' tab. Go to this panel and again run the experiment several times. Notice how we now sample from a Non-Normal Distribution for the first time. In this case we had chosen the Beta distribution (<math>\alpha=6.7, \beta=0.5</math>).

Question 2: Take a snapshot of the experiment using one of the distributions mentioned above. Verify that the standard deviation of the sampling distribution of the average is about <math>\sigma\over{\sqrt{n}}</math>, where <math>\sigma</math> is the standard deviation of the original native process/distribution.
<center>[[Image:SOCR_Activities_General_CLT_Dinov_012207_Fig4.jpg|300px]]
[[Image:SOCR_Activities_General_CLT_Dinov_012207_Fig5.jpg|300px]]</center>

===Experiment 4===
Suppose the distribution we want to sample from is not included in the list of [[About_pages_for_SOCR_Distributions | SOCR Distributions]], under the '''Distributions''' tab. We can then draw a shape for a hypothetical distribution by clicking and dragging the mouse in the top graphing canvas (Histograms and Summaries tab panel). Choose a sample size 50 or bigger. This way you can construct continuous and discontinuous, symmetric and asymmetric, unimodal and multi-modal, leptokurtic and mesokurtic and other [http://en.wikipedia.org/wiki/Probability_distribution types of distributions]. In the figure below, we had demonstrated this functionality to study differences between two data-driven estimates for the population center - sample [http://en.wikipedia.org/wiki/Mean mean] and sample [http://en.wikipedia.org/wiki/Median median]. Look how the sampling distribution of the sample-average is very close to Normal, where as the sampling distribution of the sample median is not.

Question 3: Take a snapshot of an experiment where you create your own distribution. Verify that the standard deviation of the sampling distribution of the average is about <math>\sigma\over{\sqrt{n}}</math>, where <math>\sigma</math> is the standard deviation of the original native process/distribution.

<center>[[Image:SOCR_Activities_General_CLT_Dinov_012207_Fig6.jpg|300px]]</center>

==Questions==
* What effects will asymmetry, gaps and continuity of the native distribution have on the applicability of the CLT, or on the asymptotic distribution of various sample statistics?

Answering the following questions is '''optional''':

* When can we reasonably expect statistics, other than the sample mean, to have CLT properties?
* If a native process has <math>\sigma_{X}=10</math> and we take a sample of size 10, what will be <math>\sigma_{\overline{X}}</math>? Does it depend on the shape of the original process? How large should the sample-size be so that <math>\sigma_{\overline{X}}={2\over 3}\sigma_{X}</math>?

==Applications==
The second part of this SOCR Activity demonstrates the [[SOCR_EduMaterials_Activities_GCLT_Applications | applications of the Central Limit Theorem]].

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2011_2012_Stat13_1_Lab6}}

SOCR Courses 2011 2012 Stat13 1 Lab6

2012-03-01T15:50:44Z

SeanWang: /* Experiment 2 */

== [[SOCR_Courses_2011_2012_Stat13_1 | Stats 13.1]] - Laboratory Activity 6==

===Central Limit Theorem (CLT) Activity===

This activity represents a very general demonstration of the the [http://en.wikipedia.org/wiki/Central_limit_theorem Central Limit Theorem (CLT)]. The activity is based on the [http://socr.ucla.edu/htmls/SOCR_Experiments.html SOCR Sampling Distribution CLT Experiment]. This experiment builds upon a [http://www.ruf.rice.edu/~lane/stat_sim/sampling_dist/ RVLS CLT applet] by extending the applet functionality and providing the capability of sampling from any [[About_pages_for_SOCR_Distributions | SOCR Distribution]].
----

==Goals==
The aims of this activity are to:
* Provide intuitive notion of sampling from any process with a well-defined distribution.
* Motivate and facilitate learning of the [http://en.wikipedia.org/wiki/Central_limit_theorem central limit theorem].
* Empirically validate that sample-averages of random observations (most processes) follow approximately [http://en.wikipedia.org/wiki/Normal_distribution normal distribution].
* Empirically demonstrate that the ''sample-average'' is special and other [http://en.wikipedia.org/wiki/Point_estimator sample statistics] (e.g., median, variance, range, etc.) generally do not have distributions that are normal.
* Illustrate that the expectation of the sample-average equals the population mean (and the sample-average is typically a good measure of centrality for a population/process).
* Show that the variation of the sampling distribution of the mean rapidly decreases as the sample size increases (<math> ~1\over{\sqrt{n}}</math>).
* Reinforce the concepts of a native distribution, sample, sample distribution, sampling distribution, parameter estimator and data-driven numerical parameter estimate.

==The SOCR CLT Experiment==
To start the this Experiment, go to [http://www.socr.ucla.edu/htmls/SOCR_Experiments.html SOCR Experiments] and select the SOCR Sampling Distribution CLT Experiment from the drop-down list of experiments in the left panel. The image below shows the interface to this experiment. Notice the main control widgets on this image (boxed in blue and pointed to by arrows). The generic control buttons on the top allow you to do one or multiple steps/runs, stop and reset this experiment. The two tabs in the main frame provide graphical access to the results of the experiment (Histograms and Summaries) or the Distribution selection panel (Distributions). Remember that choosing sample-sizes <= 16 will animate the samples (second graphing row), whereas larger sample-sizes (N>20) will only show the updates of the sampling distributions (bottom two graphing rows).

<center>[[Image:SOCR_Activities_General_CLT_Dinov_012207_Fig1.jpg|400px]]</center>

===Experiment 1===
Expand your Experiment panel (right panel) by clicking/dragging the vertical split-pane bar. Choose the two sample sizes for the two statistics to be 10. Press the '''step'''-button a few of times (2-5) to see the experiment run several times. Notice how data is being sampled from the native population (the distribution of the process on the top). For each step, the process of sampling 2 samples of 10 observations will generate 2 sample statistics of the 2 parameters of interest (these are defaulted to ''mean'' and ''variance''). At each step, you can see the plots of all sample values, as well as the computed sample statistics for each parameter. The sample values are shown on the second row graph, below the distribution of the process, and the two sample statistics are plotted on the bottom two rows. If we run this experiment many times, the bottom two graphs/histograms become good approximations to the corresponding sampling distributions. If we did this infinitely many times these two graphs become the sampling distributions of the chosen sample statistics (as the observations/measurements are independent within each sample and between samples). Finally, press the '''Refresh Stats Table''' button on the top to see the sample summary statistics for the native population distribution (row 1), last sample (row 2) and the two sampling distributions, in this case ''mean'' and ''variance'' (rows 3 and 4).

<center>[[Image:SOCR_Activities_General_CLT_Dinov_012207_Fig2.jpg|400px]]</center>

===Experiment 2===
For this experiment we'll look at the mean, standard deviation, skewness and kurtosis of the sample-average and the sample-variance (these two sample data-driven statistical estimates). Choose sample-sizes of 50, for both estimates (mean and variance). Select the '''Fit Normal Curve''' check-boxes for both sample distributions. '''Run''' through the experiment 10 times (by clicking the Run button and selecting Stop 10) and then click '''Refresh Stats Table''' button on the top to see the sample summary statistics. '''Run''' again 100 times by selecting Stop 100 instead of Stop 10. Try to understand and relate these sample-distribution statistics to their analogues from the native population (on the top row). For example, the mean of the multiple sample-averages is about the same as the mean of the native population, but the standard deviation of the sampling distribution of the average is about <math>\sigma\over{\sqrt{n}}</math>, where <math>\sigma</math> is the standard deviation of the original native process/distribution.

Question 1: Verify that the standard deviation of the sampling distribution of the average is about <math>\sigma\over{\sqrt{n}}</math> after running the experiment 100 times, where <math>\sigma</math> is the standard deviation of the original native process/distribution.

<center>[[Image:SOCR_Activities_General_CLT_Dinov_012207_Fig3.jpg|400px]]</center>

===Experiment 3===
Now let's select any of the [[About_pages_for_SOCR_Distributions | SOCR Distributions]], sample from it repeatedly and see if the central limit theorem is valid for the process we have selected. Try Poisson, Beta, Gamma, Cauchy and other continuous or discrete distributions. Choose a sample size 50 or bigger. Are our empirical results in agreement with the CLT? Go to the '''Distributions''' tab on the top of the graphing panel. Reset the experiments panel (button on the top). Select a distribution from the drop-down list of distributions in this list. Choose appropriate parameters for your distribution, if any, and click the '''Sample from this Current Distribution''' button to send this distribution to the graphing panel in the '''Histograms and Summaries''' tab. Go to this panel and again run the experiment several times. Notice how we now sample from a Non-Normal Distribution for the first time. In this case we had chosen the Beta distribution (<math>\alpha=6.7, \beta=0.5</math>).

Question 2: Take a snapshot of the experiment using one of the distributions mentioned above. Verify that the standard deviation of the sampling distribution of the average is about <math>\sigma\over{\sqrt{n}}</math>, where <math>\sigma</math> is the standard deviation of the original native process/distribution.
<center>[[Image:SOCR_Activities_General_CLT_Dinov_012207_Fig4.jpg|300px]]
[[Image:SOCR_Activities_General_CLT_Dinov_012207_Fig5.jpg|300px]]</center>

===Experiment 4===
Suppose the distribution we want to sample from is not included in the list of [[About_pages_for_SOCR_Distributions | SOCR Distributions]], under the '''Distributions''' tab. We can then draw a shape for a hypothetical distribution by clicking and dragging the mouse in the top graphing canvas (Histograms and Summaries tab panel). Choose a sample size 50 or bigger. This way you can construct continuous and discontinuous, symmetric and asymmetric, unimodal and multi-modal, leptokurtic and mesokurtic and other [http://en.wikipedia.org/wiki/Probability_distribution types of distributions]. In the figure below, we had demonstrated this functionality to study differences between two data-driven estimates for the population center - sample [http://en.wikipedia.org/wiki/Mean mean] and sample [http://en.wikipedia.org/wiki/Median median]. Look how the sampling distribution of the sample-average is very close to Normal, where as the sampling distribution of the sample median is not.

Question 3: Take a snapshot of an experiment where you create your own distribution. Verify that the standard deviation of the sampling distribution of the average is about <math>\sigma\over{\sqrt{n}}</math>, where <math>\sigma</math> is the standard deviation of the original native process/distribution.

<center>[[Image:SOCR_Activities_General_CLT_Dinov_012207_Fig6.jpg|300px]]</center>

==Questions==
* What effects will asymmetry, gaps and continuity of the native distribution have on the applicability of the CLT, or on the asymptotic distribution of various sample statistics?

Answering the following questions is optional:

* When can we reasonably expect statistics, other than the sample mean, to have CLT properties?
* If a native process has <math>\sigma_{X}=10</math> and we take a sample of size 10, what will be <math>\sigma_{\overline{X}}</math>? Does it depend on the shape of the original process? How large should the sample-size be so that <math>\sigma_{\overline{X}}={2\over 3}\sigma_{X}</math>?

==Applications==
The second part of this SOCR Activity demonstrates the [[SOCR_EduMaterials_Activities_GCLT_Applications | applications of the Central Limit Theorem]].

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2011_2012_Stat13_1_Lab6}}

SOCR Courses 2011 2012 Stat13 1 Lab5

2012-02-23T18:09:42Z

SeanWang: /* Exercise 1 */

== [[SOCR_Courses_2011_2012_Stat13_1 | Stats 13.1]] - Laboratory Activity 5: Confidence Interval Activity==

This is an activity to explore the confidence intervals for the population mean when the standard deviation is known.

===Description===
You can access the applet for the confidence intervals experiment at [http://www.socr.ucla.edu/htmls/SOCR_Experiments.html SOCR Experiments]. Use the scroll down button to find the [http://socr.ucla.edu/htmls/exp/Confidence_Interval_Experiment.html Confidence Interval Experiment].

The confidence interval for the population mean <math>\mu</math> when <math> \sigma </math> is known is given by (when n>30):
<math>
\bar x - z_{\frac{\alpha}{2}} \frac{\sigma}{\sqrt{n}} \le \mu \le \bar x + z_{\frac{\alpha}{2}} \frac{\sigma}{\sqrt{n}}
</math>
where <math>z_{\frac{\alpha}{2}}</math> is the value of <math>z</math> such that the area to its left (or right) is <math>\frac{\alpha}{2}</math>. For example if we choose a <math>95 \% </math> confidence level then <math>1-\alpha=0.95</math> or <math>\alpha=0.05</math> and therefore <math>\frac{\alpha}{2}=0.025</math> which gives <math>z_{\frac{\alpha}{2}}=1.96</math>. The sample mean <math>\bar x</math> is the mean of the sample of size <math>n</math>, and <math>\sigma </math> is the standard deviation. In this lab we will generate many confidence intervals based on different sample sizes. The samples in this lab are always selected from the standard normal distribution <math>N(0,1)</math>. Therefore we know that the mean is <math>\mu=0</math>, and the standard deviation <math>\sigma=1</math>. Let's pretend that <math>\mu</math> is unknown and that only <math>\sigma</math> is known. We will select many samples each one of size <math>n</math> and use it to construct a confidence interval for the population mean.

===Exercise 1===
Using the scroll down button select "Number of Experiments = 100". Select sample size <math>n=20</math>, and choose number of intervals 200. It means: You will select 200 samples and with each sample you will obtain a confidence interval. You will do this 100 times. Take a snapshot and answer the following questions.

# What do the numbers -3, -2, -1, 0, 1, 2, 3 represent?
# What do the blue lines represent?
# How is the confidence interval represented?
# What does the green dot represent?
# How many intervals (out of the 200) do you ''expect'' to miss the population mean <math>\mu=0</math>? How many ''did'' when you ran it (in the final experiment)?
# Write down the formula on which these confidence intervals are based.

===Exercise 2===
# Reset and repeat Exercise 1, answering questions 5 and 6, with <math>\alpha=0.01</math>. Will the intervals be larger, smaller, or stay the same compared to that of Exercise 1?
# Reset and repeat Exercise 1, answering questions 5 and 6, with sample size now <math>n=80</math> and <math>\alpha=0.01</math>. Will the intervals be larger, smaller, or stay the same compared to that of Exercise 1?
# Reset and repeat Exercise 1, answering questions 5 and 6, with sample size <math>n=80</math> and <math>\alpha=1.0E-4</math> (this is <math>10^{-4}</math>). Will the intervals be larger, smaller, or stay the same compared to that of Exercise 1? Take a snapshot.
# What should happen to the lengths of the intervals as <math>\alpha</math> gets larger? What about as <math>n</math> gets larger?

Below you can see a snapshot of the run of 100 intervals with <math> n=36, \ \alpha=0.05 </math>.

<center>[[Image: SOCR_Activities_Christou_christou_confint.jpg|600px]]</center>

===See also===
* [[SOCR_EduMaterials_Activities_General_CI_Experiment | General Confidence Interval Activity]] and the [http://socr.ucla.edu/htmls/exp/Confidence_Interval_Experiment_General.html General Confidence Interval Applet].

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2011_2012_Stat13_1_Lab5}}

SOCR Courses 2011 2012 Stat13 1 Lab5

2012-02-23T15:38:22Z

SeanWang: /* Exercise 2 */

== [[SOCR_Courses_2011_2012_Stat13_1 | Stats 13.1]] - Laboratory Activity 5: Confidence Interval Activity==

This is an activity to explore the confidence intervals for the population mean when the standard deviation is known.

===Description===
You can access the applet for the confidence intervals experiment at [http://www.socr.ucla.edu/htmls/SOCR_Experiments.html SOCR Experiments]. Use the scroll down button to find the [http://socr.ucla.edu/htmls/exp/Confidence_Interval_Experiment.html Confidence Interval Experiment].

The confidence interval for the population mean <math>\mu</math> when <math> \sigma </math> is known is given by (when n>30):
<math>
\bar x - z_{\frac{\alpha}{2}} \frac{\sigma}{\sqrt{n}} \le \mu \le \bar x + z_{\frac{\alpha}{2}} \frac{\sigma}{\sqrt{n}}
</math>
where <math>z_{\frac{\alpha}{2}}</math> is the value of <math>z</math> such that the area to its left (or right) is <math>\frac{\alpha}{2}</math>. For example if we choose a <math>95 \% </math> confidence level then <math>1-\alpha=0.95</math> or <math>\alpha=0.05</math> and therefore <math>\frac{\alpha}{2}=0.025</math> which gives <math>z_{\frac{\alpha}{2}}=1.96</math>. The sample mean <math>\bar x</math> is the mean of the sample of size <math>n</math>, and <math>\sigma </math> is the standard deviation. In this lab we will generate many confidence intervals based on different sample sizes. The samples in this lab are always selected from the standard normal distribution <math>N(0,1)</math>. Therefore we know that the mean is <math>\mu=0</math>, and the standard deviation <math>\sigma=1</math>. Let's pretend that <math>\mu</math> is unknown and that only <math>\sigma</math> is known. We will select many samples each one of size <math>n</math> and use it to construct a confidence interval for the population mean.

===Exercise 1===
Using the scroll down button select "Number of Experiments = 100". Select sample size <math>n=20</math>, and choose number of intervals 200. It means: You will select 200 samples and with each sample you will obtain a confidence interval. You will do this 100 times. Take a snapshot and answer the following questions.

# What do the numbers -3, -2, -1, 0, 1, 2, 3 represent?
# What do the blue lines represent?
# How is the confidence interval represented?
# What does the green dot represent?
# How many intervals (out of the 200) do you ''expect'' to miss the population mean <math>\mu=0</math>? How many ''did'' when you ran it?
# Write down the formula on which these confidence intervals are based.

===Exercise 2===
# Reset and repeat Exercise 1, answering questions 5 and 6, with <math>\alpha=0.01</math>. Will the intervals be larger, smaller, or stay the same compared to that of Exercise 1?
# Reset and repeat Exercise 1, answering questions 5 and 6, with sample size now <math>n=80</math> and <math>\alpha=0.01</math>. Will the intervals be larger, smaller, or stay the same compared to that of Exercise 1?
# Reset and repeat Exercise 1, answering questions 5 and 6, with sample size <math>n=80</math> and <math>\alpha=1.0E-4</math> (this is <math>10^{-4}</math>). Will the intervals be larger, smaller, or stay the same compared to that of Exercise 1? Take a snapshot.
# What should happen to the lengths of the intervals as <math>\alpha</math> gets larger? What about as <math>n</math> gets larger?

Below you can see a snapshot of the run of 100 intervals with <math> n=36, \ \alpha=0.05 </math>.

<center>[[Image: SOCR_Activities_Christou_christou_confint.jpg|600px]]</center>

===See also===
* [[SOCR_EduMaterials_Activities_General_CI_Experiment | General Confidence Interval Activity]] and the [http://socr.ucla.edu/htmls/exp/Confidence_Interval_Experiment_General.html General Confidence Interval Applet].

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2011_2012_Stat13_1_Lab5}}

SOCR Courses 2011 2012 Stat13 1 Lab5

2012-02-23T15:37:26Z

SeanWang: /* Exercise 2 */

== [[SOCR_Courses_2011_2012_Stat13_1 | Stats 13.1]] - Laboratory Activity 5: Confidence Interval Activity==

This is an activity to explore the confidence intervals for the population mean when the standard deviation is known.

===Description===
You can access the applet for the confidence intervals experiment at [http://www.socr.ucla.edu/htmls/SOCR_Experiments.html SOCR Experiments]. Use the scroll down button to find the [http://socr.ucla.edu/htmls/exp/Confidence_Interval_Experiment.html Confidence Interval Experiment].

The confidence interval for the population mean <math>\mu</math> when <math> \sigma </math> is known is given by (when n>30):
<math>
\bar x - z_{\frac{\alpha}{2}} \frac{\sigma}{\sqrt{n}} \le \mu \le \bar x + z_{\frac{\alpha}{2}} \frac{\sigma}{\sqrt{n}}
</math>
where <math>z_{\frac{\alpha}{2}}</math> is the value of <math>z</math> such that the area to its left (or right) is <math>\frac{\alpha}{2}</math>. For example if we choose a <math>95 \% </math> confidence level then <math>1-\alpha=0.95</math> or <math>\alpha=0.05</math> and therefore <math>\frac{\alpha}{2}=0.025</math> which gives <math>z_{\frac{\alpha}{2}}=1.96</math>. The sample mean <math>\bar x</math> is the mean of the sample of size <math>n</math>, and <math>\sigma </math> is the standard deviation. In this lab we will generate many confidence intervals based on different sample sizes. The samples in this lab are always selected from the standard normal distribution <math>N(0,1)</math>. Therefore we know that the mean is <math>\mu=0</math>, and the standard deviation <math>\sigma=1</math>. Let's pretend that <math>\mu</math> is unknown and that only <math>\sigma</math> is known. We will select many samples each one of size <math>n</math> and use it to construct a confidence interval for the population mean.

===Exercise 1===
Using the scroll down button select "Number of Experiments = 100". Select sample size <math>n=20</math>, and choose number of intervals 200. It means: You will select 200 samples and with each sample you will obtain a confidence interval. You will do this 100 times. Take a snapshot and answer the following questions.

# What do the numbers -3, -2, -1, 0, 1, 2, 3 represent?
# What do the blue lines represent?
# How is the confidence interval represented?
# What does the green dot represent?
# How many intervals (out of the 200) do you ''expect'' to miss the population mean <math>\mu=0</math>? How many ''did'' when you ran it?
# Write down the formula on which these confidence intervals are based.

===Exercise 2===
# Reset and repeat Exercise 1, answering questions 5 and 6, with <math>\alpha=0.01</math>. Will the intervals be larger, smaller, or stay the same compared to that of Exercise 1?
# Reset and repeat Exercise 1, answering questions 5 and 6, with sample size now <math>n=80</math> and <math>\alpha=0.01</math>. Will the intervals be larger, smaller, or stay the same compared to that of Exercise 1?
# Reset and repeat Exercise 1, answering questions 5 and 6, with sample size <math>n=80</math> and <math>\alpha=1.0E-4</math> (this is <math>10^{-4}</math>). Will the intervals be larger, smaller, or stay the same compared to that of Exercise 1? Take a snapshot.
# What does this tell you about what should happen when you make <math>\alpha</math> smaller or larger? What about when you make <math>n</math> smaller or larger?

Below you can see a snapshot of the run of 100 intervals with <math> n=36, \ \alpha=0.05 </math>.

<center>[[Image: SOCR_Activities_Christou_christou_confint.jpg|600px]]</center>

===See also===
* [[SOCR_EduMaterials_Activities_General_CI_Experiment | General Confidence Interval Activity]] and the [http://socr.ucla.edu/htmls/exp/Confidence_Interval_Experiment_General.html General Confidence Interval Applet].

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2011_2012_Stat13_1_Lab5}}

SOCR Courses 2011 2012 Stat13 1 Lab5

2012-02-23T15:35:05Z

SeanWang: /* Exercise 2 */

== [[SOCR_Courses_2011_2012_Stat13_1 | Stats 13.1]] - Laboratory Activity 5: Confidence Interval Activity==

This is an activity to explore the confidence intervals for the population mean when the standard deviation is known.

===Description===
You can access the applet for the confidence intervals experiment at [http://www.socr.ucla.edu/htmls/SOCR_Experiments.html SOCR Experiments]. Use the scroll down button to find the [http://socr.ucla.edu/htmls/exp/Confidence_Interval_Experiment.html Confidence Interval Experiment].

The confidence interval for the population mean <math>\mu</math> when <math> \sigma </math> is known is given by (when n>30):
<math>
\bar x - z_{\frac{\alpha}{2}} \frac{\sigma}{\sqrt{n}} \le \mu \le \bar x + z_{\frac{\alpha}{2}} \frac{\sigma}{\sqrt{n}}
</math>
where <math>z_{\frac{\alpha}{2}}</math> is the value of <math>z</math> such that the area to its left (or right) is <math>\frac{\alpha}{2}</math>. For example if we choose a <math>95 \% </math> confidence level then <math>1-\alpha=0.95</math> or <math>\alpha=0.05</math> and therefore <math>\frac{\alpha}{2}=0.025</math> which gives <math>z_{\frac{\alpha}{2}}=1.96</math>. The sample mean <math>\bar x</math> is the mean of the sample of size <math>n</math>, and <math>\sigma </math> is the standard deviation. In this lab we will generate many confidence intervals based on different sample sizes. The samples in this lab are always selected from the standard normal distribution <math>N(0,1)</math>. Therefore we know that the mean is <math>\mu=0</math>, and the standard deviation <math>\sigma=1</math>. Let's pretend that <math>\mu</math> is unknown and that only <math>\sigma</math> is known. We will select many samples each one of size <math>n</math> and use it to construct a confidence interval for the population mean.

===Exercise 1===
Using the scroll down button select "Number of Experiments = 100". Select sample size <math>n=20</math>, and choose number of intervals 200. It means: You will select 200 samples and with each sample you will obtain a confidence interval. You will do this 100 times. Take a snapshot and answer the following questions.

# What do the numbers -3, -2, -1, 0, 1, 2, 3 represent?
# What do the blue lines represent?
# How is the confidence interval represented?
# What does the green dot represent?
# How many intervals (out of the 200) do you ''expect'' to miss the population mean <math>\mu=0</math>? How many ''did'' when you ran it?
# Write down the formula on which these confidence intervals are based.

===Exercise 2===
# Reset and repeat Exercise 1, answering questions 5 and 6, with <math>\alpha=0.01</math>. Will the intervals be larger, smaller, or stay the same compared to that of Exercise 1?
# Reset and repeat Exercise 1, answering questions 5 and 6, with sample size now <math>n=80</math> and <math>\alpha=0.01</math>. Will the intervals be larger, smaller, or stay the same compared to that of Exercise 1?
# Reset and repeat Exercise 1, answering questions 5 and 6, with sample size <math>n=80</math> and <math>\alpha=1.0E-4</math> (this is <math>10^{-4}</math>). Will the intervals be larger, smaller, or stay the same compared to that of Exercise 1?

Below you can see a snapshot of the run of 100 intervals with <math> n=36, \ \alpha=0.05 </math>.

<center>[[Image: SOCR_Activities_Christou_christou_confint.jpg|600px]]</center>

===See also===
* [[SOCR_EduMaterials_Activities_General_CI_Experiment | General Confidence Interval Activity]] and the [http://socr.ucla.edu/htmls/exp/Confidence_Interval_Experiment_General.html General Confidence Interval Applet].

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2011_2012_Stat13_1_Lab5}}

SOCR Courses 2011 2012 Stat13 1 Lab5

2012-02-23T15:34:56Z

SeanWang: /* Exercise 2 */

== [[SOCR_Courses_2011_2012_Stat13_1 | Stats 13.1]] - Laboratory Activity 5: Confidence Interval Activity==

This is an activity to explore the confidence intervals for the population mean when the standard deviation is known.

===Description===
You can access the applet for the confidence intervals experiment at [http://www.socr.ucla.edu/htmls/SOCR_Experiments.html SOCR Experiments]. Use the scroll down button to find the [http://socr.ucla.edu/htmls/exp/Confidence_Interval_Experiment.html Confidence Interval Experiment].

The confidence interval for the population mean <math>\mu</math> when <math> \sigma </math> is known is given by (when n>30):
<math>
\bar x - z_{\frac{\alpha}{2}} \frac{\sigma}{\sqrt{n}} \le \mu \le \bar x + z_{\frac{\alpha}{2}} \frac{\sigma}{\sqrt{n}}
</math>
where <math>z_{\frac{\alpha}{2}}</math> is the value of <math>z</math> such that the area to its left (or right) is <math>\frac{\alpha}{2}</math>. For example if we choose a <math>95 \% </math> confidence level then <math>1-\alpha=0.95</math> or <math>\alpha=0.05</math> and therefore <math>\frac{\alpha}{2}=0.025</math> which gives <math>z_{\frac{\alpha}{2}}=1.96</math>. The sample mean <math>\bar x</math> is the mean of the sample of size <math>n</math>, and <math>\sigma </math> is the standard deviation. In this lab we will generate many confidence intervals based on different sample sizes. The samples in this lab are always selected from the standard normal distribution <math>N(0,1)</math>. Therefore we know that the mean is <math>\mu=0</math>, and the standard deviation <math>\sigma=1</math>. Let's pretend that <math>\mu</math> is unknown and that only <math>\sigma</math> is known. We will select many samples each one of size <math>n</math> and use it to construct a confidence interval for the population mean.

===Exercise 1===
Using the scroll down button select "Number of Experiments = 100". Select sample size <math>n=20</math>, and choose number of intervals 200. It means: You will select 200 samples and with each sample you will obtain a confidence interval. You will do this 100 times. Take a snapshot and answer the following questions.

# What do the numbers -3, -2, -1, 0, 1, 2, 3 represent?
# What do the blue lines represent?
# How is the confidence interval represented?
# What does the green dot represent?
# How many intervals (out of the 200) do you ''expect'' to miss the population mean <math>\mu=0</math>? How many ''did'' when you ran it?
# Write down the formula on which these confidence intervals are based.

===Exercise 2===
# Reset and repeat Exercise 1, answering questions 5 and 6, with <math>\alpha=0.01</math>. Will the intervals be larger, smaller, or stay the same compared to that of Exercise 1?
# Reset and repeat Exercise 1, answering questions 5 and 6, with sample size now <math>n=80</math> and <math>\alpha=0.01</math>. Will the intervals be larger, smaller, or stay the same compared to that of Exercise 1?
# Reset and repeat Exercise 1, answering questions 5 and 6, with sample size <math>n=80</math> and <math>\alpha=1.0E-4</math> (this is <math>10^{-4}</math>). Will the intervals be larger, smaller, or stay the same compared to that of Exercise 1?

Below you can see a snapshot of the run of 100 intervals with <math> n=36, \ \alpha=0.05 </math>.

<center>[[Image: SOCR_Activities_Christou_christou_confint.jpg|600px]]</center>

===See also===
* [[SOCR_EduMaterials_Activities_General_CI_Experiment | General Confidence Interval Activity]] and the [http://socr.ucla.edu/htmls/exp/Confidence_Interval_Experiment_General.html General Confidence Interval Applet].

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2011_2012_Stat13_1_Lab5}}

SOCR Courses 2011 2012 Stat13 1 Lab5

2012-02-23T15:34:36Z

SeanWang: /* Exercise 1 */

== [[SOCR_Courses_2011_2012_Stat13_1 | Stats 13.1]] - Laboratory Activity 5: Confidence Interval Activity==

This is an activity to explore the confidence intervals for the population mean when the standard deviation is known.

===Description===
You can access the applet for the confidence intervals experiment at [http://www.socr.ucla.edu/htmls/SOCR_Experiments.html SOCR Experiments]. Use the scroll down button to find the [http://socr.ucla.edu/htmls/exp/Confidence_Interval_Experiment.html Confidence Interval Experiment].

The confidence interval for the population mean <math>\mu</math> when <math> \sigma </math> is known is given by (when n>30):
<math>
\bar x - z_{\frac{\alpha}{2}} \frac{\sigma}{\sqrt{n}} \le \mu \le \bar x + z_{\frac{\alpha}{2}} \frac{\sigma}{\sqrt{n}}
</math>
where <math>z_{\frac{\alpha}{2}}</math> is the value of <math>z</math> such that the area to its left (or right) is <math>\frac{\alpha}{2}</math>. For example if we choose a <math>95 \% </math> confidence level then <math>1-\alpha=0.95</math> or <math>\alpha=0.05</math> and therefore <math>\frac{\alpha}{2}=0.025</math> which gives <math>z_{\frac{\alpha}{2}}=1.96</math>. The sample mean <math>\bar x</math> is the mean of the sample of size <math>n</math>, and <math>\sigma </math> is the standard deviation. In this lab we will generate many confidence intervals based on different sample sizes. The samples in this lab are always selected from the standard normal distribution <math>N(0,1)</math>. Therefore we know that the mean is <math>\mu=0</math>, and the standard deviation <math>\sigma=1</math>. Let's pretend that <math>\mu</math> is unknown and that only <math>\sigma</math> is known. We will select many samples each one of size <math>n</math> and use it to construct a confidence interval for the population mean.

===Exercise 1===
Using the scroll down button select "Number of Experiments = 100". Select sample size <math>n=20</math>, and choose number of intervals 200. It means: You will select 200 samples and with each sample you will obtain a confidence interval. You will do this 100 times. Take a snapshot and answer the following questions.

# What do the numbers -3, -2, -1, 0, 1, 2, 3 represent?
# What do the blue lines represent?
# How is the confidence interval represented?
# What does the green dot represent?
# How many intervals (out of the 200) do you ''expect'' to miss the population mean <math>\mu=0</math>? How many ''did'' when you ran it?
# Write down the formula on which these confidence intervals are based.

===Exercise 2===
# Reset and repeat Exercise 1, answering questions 1 and 6, with <math>\alpha=0.01</math>. Will the intervals be larger, smaller, or stay the same compared to that of Exercise 1?
# Reset and repeat Exercise 1, answering questions 1 and 6, with sample size now <math>n=80</math> and <math>\alpha=0.01</math>. Will the intervals be larger, smaller, or stay the same compared to that of Exercise 1?
# Reset and repeat Exercise 1, answering questions 1 and 6, with sample size <math>n=80</math> and <math>\alpha=1.0E-4</math> (this is <math>10^{-4}</math>). Will the intervals be larger, smaller, or stay the same compared to that of Exercise 1?

Below you can see a snapshot of the run of 100 intervals with <math> n=36, \ \alpha=0.05 </math>.

<center>[[Image: SOCR_Activities_Christou_christou_confint.jpg|600px]]</center>

===See also===
* [[SOCR_EduMaterials_Activities_General_CI_Experiment | General Confidence Interval Activity]] and the [http://socr.ucla.edu/htmls/exp/Confidence_Interval_Experiment_General.html General Confidence Interval Applet].

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2011_2012_Stat13_1_Lab5}}

SOCR Courses 2011 2012 Stat13 1 Lab5

2012-02-23T15:33:39Z

SeanWang: /* Exercise 1 */

== [[SOCR_Courses_2011_2012_Stat13_1 | Stats 13.1]] - Laboratory Activity 5: Confidence Interval Activity==

This is an activity to explore the confidence intervals for the population mean when the standard deviation is known.

===Description===
You can access the applet for the confidence intervals experiment at [http://www.socr.ucla.edu/htmls/SOCR_Experiments.html SOCR Experiments]. Use the scroll down button to find the [http://socr.ucla.edu/htmls/exp/Confidence_Interval_Experiment.html Confidence Interval Experiment].

The confidence interval for the population mean <math>\mu</math> when <math> \sigma </math> is known is given by (when n>30):
<math>
\bar x - z_{\frac{\alpha}{2}} \frac{\sigma}{\sqrt{n}} \le \mu \le \bar x + z_{\frac{\alpha}{2}} \frac{\sigma}{\sqrt{n}}
</math>
where <math>z_{\frac{\alpha}{2}}</math> is the value of <math>z</math> such that the area to its left (or right) is <math>\frac{\alpha}{2}</math>. For example if we choose a <math>95 \% </math> confidence level then <math>1-\alpha=0.95</math> or <math>\alpha=0.05</math> and therefore <math>\frac{\alpha}{2}=0.025</math> which gives <math>z_{\frac{\alpha}{2}}=1.96</math>. The sample mean <math>\bar x</math> is the mean of the sample of size <math>n</math>, and <math>\sigma </math> is the standard deviation. In this lab we will generate many confidence intervals based on different sample sizes. The samples in this lab are always selected from the standard normal distribution <math>N(0,1)</math>. Therefore we know that the mean is <math>\mu=0</math>, and the standard deviation <math>\sigma=1</math>. Let's pretend that <math>\mu</math> is unknown and that only <math>\sigma</math> is known. We will select many samples each one of size <math>n</math> and use it to construct a confidence interval for the population mean.

===Exercise 1===
Using the scroll down button select "Number of Experiments = 100". Select sample size <math>n=20</math>, and choose number of intervals 200. It means: You will select 200 samples and with each sample you will obtain a confidence interval. You will do this 100 times. Take a snapshot and answer the following questions.

# How many intervals (out of the 200) do you ''expect'' to miss the population mean <math>\mu=0</math>? How many ''did'' when you ran it?
# What do the numbers -3, -2, -1, 0, 1, 2, 3 represent?
# What do the blue lines represent?
# How is the confidence interval represented?
# What does the green dot represent?
# Write down the formula on which these confidence intervals are based.

===Exercise 2===
# Reset and repeat Exercise 1, answering questions 1 and 6, with <math>\alpha=0.01</math>. Will the intervals be larger, smaller, or stay the same compared to that of Exercise 1?
# Reset and repeat Exercise 1, answering questions 1 and 6, with sample size now <math>n=80</math> and <math>\alpha=0.01</math>. Will the intervals be larger, smaller, or stay the same compared to that of Exercise 1?
# Reset and repeat Exercise 1, answering questions 1 and 6, with sample size <math>n=80</math> and <math>\alpha=1.0E-4</math> (this is <math>10^{-4}</math>). Will the intervals be larger, smaller, or stay the same compared to that of Exercise 1?

Below you can see a snapshot of the run of 100 intervals with <math> n=36, \ \alpha=0.05 </math>.

<center>[[Image: SOCR_Activities_Christou_christou_confint.jpg|600px]]</center>

===See also===
* [[SOCR_EduMaterials_Activities_General_CI_Experiment | General Confidence Interval Activity]] and the [http://socr.ucla.edu/htmls/exp/Confidence_Interval_Experiment_General.html General Confidence Interval Applet].

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2011_2012_Stat13_1_Lab5}}

SOCR Courses 2011 2012 Stat13 1 Lab5

2012-02-23T15:33:09Z

SeanWang: /* Exercise 1 */

== [[SOCR_Courses_2011_2012_Stat13_1 | Stats 13.1]] - Laboratory Activity 5: Confidence Interval Activity==

This is an activity to explore the confidence intervals for the population mean when the standard deviation is known.

===Description===
You can access the applet for the confidence intervals experiment at [http://www.socr.ucla.edu/htmls/SOCR_Experiments.html SOCR Experiments]. Use the scroll down button to find the [http://socr.ucla.edu/htmls/exp/Confidence_Interval_Experiment.html Confidence Interval Experiment].

The confidence interval for the population mean <math>\mu</math> when <math> \sigma </math> is known is given by (when n>30):
<math>
\bar x - z_{\frac{\alpha}{2}} \frac{\sigma}{\sqrt{n}} \le \mu \le \bar x + z_{\frac{\alpha}{2}} \frac{\sigma}{\sqrt{n}}
</math>
where <math>z_{\frac{\alpha}{2}}</math> is the value of <math>z</math> such that the area to its left (or right) is <math>\frac{\alpha}{2}</math>. For example if we choose a <math>95 \% </math> confidence level then <math>1-\alpha=0.95</math> or <math>\alpha=0.05</math> and therefore <math>\frac{\alpha}{2}=0.025</math> which gives <math>z_{\frac{\alpha}{2}}=1.96</math>. The sample mean <math>\bar x</math> is the mean of the sample of size <math>n</math>, and <math>\sigma </math> is the standard deviation. In this lab we will generate many confidence intervals based on different sample sizes. The samples in this lab are always selected from the standard normal distribution <math>N(0,1)</math>. Therefore we know that the mean is <math>\mu=0</math>, and the standard deviation <math>\sigma=1</math>. Let's pretend that <math>\mu</math> is unknown and that only <math>\sigma</math> is known. We will select many samples each one of size <math>n</math> and use it to construct a confidence interval for the population mean.

===Exercise 1===
Using the scroll down button select "Number of Experiments = 100". Select sample size <math>n=20</math>, and choose number of intervals 200. It means: You will select 200 samples and with each sample you will obtain a confidence interval. You will do this 100 times. Take a snapshot and answer the following questions.

# How many intervals (out of the 200) do you ''expect'' to miss the population mean <math>\mu=0</math>? How many ''did'' when you ran it?
# What do the number -3, -2, -1, 0, 1, 2, 3 represent?
# What do the blue lines represent?
# How is the confidence interval represented?
# What does the green dot represent?
# Write down the formula on which these confidence intervals are based.

===Exercise 2===
# Reset and repeat Exercise 1, answering questions 1 and 6, with <math>\alpha=0.01</math>. Will the intervals be larger, smaller, or stay the same compared to that of Exercise 1?
# Reset and repeat Exercise 1, answering questions 1 and 6, with sample size now <math>n=80</math> and <math>\alpha=0.01</math>. Will the intervals be larger, smaller, or stay the same compared to that of Exercise 1?
# Reset and repeat Exercise 1, answering questions 1 and 6, with sample size <math>n=80</math> and <math>\alpha=1.0E-4</math> (this is <math>10^{-4}</math>). Will the intervals be larger, smaller, or stay the same compared to that of Exercise 1?

Below you can see a snapshot of the run of 100 intervals with <math> n=36, \ \alpha=0.05 </math>.

<center>[[Image: SOCR_Activities_Christou_christou_confint.jpg|600px]]</center>

===See also===
* [[SOCR_EduMaterials_Activities_General_CI_Experiment | General Confidence Interval Activity]] and the [http://socr.ucla.edu/htmls/exp/Confidence_Interval_Experiment_General.html General Confidence Interval Applet].

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2011_2012_Stat13_1_Lab5}}

SOCR Courses 2011 2012 Stat13 1 Lab5

2012-02-23T15:32:22Z

SeanWang: /* Exercise 1 */

== [[SOCR_Courses_2011_2012_Stat13_1 | Stats 13.1]] - Laboratory Activity 5: Confidence Interval Activity==

This is an activity to explore the confidence intervals for the population mean when the standard deviation is known.

===Description===
You can access the applet for the confidence intervals experiment at [http://www.socr.ucla.edu/htmls/SOCR_Experiments.html SOCR Experiments]. Use the scroll down button to find the [http://socr.ucla.edu/htmls/exp/Confidence_Interval_Experiment.html Confidence Interval Experiment].

The confidence interval for the population mean <math>\mu</math> when <math> \sigma </math> is known is given by (when n>30):
<math>
\bar x - z_{\frac{\alpha}{2}} \frac{\sigma}{\sqrt{n}} \le \mu \le \bar x + z_{\frac{\alpha}{2}} \frac{\sigma}{\sqrt{n}}
</math>
where <math>z_{\frac{\alpha}{2}}</math> is the value of <math>z</math> such that the area to its left (or right) is <math>\frac{\alpha}{2}</math>. For example if we choose a <math>95 \% </math> confidence level then <math>1-\alpha=0.95</math> or <math>\alpha=0.05</math> and therefore <math>\frac{\alpha}{2}=0.025</math> which gives <math>z_{\frac{\alpha}{2}}=1.96</math>. The sample mean <math>\bar x</math> is the mean of the sample of size <math>n</math>, and <math>\sigma </math> is the standard deviation. In this lab we will generate many confidence intervals based on different sample sizes. The samples in this lab are always selected from the standard normal distribution <math>N(0,1)</math>. Therefore we know that the mean is <math>\mu=0</math>, and the standard deviation <math>\sigma=1</math>. Let's pretend that <math>\mu</math> is unknown and that only <math>\sigma</math> is known. We will select many samples each one of size <math>n</math> and use it to construct a confidence interval for the population mean.

===Exercise 1===
Using the scroll down button select "Number of Experiments = 100". Select sample size <math>n=20</math>, and choose number of intervals 200. It means: You will select 200 samples and with each sample you will obtain a confidence interval. You will do this 100 times. Take a snapshot and answer the following questions.

# How many intervals (out of the 200) do you expect to miss the population mean <math>\mu=0</math>?
# What do the number -3, -2, -1, 0, 1, 2, 3 represent?
# What do the blue lines represent?
# How is the confidence interval represented?
# What does the green dot represent?
# Write down the formula on which these confidence intervals are based.

===Exercise 2===
# Reset and repeat Exercise 1, answering questions 1 and 6, with <math>\alpha=0.01</math>. Will the intervals be larger, smaller, or stay the same compared to that of Exercise 1?
# Reset and repeat Exercise 1, answering questions 1 and 6, with sample size now <math>n=80</math> and <math>\alpha=0.01</math>. Will the intervals be larger, smaller, or stay the same compared to that of Exercise 1?
# Reset and repeat Exercise 1, answering questions 1 and 6, with sample size <math>n=80</math> and <math>\alpha=1.0E-4</math> (this is <math>10^{-4}</math>). Will the intervals be larger, smaller, or stay the same compared to that of Exercise 1?

Below you can see a snapshot of the run of 100 intervals with <math> n=36, \ \alpha=0.05 </math>.

<center>[[Image: SOCR_Activities_Christou_christou_confint.jpg|600px]]</center>

===See also===
* [[SOCR_EduMaterials_Activities_General_CI_Experiment | General Confidence Interval Activity]] and the [http://socr.ucla.edu/htmls/exp/Confidence_Interval_Experiment_General.html General Confidence Interval Applet].

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2011_2012_Stat13_1_Lab5}}

SOCR Courses 2011 2012 Stat13 1 Lab4

2012-02-16T13:26:27Z

SeanWang: /* Exercise 4 */

== [[SOCR_Courses_2011_2012_Stat13_1 | Stats 13.1]] - Laboratory Activity 4==

=== Normal Probability Distribution Activity ===

Note: If at the end of this lab you feel like you could use some more examples, see: [[SOCR EduMaterials Activities Normal Probability examples]]

For the purposes of this lab, we are looking at <math> N(\mu, \sigma) </math>.

'''Description''': You can access the [http://www.socr.ucla.edu/htmls/SOCR_Distributions.html SOCR distribution applets here].

====Exercise 1====
Use SOCR to graph and print the distribution of <math> X \sim N(20, 3) </math>. Show on the graph the following points: <math>\mu \pm 1 \sigma, \mu \pm 2 \sigma, \mu \pm 3 \sigma </math> by drawing hashmarks of where they are on the printout. How many standard deviations from the mean is the value <math> x=27.5 </math>?

====Exercise 2====
Graph the distribution of <math> X \sim N(40, 10)</math>.
# Find <math>P(X > 49)</math>.
# Find <math>P(X < 22)</math>.
# Find <math>P(12 < X < 37)</math>. Submit a printout.
# Use the mouse or the left cut off or right cut off points to find the <math>8^{th}, 20^{th}, 45^{th}, 55^{th}, 70^{th}, 95^{th} </math> percentiles. After you find these percentiles submit a printout for the <math>95^{th}</math> percentile.
# Make sure you know how to answer the above questions using the <math>z</math> score <math>z=\frac{x-\mu}{\sigma}</math> and your <math>z</math> table from the handout! You do '''not''' need to submit anything here.

====Exercise 3====
The lifetime of tires of brand <math>A</math> follows the normal distribution with mean 40000 miles and standard deviation 4000 miles.
# Use <math>SOCR</math> to find the probability that a tire will last between 40000 and 46000 miles.
# Given that a tire will last more than 46000 miles, what is the probability that it will last more than 50000 miles? Explain how you get the answer.
# Given that a tire will last more than 46000 miles, what is the probability that it will last less than 50000 miles? Submit a printout and explain how you get the answer.

====Exercise 4====
# The probability that a student is admitted in the Math Department Major at a college is <math>45 \%</math>. Suppose that this year 100 students will apply for admission into the Math major.
# What is the distribution of the number of students admitted? Use SOCR to graph and print this distribution. What is the shape of this distribution? What is the mean and standard deviation of this distribution?
# Write an expression for the '''exact''' probability that among the 100 students at least 55 will be admitted.
# Use the appropriate applet in SOCR to compute the '''exact''' probability of part (3).
# Use the normal distribution applet in SOCR to approximate the probability of part (3) (do not forget the continuity correction). What is the error of the approximation (i.e., how much does this differ from part (4))?

Below you can see the distribution of a normal random variable <math> X </math> with <math> \mu=50, \sigma=5 </math>. In this graph you can also see the probability that <math> X </math> is between 53 and 60.

<center>[[Image: SOCR_Activities_Christou_normal.jpg|600px]]</center>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2011_2012_Stat13_1_Lab4}}

SOCR Courses 2011 2012 Stat13 1 Lab4

2012-02-16T13:25:24Z

SeanWang: /* Exercise 4 */

== [[SOCR_Courses_2011_2012_Stat13_1 | Stats 13.1]] - Laboratory Activity 4==

=== Normal Probability Distribution Activity ===

Note: If at the end of this lab you feel like you could use some more examples, see: [[SOCR EduMaterials Activities Normal Probability examples]]

For the purposes of this lab, we are looking at <math> N(\mu, \sigma) </math>.

'''Description''': You can access the [http://www.socr.ucla.edu/htmls/SOCR_Distributions.html SOCR distribution applets here].

====Exercise 1====
Use SOCR to graph and print the distribution of <math> X \sim N(20, 3) </math>. Show on the graph the following points: <math>\mu \pm 1 \sigma, \mu \pm 2 \sigma, \mu \pm 3 \sigma </math> by drawing hashmarks of where they are on the printout. How many standard deviations from the mean is the value <math> x=27.5 </math>?

====Exercise 2====
Graph the distribution of <math> X \sim N(40, 10)</math>.
# Find <math>P(X > 49)</math>.
# Find <math>P(X < 22)</math>.
# Find <math>P(12 < X < 37)</math>. Submit a printout.
# Use the mouse or the left cut off or right cut off points to find the <math>8^{th}, 20^{th}, 45^{th}, 55^{th}, 70^{th}, 95^{th} </math> percentiles. After you find these percentiles submit a printout for the <math>95^{th}</math> percentile.
# Make sure you know how to answer the above questions using the <math>z</math> score <math>z=\frac{x-\mu}{\sigma}</math> and your <math>z</math> table from the handout! You do '''not''' need to submit anything here.

====Exercise 3====
The lifetime of tires of brand <math>A</math> follows the normal distribution with mean 40000 miles and standard deviation 4000 miles.
# Use <math>SOCR</math> to find the probability that a tire will last between 40000 and 46000 miles.
# Given that a tire will last more than 46000 miles, what is the probability that it will last more than 50000 miles? Explain how you get the answer.
# Given that a tire will last more than 46000 miles, what is the probability that it will last less than 50000 miles? Submit a printout and explain how you get the answer.

====Exercise 4====
# The probability that a student is admitted in the Math Department Major at a college is <math>45 \%</math>. Suppose that this year 100 students will apply for admission into the Math major.
# What is the distribution of the number of students admitted? Use SOCR to graph and print this distribution. What is the shape of this distribution? What is the mean and standard deviation of this distribution?
# Write an expression for the exact probability that among the 100 students at least 55 will be admitted.
# Use the binomial distribution applet in SOCR to compute the probability of part (3).
# Use the normal distribution applet in SOCR to approximate the probability of part (3) (do not forget the continuity correction). What is the error of the approximation (i.e., how much does this differ from part (4))?

Below you can see the distribution of a normal random variable <math> X </math> with <math> \mu=50, \sigma=5 </math>. In this graph you can also see the probability that <math> X </math> is between 53 and 60.

<center>[[Image: SOCR_Activities_Christou_normal.jpg|600px]]</center>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2011_2012_Stat13_1_Lab4}}

SOCR Courses 2011 2012 Stat13 1 Lab4

2012-02-16T13:24:45Z

SeanWang: /* Exercise 4 */

== [[SOCR_Courses_2011_2012_Stat13_1 | Stats 13.1]] - Laboratory Activity 4==

=== Normal Probability Distribution Activity ===

Note: If at the end of this lab you feel like you could use some more examples, see: [[SOCR EduMaterials Activities Normal Probability examples]]

For the purposes of this lab, we are looking at <math> N(\mu, \sigma) </math>.

'''Description''': You can access the [http://www.socr.ucla.edu/htmls/SOCR_Distributions.html SOCR distribution applets here].

====Exercise 1====
Use SOCR to graph and print the distribution of <math> X \sim N(20, 3) </math>. Show on the graph the following points: <math>\mu \pm 1 \sigma, \mu \pm 2 \sigma, \mu \pm 3 \sigma </math> by drawing hashmarks of where they are on the printout. How many standard deviations from the mean is the value <math> x=27.5 </math>?

====Exercise 2====
Graph the distribution of <math> X \sim N(40, 10)</math>.
# Find <math>P(X > 49)</math>.
# Find <math>P(X < 22)</math>.
# Find <math>P(12 < X < 37)</math>. Submit a printout.
# Use the mouse or the left cut off or right cut off points to find the <math>8^{th}, 20^{th}, 45^{th}, 55^{th}, 70^{th}, 95^{th} </math> percentiles. After you find these percentiles submit a printout for the <math>95^{th}</math> percentile.
# Make sure you know how to answer the above questions using the <math>z</math> score <math>z=\frac{x-\mu}{\sigma}</math> and your <math>z</math> table from the handout! You do '''not''' need to submit anything here.

====Exercise 3====
The lifetime of tires of brand <math>A</math> follows the normal distribution with mean 40000 miles and standard deviation 4000 miles.
# Use <math>SOCR</math> to find the probability that a tire will last between 40000 and 46000 miles.
# Given that a tire will last more than 46000 miles, what is the probability that it will last more than 50000 miles? Explain how you get the answer.
# Given that a tire will last more than 46000 miles, what is the probability that it will last less than 50000 miles? Submit a printout and explain how you get the answer.

====Exercise 4====
# The probability that a student is admitted in the Math Department Major at a college is <math>45 \%</math>. Suppose that this year 100 students will apply for admission into the Math major.
# What is the distribution of the number of students admitted? Use SOCR to graph and print this distribution. What is the shape of this distribution? What is the mean and standard deviation of this distribution?
# Write an expression for the exact probability that among the 100 students at least 55 will be admitted.
# Use the binomial distribution applet in SOCR to compute the probability of part (3).
# Use the normal distribution applet in SOCR to approximate the probability of part (3) (do not forget the continuity correction). What is the error of the approximation?

Below you can see the distribution of a normal random variable <math> X </math> with <math> \mu=50, \sigma=5 </math>. In this graph you can also see the probability that <math> X </math> is between 53 and 60.

<center>[[Image: SOCR_Activities_Christou_normal.jpg|600px]]</center>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2011_2012_Stat13_1_Lab4}}

SOCR Courses 2011 2012 Stat13 1 Lab4

2012-02-16T13:23:05Z

SeanWang: /* Exercise 4 */

== [[SOCR_Courses_2011_2012_Stat13_1 | Stats 13.1]] - Laboratory Activity 4==

=== Normal Probability Distribution Activity ===

Note: If at the end of this lab you feel like you could use some more examples, see: [[SOCR EduMaterials Activities Normal Probability examples]]

For the purposes of this lab, we are looking at <math> N(\mu, \sigma) </math>.

'''Description''': You can access the [http://www.socr.ucla.edu/htmls/SOCR_Distributions.html SOCR distribution applets here].

====Exercise 1====
Use SOCR to graph and print the distribution of <math> X \sim N(20, 3) </math>. Show on the graph the following points: <math>\mu \pm 1 \sigma, \mu \pm 2 \sigma, \mu \pm 3 \sigma </math> by drawing hashmarks of where they are on the printout. How many standard deviations from the mean is the value <math> x=27.5 </math>?

====Exercise 2====
Graph the distribution of <math> X \sim N(40, 10)</math>.
# Find <math>P(X > 49)</math>.
# Find <math>P(X < 22)</math>.
# Find <math>P(12 < X < 37)</math>. Submit a printout.
# Use the mouse or the left cut off or right cut off points to find the <math>8^{th}, 20^{th}, 45^{th}, 55^{th}, 70^{th}, 95^{th} </math> percentiles. After you find these percentiles submit a printout for the <math>95^{th}</math> percentile.
# Make sure you know how to answer the above questions using the <math>z</math> score <math>z=\frac{x-\mu}{\sigma}</math> and your <math>z</math> table from the handout! You do '''not''' need to submit anything here.

====Exercise 3====
The lifetime of tires of brand <math>A</math> follows the normal distribution with mean 40000 miles and standard deviation 4000 miles.
# Use <math>SOCR</math> to find the probability that a tire will last between 40000 and 46000 miles.
# Given that a tire will last more than 46000 miles, what is the probability that it will last more than 50000 miles? Explain how you get the answer.
# Given that a tire will last more than 46000 miles, what is the probability that it will last less than 50000 miles? Submit a printout and explain how you get the answer.

====Exercise 4====
# The probability that a student is admitted in the Math Department Major at a college is <math>45 \%</math>. Suppose that this year 100 students will apply for admission into the Math major.
# What is the distribution of the number of students admitted? Use <math>SOCR</math> to graph and print this distribution. What is the shape of this distribution? What is the mean and standard deviation of this distribution?
# Write an expression for the exact probability that among the 100 students at least 55 will be admitted.
# Use SOCR to compute the probability of part (3).
# Use the normal distribution applet in SOCR to approximate the probability of part (3) (do not forget the continuity correction). What is the error of the approximation?

Below you can see the distribution of a normal random variable <math> X </math> with <math> \mu=50, \sigma=5 </math>. In this graph you can also see the probability that <math> X </math> is between 53 and 60.

<center>[[Image: SOCR_Activities_Christou_normal.jpg|600px]]</center>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2011_2012_Stat13_1_Lab4}}

SOCR Courses 2011 2012 Stat13 1 Lab4

2012-02-16T13:22:51Z

SeanWang: /* Exercise 4 */

== [[SOCR_Courses_2011_2012_Stat13_1 | Stats 13.1]] - Laboratory Activity 4==

=== Normal Probability Distribution Activity ===

Note: If at the end of this lab you feel like you could use some more examples, see: [[SOCR EduMaterials Activities Normal Probability examples]]

For the purposes of this lab, we are looking at <math> N(\mu, \sigma) </math>.

'''Description''': You can access the [http://www.socr.ucla.edu/htmls/SOCR_Distributions.html SOCR distribution applets here].

====Exercise 1====
Use SOCR to graph and print the distribution of <math> X \sim N(20, 3) </math>. Show on the graph the following points: <math>\mu \pm 1 \sigma, \mu \pm 2 \sigma, \mu \pm 3 \sigma </math> by drawing hashmarks of where they are on the printout. How many standard deviations from the mean is the value <math> x=27.5 </math>?

====Exercise 2====
Graph the distribution of <math> X \sim N(40, 10)</math>.
# Find <math>P(X > 49)</math>.
# Find <math>P(X < 22)</math>.
# Find <math>P(12 < X < 37)</math>. Submit a printout.
# Use the mouse or the left cut off or right cut off points to find the <math>8^{th}, 20^{th}, 45^{th}, 55^{th}, 70^{th}, 95^{th} </math> percentiles. After you find these percentiles submit a printout for the <math>95^{th}</math> percentile.
# Make sure you know how to answer the above questions using the <math>z</math> score <math>z=\frac{x-\mu}{\sigma}</math> and your <math>z</math> table from the handout! You do '''not''' need to submit anything here.

====Exercise 3====
The lifetime of tires of brand <math>A</math> follows the normal distribution with mean 40000 miles and standard deviation 4000 miles.
# Use <math>SOCR</math> to find the probability that a tire will last between 40000 and 46000 miles.
# Given that a tire will last more than 46000 miles, what is the probability that it will last more than 50000 miles? Explain how you get the answer.
# Given that a tire will last more than 46000 miles, what is the probability that it will last less than 50000 miles? Submit a printout and explain how you get the answer.

====Exercise 4====
# The probability that a student is admitted in the Math Department Major at a college is <math>45 \%</math>. Suppose that this year 100 students will apply for admission into the Math major.
# What is the distribution of the number of students admitted? Use <math>SOCR</math> to graph and print this distribution. What is the shape of this distribution? What is the mean and standard deviation of this distribution?
# Write an expression for the exact probability that among the 100 students at least 55 will be admitted.
# Use SOCR to compute the probability of part (3).
# Use the normal distribution applet in SOCR to approximate the probability of part (3) (do not forget the continuity correction). What is the error of the approximation?

<math>X \sim N(\mu, \sigma) </math>

<math>X \sim N(10, 3) </math>

Below you can see the distribution of a normal random variable <math> X </math> with <math> \mu=50, \sigma=5 </math>. In this graph you can also see the probability that <math> X </math> is between 53 and 60.

<center>[[Image: SOCR_Activities_Christou_normal.jpg|600px]]</center>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2011_2012_Stat13_1_Lab4}}

SOCR Courses 2011 2012 Stat13 1 Lab4

2012-02-16T13:22:34Z

SeanWang: /* Exercise 4 */

== [[SOCR_Courses_2011_2012_Stat13_1 | Stats 13.1]] - Laboratory Activity 4==

=== Normal Probability Distribution Activity ===

Note: If at the end of this lab you feel like you could use some more examples, see: [[SOCR EduMaterials Activities Normal Probability examples]]

For the purposes of this lab, we are looking at <math> N(\mu, \sigma) </math>.

'''Description''': You can access the [http://www.socr.ucla.edu/htmls/SOCR_Distributions.html SOCR distribution applets here].

====Exercise 1====
Use SOCR to graph and print the distribution of <math> X \sim N(20, 3) </math>. Show on the graph the following points: <math>\mu \pm 1 \sigma, \mu \pm 2 \sigma, \mu \pm 3 \sigma </math> by drawing hashmarks of where they are on the printout. How many standard deviations from the mean is the value <math> x=27.5 </math>?

====Exercise 2====
Graph the distribution of <math> X \sim N(40, 10)</math>.
# Find <math>P(X > 49)</math>.
# Find <math>P(X < 22)</math>.
# Find <math>P(12 < X < 37)</math>. Submit a printout.
# Use the mouse or the left cut off or right cut off points to find the <math>8^{th}, 20^{th}, 45^{th}, 55^{th}, 70^{th}, 95^{th} </math> percentiles. After you find these percentiles submit a printout for the <math>95^{th}</math> percentile.
# Make sure you know how to answer the above questions using the <math>z</math> score <math>z=\frac{x-\mu}{\sigma}</math> and your <math>z</math> table from the handout! You do '''not''' need to submit anything here.

====Exercise 3====
The lifetime of tires of brand <math>A</math> follows the normal distribution with mean 40000 miles and standard deviation 4000 miles.
# Use <math>SOCR</math> to find the probability that a tire will last between 40000 and 46000 miles.
# Given that a tire will last more than 46000 miles, what is the probability that it will last more than 50000 miles? Explain how you get the answer.
# Given that a tire will last more than 46000 miles, what is the probability that it will last less than 50000 miles? Submit a printout and explain how you get the answer.

====Exercise 4====
# The probability that a student is admitted in the Math Department Major at a college is <math>45 \%</math>. Suppose that this year 100 students will apply for admission into the Math major.
# What is the distribution of the number of students admitted? Use <math>SOCR</math> to graph and print this distribution. What is the shape of this distribution? What is the mean and standard deviation of this distribution?
# Write an expression for the exact probability that among the 100 students at least 55 will be admitted.
# Use SOCR to compute the probability of part (3).
# Use the normal distribution applet in SOCR to approximate the probability of part (3) (do not forget the continuity correction). What is the error of the approximation?

<math> N(\mu, \sigma) </math>

<math> N(10, 3) </math>

Below you can see the distribution of a normal random variable <math> X </math> with <math> \mu=50, \sigma=5 </math>. In this graph you can also see the probability that <math> X </math> is between 53 and 60.

<center>[[Image: SOCR_Activities_Christou_normal.jpg|600px]]</center>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2011_2012_Stat13_1_Lab4}}

SOCR Courses 2011 2012 Stat13 1 Lab4

2012-02-16T13:22:23Z

SeanWang: /* Exercise 4 */

== [[SOCR_Courses_2011_2012_Stat13_1 | Stats 13.1]] - Laboratory Activity 4==

=== Normal Probability Distribution Activity ===

Note: If at the end of this lab you feel like you could use some more examples, see: [[SOCR EduMaterials Activities Normal Probability examples]]

For the purposes of this lab, we are looking at <math> N(\mu, \sigma) </math>.

'''Description''': You can access the [http://www.socr.ucla.edu/htmls/SOCR_Distributions.html SOCR distribution applets here].

====Exercise 1====
Use SOCR to graph and print the distribution of <math> X \sim N(20, 3) </math>. Show on the graph the following points: <math>\mu \pm 1 \sigma, \mu \pm 2 \sigma, \mu \pm 3 \sigma </math> by drawing hashmarks of where they are on the printout. How many standard deviations from the mean is the value <math> x=27.5 </math>?

====Exercise 2====
Graph the distribution of <math> X \sim N(40, 10)</math>.
# Find <math>P(X > 49)</math>.
# Find <math>P(X < 22)</math>.
# Find <math>P(12 < X < 37)</math>. Submit a printout.
# Use the mouse or the left cut off or right cut off points to find the <math>8^{th}, 20^{th}, 45^{th}, 55^{th}, 70^{th}, 95^{th} </math> percentiles. After you find these percentiles submit a printout for the <math>95^{th}</math> percentile.
# Make sure you know how to answer the above questions using the <math>z</math> score <math>z=\frac{x-\mu}{\sigma}</math> and your <math>z</math> table from the handout! You do '''not''' need to submit anything here.

====Exercise 3====
The lifetime of tires of brand <math>A</math> follows the normal distribution with mean 40000 miles and standard deviation 4000 miles.
# Use <math>SOCR</math> to find the probability that a tire will last between 40000 and 46000 miles.
# Given that a tire will last more than 46000 miles, what is the probability that it will last more than 50000 miles? Explain how you get the answer.
# Given that a tire will last more than 46000 miles, what is the probability that it will last less than 50000 miles? Submit a printout and explain how you get the answer.

====Exercise 4====
# The probability that a student is admitted in the Math Department Major at a college is <math>45 \%</math>. Suppose that this year 100 students will apply for admission into the Math major.
# What is the distribution of the number of students admitted? Use <math>SOCR</math> to graph and print this distribution. What is the shape of this distribution? What is the mean and standard deviation of this distribution?
# Write an expression for the exact probability that among the 100 students at least 55 will be admitted.
# Use SOCR to compute the probability of part (3).
# Use the normal distribution applet in SOCR to approximate the probability of part (3) (do not forget the continuity correction). What is the error of the approximation?

<math> N(\mu, \sigma) </math>
<math> N(10, 3) </math>

Below you can see the distribution of a normal random variable <math> X </math> with <math> \mu=50, \sigma=5 </math>. In this graph you can also see the probability that <math> X </math> is between 53 and 60.

<center>[[Image: SOCR_Activities_Christou_normal.jpg|600px]]</center>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2011_2012_Stat13_1_Lab4}}

SOCR Courses 2011 2012 Stat13 1 Lab4

2012-02-16T13:21:33Z

SeanWang: /* Exercise 1 */

== [[SOCR_Courses_2011_2012_Stat13_1 | Stats 13.1]] - Laboratory Activity 4==

=== Normal Probability Distribution Activity ===

Note: If at the end of this lab you feel like you could use some more examples, see: [[SOCR EduMaterials Activities Normal Probability examples]]

For the purposes of this lab, we are looking at <math> N(\mu, \sigma) </math>.

'''Description''': You can access the [http://www.socr.ucla.edu/htmls/SOCR_Distributions.html SOCR distribution applets here].

====Exercise 1====
Use SOCR to graph and print the distribution of <math> X \sim N(20, 3) </math>. Show on the graph the following points: <math>\mu \pm 1 \sigma, \mu \pm 2 \sigma, \mu \pm 3 \sigma </math> by drawing hashmarks of where they are on the printout. How many standard deviations from the mean is the value <math> x=27.5 </math>?

====Exercise 2====
Graph the distribution of <math> X \sim N(40, 10)</math>.
# Find <math>P(X > 49)</math>.
# Find <math>P(X < 22)</math>.
# Find <math>P(12 < X < 37)</math>. Submit a printout.
# Use the mouse or the left cut off or right cut off points to find the <math>8^{th}, 20^{th}, 45^{th}, 55^{th}, 70^{th}, 95^{th} </math> percentiles. After you find these percentiles submit a printout for the <math>95^{th}</math> percentile.
# Make sure you know how to answer the above questions using the <math>z</math> score <math>z=\frac{x-\mu}{\sigma}</math> and your <math>z</math> table from the handout! You do '''not''' need to submit anything here.

====Exercise 3====
The lifetime of tires of brand <math>A</math> follows the normal distribution with mean 40000 miles and standard deviation 4000 miles.
# Use <math>SOCR</math> to find the probability that a tire will last between 40000 and 46000 miles.
# Given that a tire will last more than 46000 miles, what is the probability that it will last more than 50000 miles? Explain how you get the answer.
# Given that a tire will last more than 46000 miles, what is the probability that it will last less than 50000 miles? Submit a printout and explain how you get the answer.

====Exercise 4====
# The probability that a student is admitted in the Math Department Major at a college is <math>45 \%</math>. Suppose that this year 100 students will apply for admission into the Math major.
# What is the distribution of the number of students admitted? Use <math>SOCR</math> to graph and print this distribution. What is the shape of this distribution? What is the mean and standard deviation of this distribution?
# Write an expression for the exact probability that among the 100 students at least 55 will be admitted.
# Use SOCR to compute the probability of part (3).
# Use the normal distribution applet in SOCR to approximate the probability of part (3) (do not forget the continuity correction). What is the error of the approximation?

Below you can see the distribution of a normal random variable <math> X </math> with <math> \mu=50, \sigma=5 </math>. In this graph you can also see the probability that <math> X </math> is between 53 and 60.

<center>[[Image: SOCR_Activities_Christou_normal.jpg|600px]]</center>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2011_2012_Stat13_1_Lab4}}

SOCR Courses 2011 2012 Stat13 1 Lab4

2012-02-16T13:19:04Z

SeanWang: /* Exercise 3 */

== [[SOCR_Courses_2011_2012_Stat13_1 | Stats 13.1]] - Laboratory Activity 4==

=== Normal Probability Distribution Activity ===

Note: If at the end of this lab you feel like you could use some more examples, see: [[SOCR EduMaterials Activities Normal Probability examples]]

For the purposes of this lab, we are looking at <math> N(\mu, \sigma) </math>.

'''Description''': You can access the [http://www.socr.ucla.edu/htmls/SOCR_Distributions.html SOCR distribution applets here].

====Exercise 1====
Use SOCR to graph and print the distribution of <math> X \sim N(20, 3) </math>. Show on the graph the following points: <math>\mu \pm 1 \sigma, \mu \pm 2 \sigma, \mu \pm 3 \sigma </math>. How many standard deviations from the mean is the value <math> x=27.5 </math>?

====Exercise 2====
Graph the distribution of <math> X \sim N(40, 10)</math>.
# Find <math>P(X > 49)</math>.
# Find <math>P(X < 22)</math>.
# Find <math>P(12 < X < 37)</math>. Submit a printout.
# Use the mouse or the left cut off or right cut off points to find the <math>8^{th}, 20^{th}, 45^{th}, 55^{th}, 70^{th}, 95^{th} </math> percentiles. After you find these percentiles submit a printout for the <math>95^{th}</math> percentile.
# Make sure you know how to answer the above questions using the <math>z</math> score <math>z=\frac{x-\mu}{\sigma}</math> and your <math>z</math> table from the handout! You do '''not''' need to submit anything here.

====Exercise 3====
The lifetime of tires of brand <math>A</math> follows the normal distribution with mean 40000 miles and standard deviation 4000 miles.
# Use <math>SOCR</math> to find the probability that a tire will last between 40000 and 46000 miles.
# Given that a tire will last more than 46000 miles, what is the probability that it will last more than 50000 miles? Explain how you get the answer.
# Given that a tire will last more than 46000 miles, what is the probability that it will last less than 50000 miles? Submit a printout and explain how you get the answer.

====Exercise 4====
# The probability that a student is admitted in the Math Department Major at a college is <math>45 \%</math>. Suppose that this year 100 students will apply for admission into the Math major.
# What is the distribution of the number of students admitted? Use <math>SOCR</math> to graph and print this distribution. What is the shape of this distribution? What is the mean and standard deviation of this distribution?
# Write an expression for the exact probability that among the 100 students at least 55 will be admitted.
# Use SOCR to compute the probability of part (3).
# Use the normal distribution applet in SOCR to approximate the probability of part (3) (do not forget the continuity correction). What is the error of the approximation?

Below you can see the distribution of a normal random variable <math> X </math> with <math> \mu=50, \sigma=5 </math>. In this graph you can also see the probability that <math> X </math> is between 53 and 60.

<center>[[Image: SOCR_Activities_Christou_normal.jpg|600px]]</center>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2011_2012_Stat13_1_Lab4}}

SOCR Courses 2011 2012 Stat13 1 Lab4

2012-02-16T13:17:58Z

SeanWang: /* Exercise 4 */

== [[SOCR_Courses_2011_2012_Stat13_1 | Stats 13.1]] - Laboratory Activity 4==

=== Normal Probability Distribution Activity ===

Note: If at the end of this lab you feel like you could use some more examples, see: [[SOCR EduMaterials Activities Normal Probability examples]]

For the purposes of this lab, we are looking at <math> N(\mu, \sigma) </math>.

'''Description''': You can access the [http://www.socr.ucla.edu/htmls/SOCR_Distributions.html SOCR distribution applets here].

====Exercise 1====
Use SOCR to graph and print the distribution of <math> X \sim N(20, 3) </math>. Show on the graph the following points: <math>\mu \pm 1 \sigma, \mu \pm 2 \sigma, \mu \pm 3 \sigma </math>. How many standard deviations from the mean is the value <math> x=27.5 </math>?

====Exercise 2====
Graph the distribution of <math> X \sim N(40, 10)</math>.
# Find <math>P(X > 49)</math>.
# Find <math>P(X < 22)</math>.
# Find <math>P(12 < X < 37)</math>. Submit a printout.
# Use the mouse or the left cut off or right cut off points to find the <math>8^{th}, 20^{th}, 45^{th}, 55^{th}, 70^{th}, 95^{th} </math> percentiles. After you find these percentiles submit a printout for the <math>95^{th}</math> percentile.
# Make sure you know how to answer the above questions using the <math>z</math> score <math>z=\frac{x-\mu}{\sigma}</math> and your <math>z</math> table from the handout! You do '''not''' need to submit anything here.

====Exercise 3====
The lifetime of tires of brand <math>A</math> follows the normal distribution with mean 40000 miles and standard deviation 4000 miles.
# Use <math>SOCR</math> to find the probability that a tire will last between 40000 and 46000 miles.
# Given that a tire will last more than 46000 miles, what is the probability that it will last more than 50000 miles? Submit a printout and explain how you get the answer.
# Given that a tire will last more than 46000 miles, what is the probability that it will last less than 50000 miles? Submit a printout and explain how you get the answer.

====Exercise 4====
# The probability that a student is admitted in the Math Department Major at a college is <math>45 \%</math>. Suppose that this year 100 students will apply for admission into the Math major.
# What is the distribution of the number of students admitted? Use <math>SOCR</math> to graph and print this distribution. What is the shape of this distribution? What is the mean and standard deviation of this distribution?
# Write an expression for the exact probability that among the 100 students at least 55 will be admitted.
# Use SOCR to compute the probability of part (3).
# Use the normal distribution applet in SOCR to approximate the probability of part (3) (do not forget the continuity correction). What is the error of the approximation?

Below you can see the distribution of a normal random variable <math> X </math> with <math> \mu=50, \sigma=5 </math>. In this graph you can also see the probability that <math> X </math> is between 53 and 60.

<center>[[Image: SOCR_Activities_Christou_normal.jpg|600px]]</center>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2011_2012_Stat13_1_Lab4}}

SOCR Courses 2011 2012 Stat13 1 Lab4

2012-02-16T13:17:47Z

SeanWang: /* Exercise 4 */

== [[SOCR_Courses_2011_2012_Stat13_1 | Stats 13.1]] - Laboratory Activity 4==

=== Normal Probability Distribution Activity ===

Note: If at the end of this lab you feel like you could use some more examples, see: [[SOCR EduMaterials Activities Normal Probability examples]]

For the purposes of this lab, we are looking at <math> N(\mu, \sigma) </math>.

'''Description''': You can access the [http://www.socr.ucla.edu/htmls/SOCR_Distributions.html SOCR distribution applets here].

====Exercise 1====
Use SOCR to graph and print the distribution of <math> X \sim N(20, 3) </math>. Show on the graph the following points: <math>\mu \pm 1 \sigma, \mu \pm 2 \sigma, \mu \pm 3 \sigma </math>. How many standard deviations from the mean is the value <math> x=27.5 </math>?

====Exercise 2====
Graph the distribution of <math> X \sim N(40, 10)</math>.
# Find <math>P(X > 49)</math>.
# Find <math>P(X < 22)</math>.
# Find <math>P(12 < X < 37)</math>. Submit a printout.
# Use the mouse or the left cut off or right cut off points to find the <math>8^{th}, 20^{th}, 45^{th}, 55^{th}, 70^{th}, 95^{th} </math> percentiles. After you find these percentiles submit a printout for the <math>95^{th}</math> percentile.
# Make sure you know how to answer the above questions using the <math>z</math> score <math>z=\frac{x-\mu}{\sigma}</math> and your <math>z</math> table from the handout! You do '''not''' need to submit anything here.

====Exercise 3====
The lifetime of tires of brand <math>A</math> follows the normal distribution with mean 40000 miles and standard deviation 4000 miles.
# Use <math>SOCR</math> to find the probability that a tire will last between 40000 and 46000 miles.
# Given that a tire will last more than 46000 miles, what is the probability that it will last more than 50000 miles? Submit a printout and explain how you get the answer.
# Given that a tire will last more than 46000 miles, what is the probability that it will last less than 50000 miles? Submit a printout and explain how you get the answer.

====Exercise 4====
# The probability that a student is admitted in the Math Department Major at a college is <math>45 \%</math>. Suppose that this year 100 students will apply for admission into the Math major.
# What is the distribution of the number of students admitted? Use <math>SOCR</math> to graph and print this distribution. What is the shape of this distribution? What is the mean and standard deviation of this distribution?
# Write an expression for the exact probability that among the 100 students at least 55 will be admitted.
# Use SOCR to compute the probability of part (3).
# Use the normal distribution applet in SOCR to approximate the probability of part (3) (do not forget the continuity correction). What is the error of the approximation?

Below you can see the distribution of a normal random variable <math> X </math> with <math> \mu=50, \sigma=5 </math>. In this graph you can also see the probability that <math> X </math> is between 53 and 60.

<center>[[Image: SOCR_Activities_Christou_normal.jpg|600px]]</center>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2011_2012_Stat13_1_Lab4}}

SOCR Courses 2011 2012 Stat13 1 Lab4

2012-02-16T13:15:03Z

SeanWang: /* Exercise 3 */

== [[SOCR_Courses_2011_2012_Stat13_1 | Stats 13.1]] - Laboratory Activity 4==

=== Normal Probability Distribution Activity ===

Note: If at the end of this lab you feel like you could use some more examples, see: [[SOCR EduMaterials Activities Normal Probability examples]]

For the purposes of this lab, we are looking at <math> N(\mu, \sigma) </math>.

'''Description''': You can access the [http://www.socr.ucla.edu/htmls/SOCR_Distributions.html SOCR distribution applets here].

====Exercise 1====
Use SOCR to graph and print the distribution of <math> X \sim N(20, 3) </math>. Show on the graph the following points: <math>\mu \pm 1 \sigma, \mu \pm 2 \sigma, \mu \pm 3 \sigma </math>. How many standard deviations from the mean is the value <math> x=27.5 </math>?

====Exercise 2====
Graph the distribution of <math> X \sim N(40, 10)</math>.
# Find <math>P(X > 49)</math>.
# Find <math>P(X < 22)</math>.
# Find <math>P(12 < X < 37)</math>. Submit a printout.
# Use the mouse or the left cut off or right cut off points to find the <math>8^{th}, 20^{th}, 45^{th}, 55^{th}, 70^{th}, 95^{th} </math> percentiles. After you find these percentiles submit a printout for the <math>95^{th}</math> percentile.
# Make sure you know how to answer the above questions using the <math>z</math> score <math>z=\frac{x-\mu}{\sigma}</math> and your <math>z</math> table from the handout! You do '''not''' need to submit anything here.

====Exercise 3====
The lifetime of tires of brand <math>A</math> follows the normal distribution with mean 40000 miles and standard deviation 4000 miles.
# Use <math>SOCR</math> to find the probability that a tire will last between 40000 and 46000 miles.
# Given that a tire will last more than 46000 miles, what is the probability that it will last more than 50000 miles? Submit a printout and explain how you get the answer.
# Given that a tire will last more than 46000 miles, what is the probability that it will last less than 50000 miles? Submit a printout and explain how you get the answer.

====Exercise 4====
# The probability that a student is admitted in the Math Department Major at a college is <math>45 \%</math>. Suppose that this year 100 students will apply for admission into the Math major.
# What is the distribution of the number of students admitted? Use <math>SOCR</math> to graph and print this distribution. What is the shape of this distribution? What is the mean and standard deviation of this distribution?
# Write an expression for the exact probability that among the 100 students at least 55 will be admitted.
# Use SOCR to compute the probability of part (3).
# Use the normal distribution applet in SOCR to approximate the probability of part (3) (do not forget the continuity correction). What is the error of the approximation?

Below you can see the distribution of a normal random variable <math> X </math> with <math> \mu=50, \sigma=5 </math>. In this graph you can also see the probability that <math> X </math> is between 53 and 60.

<center>[[Image: SOCR_Activities_Christou_normal.jpg|600px]]</center>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2011_2012_Stat13_1_Lab4}}

SOCR Courses 2011 2012 Stat13 1 Lab4

2012-02-16T13:13:46Z

SeanWang: /* Normal Probability Distribution Activity */

== [[SOCR_Courses_2011_2012_Stat13_1 | Stats 13.1]] - Laboratory Activity 4==

=== Normal Probability Distribution Activity ===

Note: If at the end of this lab you feel like you could use some more examples, see: [[SOCR EduMaterials Activities Normal Probability examples]]

For the purposes of this lab, we are looking at <math> N(\mu, \sigma) </math>.

'''Description''': You can access the [http://www.socr.ucla.edu/htmls/SOCR_Distributions.html SOCR distribution applets here].

====Exercise 1====
Use SOCR to graph and print the distribution of <math> X \sim N(20, 3) </math>. Show on the graph the following points: <math>\mu \pm 1 \sigma, \mu \pm 2 \sigma, \mu \pm 3 \sigma </math>. How many standard deviations from the mean is the value <math> x=27.5 </math>?

====Exercise 2====
Graph the distribution of <math> X \sim N(40, 10)</math>.
# Find <math>P(X > 49)</math>.
# Find <math>P(X < 22)</math>.
# Find <math>P(12 < X < 37)</math>. Submit a printout.
# Use the mouse or the left cut off or right cut off points to find the <math>8^{th}, 20^{th}, 45^{th}, 55^{th}, 70^{th}, 95^{th} </math> percentiles. After you find these percentiles submit a printout for the <math>95^{th}</math> percentile.
# Make sure you know how to answer the above questions using the <math>z</math> score <math>z=\frac{x-\mu}{\sigma}</math> and your <math>z</math> table from the handout! You do '''not''' need to submit anything here.

====Exercise 3====
The lifetime of tires of brand <math>A</math> follows the normal distribution with mean 40000 miles and standard deviation 4000 miles.
# Use <math>SOCR</math> to find the probability that a tire will last between 40000 and 46000 miles.
# Given that a tire will last more than 46000 miles what is the probability that it will last more than 50000 miles? Submit a printout and explain how you get the answer.
# Given that a tire will last more than 46000 miles what is the probability that it will last less than 50000 miles? Submit a printout and explain how you get the answer.

====Exercise 4====
# The probability that a student is admitted in the Math Department Major at a college is <math>45 \%</math>. Suppose that this year 100 students will apply for admission into the Math major.
# What is the distribution of the number of students admitted? Use <math>SOCR</math> to graph and print this distribution. What is the shape of this distribution? What is the mean and standard deviation of this distribution?
# Write an expression for the exact probability that among the 100 students at least 55 will be admitted.
# Use SOCR to compute the probability of part (3).
# Use the normal distribution applet in SOCR to approximate the probability of part (3) (do not forget the continuity correction). What is the error of the approximation?

Below you can see the distribution of a normal random variable <math> X </math> with <math> \mu=50, \sigma=5 </math>. In this graph you can also see the probability that <math> X </math> is between 53 and 60.

<center>[[Image: SOCR_Activities_Christou_normal.jpg|600px]]</center>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2011_2012_Stat13_1_Lab4}}

SOCR Courses 2011 2012 Stat13 1 Lab4

2012-02-16T13:13:36Z

SeanWang: /* Normal Probability Distribution Activity */

== [[SOCR_Courses_2011_2012_Stat13_1 | Stats 13.1]] - Laboratory Activity 4==

=== Normal Probability Distribution Activity ===

Note: If at the end of this lab you feel like you could use some more examples, see: [[SOCR EduMaterials Activities Normal Probability examples]]

For the purposes of this lab, we are looking at <math> \sim N(\mu, \sigma) </math>.

'''Description''': You can access the [http://www.socr.ucla.edu/htmls/SOCR_Distributions.html SOCR distribution applets here].

====Exercise 1====
Use SOCR to graph and print the distribution of <math> X \sim N(20, 3) </math>. Show on the graph the following points: <math>\mu \pm 1 \sigma, \mu \pm 2 \sigma, \mu \pm 3 \sigma </math>. How many standard deviations from the mean is the value <math> x=27.5 </math>?

====Exercise 2====
Graph the distribution of <math> X \sim N(40, 10)</math>.
# Find <math>P(X > 49)</math>.
# Find <math>P(X < 22)</math>.
# Find <math>P(12 < X < 37)</math>. Submit a printout.
# Use the mouse or the left cut off or right cut off points to find the <math>8^{th}, 20^{th}, 45^{th}, 55^{th}, 70^{th}, 95^{th} </math> percentiles. After you find these percentiles submit a printout for the <math>95^{th}</math> percentile.
# Make sure you know how to answer the above questions using the <math>z</math> score <math>z=\frac{x-\mu}{\sigma}</math> and your <math>z</math> table from the handout! You do '''not''' need to submit anything here.

====Exercise 3====
The lifetime of tires of brand <math>A</math> follows the normal distribution with mean 40000 miles and standard deviation 4000 miles.
# Use <math>SOCR</math> to find the probability that a tire will last between 40000 and 46000 miles.
# Given that a tire will last more than 46000 miles what is the probability that it will last more than 50000 miles? Submit a printout and explain how you get the answer.
# Given that a tire will last more than 46000 miles what is the probability that it will last less than 50000 miles? Submit a printout and explain how you get the answer.

====Exercise 4====
# The probability that a student is admitted in the Math Department Major at a college is <math>45 \%</math>. Suppose that this year 100 students will apply for admission into the Math major.
# What is the distribution of the number of students admitted? Use <math>SOCR</math> to graph and print this distribution. What is the shape of this distribution? What is the mean and standard deviation of this distribution?
# Write an expression for the exact probability that among the 100 students at least 55 will be admitted.
# Use SOCR to compute the probability of part (3).
# Use the normal distribution applet in SOCR to approximate the probability of part (3) (do not forget the continuity correction). What is the error of the approximation?

Below you can see the distribution of a normal random variable <math> X </math> with <math> \mu=50, \sigma=5 </math>. In this graph you can also see the probability that <math> X </math> is between 53 and 60.

<center>[[Image: SOCR_Activities_Christou_normal.jpg|600px]]</center>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2011_2012_Stat13_1_Lab4}}

SOCR Courses 2011 2012 Stat13 1 Lab4

2012-02-16T13:13:24Z

SeanWang: /* Normal Probability Distribution Activity */

== [[SOCR_Courses_2011_2012_Stat13_1 | Stats 13.1]] - Laboratory Activity 4==

=== Normal Probability Distribution Activity ===

Note: If at the end of this lab you feel like you could use some more examples, see: [[SOCR EduMaterials Activities Normal Probability examples]]

For the purposes of this lab, we are looking at <math> N(\mu, \sigma) </math>.

'''Description''': You can access the [http://www.socr.ucla.edu/htmls/SOCR_Distributions.html SOCR distribution applets here].

====Exercise 1====
Use SOCR to graph and print the distribution of <math> X \sim N(20, 3) </math>. Show on the graph the following points: <math>\mu \pm 1 \sigma, \mu \pm 2 \sigma, \mu \pm 3 \sigma </math>. How many standard deviations from the mean is the value <math> x=27.5 </math>?

====Exercise 2====
Graph the distribution of <math> X \sim N(40, 10)</math>.
# Find <math>P(X > 49)</math>.
# Find <math>P(X < 22)</math>.
# Find <math>P(12 < X < 37)</math>. Submit a printout.
# Use the mouse or the left cut off or right cut off points to find the <math>8^{th}, 20^{th}, 45^{th}, 55^{th}, 70^{th}, 95^{th} </math> percentiles. After you find these percentiles submit a printout for the <math>95^{th}</math> percentile.
# Make sure you know how to answer the above questions using the <math>z</math> score <math>z=\frac{x-\mu}{\sigma}</math> and your <math>z</math> table from the handout! You do '''not''' need to submit anything here.

====Exercise 3====
The lifetime of tires of brand <math>A</math> follows the normal distribution with mean 40000 miles and standard deviation 4000 miles.
# Use <math>SOCR</math> to find the probability that a tire will last between 40000 and 46000 miles.
# Given that a tire will last more than 46000 miles what is the probability that it will last more than 50000 miles? Submit a printout and explain how you get the answer.
# Given that a tire will last more than 46000 miles what is the probability that it will last less than 50000 miles? Submit a printout and explain how you get the answer.

====Exercise 4====
# The probability that a student is admitted in the Math Department Major at a college is <math>45 \%</math>. Suppose that this year 100 students will apply for admission into the Math major.
# What is the distribution of the number of students admitted? Use <math>SOCR</math> to graph and print this distribution. What is the shape of this distribution? What is the mean and standard deviation of this distribution?
# Write an expression for the exact probability that among the 100 students at least 55 will be admitted.
# Use SOCR to compute the probability of part (3).
# Use the normal distribution applet in SOCR to approximate the probability of part (3) (do not forget the continuity correction). What is the error of the approximation?

Below you can see the distribution of a normal random variable <math> X </math> with <math> \mu=50, \sigma=5 </math>. In this graph you can also see the probability that <math> X </math> is between 53 and 60.

<center>[[Image: SOCR_Activities_Christou_normal.jpg|600px]]</center>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2011_2012_Stat13_1_Lab4}}

SOCR Courses 2011 2012 Stat13 1 Lab4

2012-02-16T13:13:04Z

SeanWang: /* Normal Probability Distribution Activity */

== [[SOCR_Courses_2011_2012_Stat13_1 | Stats 13.1]] - Laboratory Activity 4==

=== Normal Probability Distribution Activity ===

Note: If at the end of this lab you feel like you could use some more examples, see: [[SOCR EduMaterials Activities Normal Probability examples]]

For the purposes of this lab, we are looking at <math> N(20, 3) </math>.

'''Description''': You can access the [http://www.socr.ucla.edu/htmls/SOCR_Distributions.html SOCR distribution applets here].

====Exercise 1====
Use SOCR to graph and print the distribution of <math> X \sim N(20, 3) </math>. Show on the graph the following points: <math>\mu \pm 1 \sigma, \mu \pm 2 \sigma, \mu \pm 3 \sigma </math>. How many standard deviations from the mean is the value <math> x=27.5 </math>?

====Exercise 2====
Graph the distribution of <math> X \sim N(40, 10)</math>.
# Find <math>P(X > 49)</math>.
# Find <math>P(X < 22)</math>.
# Find <math>P(12 < X < 37)</math>. Submit a printout.
# Use the mouse or the left cut off or right cut off points to find the <math>8^{th}, 20^{th}, 45^{th}, 55^{th}, 70^{th}, 95^{th} </math> percentiles. After you find these percentiles submit a printout for the <math>95^{th}</math> percentile.
# Make sure you know how to answer the above questions using the <math>z</math> score <math>z=\frac{x-\mu}{\sigma}</math> and your <math>z</math> table from the handout! You do '''not''' need to submit anything here.

====Exercise 3====
The lifetime of tires of brand <math>A</math> follows the normal distribution with mean 40000 miles and standard deviation 4000 miles.
# Use <math>SOCR</math> to find the probability that a tire will last between 40000 and 46000 miles.
# Given that a tire will last more than 46000 miles what is the probability that it will last more than 50000 miles? Submit a printout and explain how you get the answer.
# Given that a tire will last more than 46000 miles what is the probability that it will last less than 50000 miles? Submit a printout and explain how you get the answer.

====Exercise 4====
# The probability that a student is admitted in the Math Department Major at a college is <math>45 \%</math>. Suppose that this year 100 students will apply for admission into the Math major.
# What is the distribution of the number of students admitted? Use <math>SOCR</math> to graph and print this distribution. What is the shape of this distribution? What is the mean and standard deviation of this distribution?
# Write an expression for the exact probability that among the 100 students at least 55 will be admitted.
# Use SOCR to compute the probability of part (3).
# Use the normal distribution applet in SOCR to approximate the probability of part (3) (do not forget the continuity correction). What is the error of the approximation?

Below you can see the distribution of a normal random variable <math> X </math> with <math> \mu=50, \sigma=5 </math>. In this graph you can also see the probability that <math> X </math> is between 53 and 60.

<center>[[Image: SOCR_Activities_Christou_normal.jpg|600px]]</center>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2011_2012_Stat13_1_Lab4}}

SOCR Courses 2011 2012 Stat13 1 Lab4

2012-02-16T13:12:22Z

SeanWang: /* Exercise 2 */

== [[SOCR_Courses_2011_2012_Stat13_1 | Stats 13.1]] - Laboratory Activity 4==

=== Normal Probability Distribution Activity ===

Note: If at the end of this lab you feel like you could use some more examples, see: [[SOCR EduMaterials Activities Normal Probability examples]]

For the purposes of this lab, we are looking at <math> N(\mu, \sigma) </math>.

'''Description''': You can access the [http://www.socr.ucla.edu/htmls/SOCR_Distributions.html SOCR distribution applets here].

====Exercise 1====
Use SOCR to graph and print the distribution of <math> X \sim N(20, 3) </math>. Show on the graph the following points: <math>\mu \pm 1 \sigma, \mu \pm 2 \sigma, \mu \pm 3 \sigma </math>. How many standard deviations from the mean is the value <math> x=27.5 </math>?

====Exercise 2====
Graph the distribution of <math> X \sim N(40, 10)</math>.
# Find <math>P(X > 49)</math>.
# Find <math>P(X < 22)</math>.
# Find <math>P(12 < X < 37)</math>. Submit a printout.
# Use the mouse or the left cut off or right cut off points to find the <math>8^{th}, 20^{th}, 45^{th}, 55^{th}, 70^{th}, 95^{th} </math> percentiles. After you find these percentiles submit a printout for the <math>95^{th}</math> percentile.
# Make sure you know how to answer the above questions using the <math>z</math> score <math>z=\frac{x-\mu}{\sigma}</math> and your <math>z</math> table from the handout! You do '''not''' need to submit anything here.

====Exercise 3====
The lifetime of tires of brand <math>A</math> follows the normal distribution with mean 40000 miles and standard deviation 4000 miles.
# Use <math>SOCR</math> to find the probability that a tire will last between 40000 and 46000 miles.
# Given that a tire will last more than 46000 miles what is the probability that it will last more than 50000 miles? Submit a printout and explain how you get the answer.
# Given that a tire will last more than 46000 miles what is the probability that it will last less than 50000 miles? Submit a printout and explain how you get the answer.

====Exercise 4====
# The probability that a student is admitted in the Math Department Major at a college is <math>45 \%</math>. Suppose that this year 100 students will apply for admission into the Math major.
# What is the distribution of the number of students admitted? Use <math>SOCR</math> to graph and print this distribution. What is the shape of this distribution? What is the mean and standard deviation of this distribution?
# Write an expression for the exact probability that among the 100 students at least 55 will be admitted.
# Use SOCR to compute the probability of part (3).
# Use the normal distribution applet in SOCR to approximate the probability of part (3) (do not forget the continuity correction). What is the error of the approximation?

Below you can see the distribution of a normal random variable <math> X </math> with <math> \mu=50, \sigma=5 </math>. In this graph you can also see the probability that <math> X </math> is between 53 and 60.

<center>[[Image: SOCR_Activities_Christou_normal.jpg|600px]]</center>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2011_2012_Stat13_1_Lab4}}

SOCR Courses 2011 2012 Stat13 1 Lab4

2012-02-16T13:12:11Z

SeanWang: /* Exercise 2 */

== [[SOCR_Courses_2011_2012_Stat13_1 | Stats 13.1]] - Laboratory Activity 4==

=== Normal Probability Distribution Activity ===

Note: If at the end of this lab you feel like you could use some more examples, see: [[SOCR EduMaterials Activities Normal Probability examples]]

For the purposes of this lab, we are looking at <math> N(\mu, \sigma) </math>.

'''Description''': You can access the [http://www.socr.ucla.edu/htmls/SOCR_Distributions.html SOCR distribution applets here].

====Exercise 1====
Use SOCR to graph and print the distribution of <math> X \sim N(20, 3) </math>. Show on the graph the following points: <math>\mu \pm 1 \sigma, \mu \pm 2 \sigma, \mu \pm 3 \sigma </math>. How many standard deviations from the mean is the value <math> x=27.5 </math>?

====Exercise 2====
Graph the distribution of <math> X \sim N(40, 10)</math>.
# Find <math>P(X > 49)</math>.
# Find <math>P(X < 22)</math>.
# Find <math>P(12 < X < 37)</math> Submit a printout.
# Use the mouse or the left cut off or right cut off points to find the <math>8^{th}, 20^{th}, 45^{th}, 55^{th}, 70^{th}, 95^{th} </math> percentiles. After you find these percentiles submit a printout for the <math>95^{th}</math> percentile.
# Make sure you know how to answer the above questions using the <math>z</math> score <math>z=\frac{x-\mu}{\sigma}</math> and your <math>z</math> table from the handout! You do '''not''' need to submit anything here.

====Exercise 3====
The lifetime of tires of brand <math>A</math> follows the normal distribution with mean 40000 miles and standard deviation 4000 miles.
# Use <math>SOCR</math> to find the probability that a tire will last between 40000 and 46000 miles.
# Given that a tire will last more than 46000 miles what is the probability that it will last more than 50000 miles? Submit a printout and explain how you get the answer.
# Given that a tire will last more than 46000 miles what is the probability that it will last less than 50000 miles? Submit a printout and explain how you get the answer.

====Exercise 4====
# The probability that a student is admitted in the Math Department Major at a college is <math>45 \%</math>. Suppose that this year 100 students will apply for admission into the Math major.
# What is the distribution of the number of students admitted? Use <math>SOCR</math> to graph and print this distribution. What is the shape of this distribution? What is the mean and standard deviation of this distribution?
# Write an expression for the exact probability that among the 100 students at least 55 will be admitted.
# Use SOCR to compute the probability of part (3).
# Use the normal distribution applet in SOCR to approximate the probability of part (3) (do not forget the continuity correction). What is the error of the approximation?

Below you can see the distribution of a normal random variable <math> X </math> with <math> \mu=50, \sigma=5 </math>. In this graph you can also see the probability that <math> X </math> is between 53 and 60.

<center>[[Image: SOCR_Activities_Christou_normal.jpg|600px]]</center>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2011_2012_Stat13_1_Lab4}}

SOCR Courses 2011 2012 Stat13 1 Lab4

2012-02-16T13:11:25Z

SeanWang: /* Exercise 2 */

== [[SOCR_Courses_2011_2012_Stat13_1 | Stats 13.1]] - Laboratory Activity 4==

=== Normal Probability Distribution Activity ===

Note: If at the end of this lab you feel like you could use some more examples, see: [[SOCR EduMaterials Activities Normal Probability examples]]

For the purposes of this lab, we are looking at <math> N(\mu, \sigma) </math>.

'''Description''': You can access the [http://www.socr.ucla.edu/htmls/SOCR_Distributions.html SOCR distribution applets here].

====Exercise 1====
Use SOCR to graph and print the distribution of <math> X \sim N(20, 3) </math>. Show on the graph the following points: <math>\mu \pm 1 \sigma, \mu \pm 2 \sigma, \mu \pm 3 \sigma </math>. How many standard deviations from the mean is the value <math> x=27.5 </math>?

====Exercise 2====
Graph the distribution of <math> X \sim N(40, 10)</math>.
# Find <math>P(X > 49)</math>.
# Find <math>P(X < 22)</math>.
# Find <math>P(12 < X < 37)</math> Submit a printout.
# Use the mouse or the left cut off or right cut off points to find the <math>8^{th}, 20^{th}, 45^{th}, 55^{th}, 70^{th}, 95^{th} </math> percentiles. After you find these percentiles submit a printout for each one of them.
# Make sure you know how to answer the above questions using the <math>z</math> score <math>z=\frac{x-\mu}{\sigma}</math> and your <math>z</math> table from the handout! You do '''not''' need to submit anything here.

====Exercise 3====
The lifetime of tires of brand <math>A</math> follows the normal distribution with mean 40000 miles and standard deviation 4000 miles.
# Use <math>SOCR</math> to find the probability that a tire will last between 40000 and 46000 miles.
# Given that a tire will last more than 46000 miles what is the probability that it will last more than 50000 miles? Submit a printout and explain how you get the answer.
# Given that a tire will last more than 46000 miles what is the probability that it will last less than 50000 miles? Submit a printout and explain how you get the answer.

====Exercise 4====
# The probability that a student is admitted in the Math Department Major at a college is <math>45 \%</math>. Suppose that this year 100 students will apply for admission into the Math major.
# What is the distribution of the number of students admitted? Use <math>SOCR</math> to graph and print this distribution. What is the shape of this distribution? What is the mean and standard deviation of this distribution?
# Write an expression for the exact probability that among the 100 students at least 55 will be admitted.
# Use SOCR to compute the probability of part (3).
# Use the normal distribution applet in SOCR to approximate the probability of part (3) (do not forget the continuity correction). What is the error of the approximation?

Below you can see the distribution of a normal random variable <math> X </math> with <math> \mu=50, \sigma=5 </math>. In this graph you can also see the probability that <math> X </math> is between 53 and 60.

<center>[[Image: SOCR_Activities_Christou_normal.jpg|600px]]</center>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2011_2012_Stat13_1_Lab4}}

SOCR Courses 2011 2012 Stat13 1 Lab4

2012-02-16T13:08:47Z

SeanWang: /* Exercise 2 */

== [[SOCR_Courses_2011_2012_Stat13_1 | Stats 13.1]] - Laboratory Activity 4==

=== Normal Probability Distribution Activity ===

Note: If at the end of this lab you feel like you could use some more examples, see: [[SOCR EduMaterials Activities Normal Probability examples]]

For the purposes of this lab, we are looking at <math> N(\mu, \sigma) </math>.

'''Description''': You can access the [http://www.socr.ucla.edu/htmls/SOCR_Distributions.html SOCR distribution applets here].

====Exercise 1====
Use SOCR to graph and print the distribution of <math> X \sim N(20, 3) </math>. Show on the graph the following points: <math>\mu \pm 1 \sigma, \mu \pm 2 \sigma, \mu \pm 3 \sigma </math>. How many standard deviations from the mean is the value <math> x=27.5 </math>?

====Exercise 2====
Graph the distribution of <math> X \sim N(40, 10)</math>.
# Find <math>P(X > 49)</math>.
# Find <math>P(X < 22)</math>.
# Find <math>P(12 < X < 37)</math> Submit a printout.
# Use the mouse or the left cut off or right cut off points to find the <math>8^{th}, 20^{th}, 45^{th}, 55^{th}, 70^{th}, 95^{th} </math> percentiles. After you find these percentiles submit a printout for each one of them.
# Make sure you know how to answer the above questions using the <math>z</math> score <math>z=\frac{x-\mu}{\sigma}</math> and your <math>z</math> table from the handout! You do not need to submit anything here.

====Exercise 3====
The lifetime of tires of brand <math>A</math> follows the normal distribution with mean 40000 miles and standard deviation 4000 miles.
# Use <math>SOCR</math> to find the probability that a tire will last between 40000 and 46000 miles.
# Given that a tire will last more than 46000 miles what is the probability that it will last more than 50000 miles? Submit a printout and explain how you get the answer.
# Given that a tire will last more than 46000 miles what is the probability that it will last less than 50000 miles? Submit a printout and explain how you get the answer.

====Exercise 4====
# The probability that a student is admitted in the Math Department Major at a college is <math>45 \%</math>. Suppose that this year 100 students will apply for admission into the Math major.
# What is the distribution of the number of students admitted? Use <math>SOCR</math> to graph and print this distribution. What is the shape of this distribution? What is the mean and standard deviation of this distribution?
# Write an expression for the exact probability that among the 100 students at least 55 will be admitted.
# Use SOCR to compute the probability of part (3).
# Use the normal distribution applet in SOCR to approximate the probability of part (3) (do not forget the continuity correction). What is the error of the approximation?

Below you can see the distribution of a normal random variable <math> X </math> with <math> \mu=50, \sigma=5 </math>. In this graph you can also see the probability that <math> X </math> is between 53 and 60.

<center>[[Image: SOCR_Activities_Christou_normal.jpg|600px]]</center>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2011_2012_Stat13_1_Lab4}}

SOCR Courses 2011 2012 Stat13 1 Lab4

2012-02-16T13:08:18Z

SeanWang: /* Exercise 2 */

== [[SOCR_Courses_2011_2012_Stat13_1 | Stats 13.1]] - Laboratory Activity 4==

=== Normal Probability Distribution Activity ===

Note: If at the end of this lab you feel like you could use some more examples, see: [[SOCR EduMaterials Activities Normal Probability examples]]

For the purposes of this lab, we are looking at <math> N(\mu, \sigma) </math>.

'''Description''': You can access the [http://www.socr.ucla.edu/htmls/SOCR_Distributions.html SOCR distribution applets here].

====Exercise 1====
Use SOCR to graph and print the distribution of <math> X \sim N(20, 3) </math>. Show on the graph the following points: <math>\mu \pm 1 \sigma, \mu \pm 2 \sigma, \mu \pm 3 \sigma </math>. How many standard deviations from the mean is the value <math> x=27.5 </math>?

====Exercise 2====
Graph the distribution of <math> X \sim N(40, 10)</math>.
# Find <math>P(X>49) </math>
# Find <math>P(X<22) </math>
# Find <math>P(12<X<37) </math> Submit a printout.
# Use the mouse or the left cut off or right cut off points to find the <math>8^{th}, 20^{th}, 45^{th}, 55^{th}, 70^{th}, 95^{th} </math> percentiles. After you find these percentiles submit a printout for each one of them.
# Make sure you know how to answer the above questions using the <math>z</math> score <math>z=\frac{x-\mu}{\sigma}</math> and your <math>z</math> table from the handout! You do not need to submit anything here.

====Exercise 3====
The lifetime of tires of brand <math>A</math> follows the normal distribution with mean 40000 miles and standard deviation 4000 miles.
# Use <math>SOCR</math> to find the probability that a tire will last between 40000 and 46000 miles.
# Given that a tire will last more than 46000 miles what is the probability that it will last more than 50000 miles? Submit a printout and explain how you get the answer.
# Given that a tire will last more than 46000 miles what is the probability that it will last less than 50000 miles? Submit a printout and explain how you get the answer.

====Exercise 4====
# The probability that a student is admitted in the Math Department Major at a college is <math>45 \%</math>. Suppose that this year 100 students will apply for admission into the Math major.
# What is the distribution of the number of students admitted? Use <math>SOCR</math> to graph and print this distribution. What is the shape of this distribution? What is the mean and standard deviation of this distribution?
# Write an expression for the exact probability that among the 100 students at least 55 will be admitted.
# Use SOCR to compute the probability of part (3).
# Use the normal distribution applet in SOCR to approximate the probability of part (3) (do not forget the continuity correction). What is the error of the approximation?

Below you can see the distribution of a normal random variable <math> X </math> with <math> \mu=50, \sigma=5 </math>. In this graph you can also see the probability that <math> X </math> is between 53 and 60.

<center>[[Image: SOCR_Activities_Christou_normal.jpg|600px]]</center>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2011_2012_Stat13_1_Lab4}}

SOCR Courses 2011 2012 Stat13 1 Lab4

2012-02-16T13:06:53Z

SeanWang: /* Normal Probability Distribution Activity */

== [[SOCR_Courses_2011_2012_Stat13_1 | Stats 13.1]] - Laboratory Activity 4==

=== Normal Probability Distribution Activity ===

Note: If at the end of this lab you feel like you could use some more examples, see: [[SOCR EduMaterials Activities Normal Probability examples]]

For the purposes of this lab, we are looking at <math> N(\mu, \sigma) </math>.

'''Description''': You can access the [http://www.socr.ucla.edu/htmls/SOCR_Distributions.html SOCR distribution applets here].

====Exercise 1====
Use SOCR to graph and print the distribution of <math> X \sim N(20, 3) </math>. Show on the graph the following points: <math>\mu \pm 1 \sigma, \mu \pm 2 \sigma, \mu \pm 3 \sigma </math>. How many standard deviations from the mean is the value <math> x=27.5 </math>?

====Exercise 2====
Graph the distribution of <math> X \sim N(40, 10)</math>.
# Find <math>P(X>49) </math> Submit a printout.
# Find <math>P(X<22) </math> Submit a printout.
# Find <math>P(12<X<37) </math> Submit a printout.
# Use the mouse or the left cut off or right cut off points to find the <math>8^{th}, 20^{th}, 45^{th}, 55^{th}, 70^{th}, 95^{th} </math> percentiles. After you find these percentiles submit a printout for each one of them.
# Make sure you know how to answer the above questions using the <math>z</math> score <math>z=\frac{x-\mu}{\sigma}</math> and your <math>z</math> table from the handout! You do not need to submit anything here.

====Exercise 3====
The lifetime of tires of brand <math>A</math> follows the normal distribution with mean 40000 miles and standard deviation 4000 miles.
# Use <math>SOCR</math> to find the probability that a tire will last between 40000 and 46000 miles.
# Given that a tire will last more than 46000 miles what is the probability that it will last more than 50000 miles? Submit a printout and explain how you get the answer.
# Given that a tire will last more than 46000 miles what is the probability that it will last less than 50000 miles? Submit a printout and explain how you get the answer.

====Exercise 4====
# The probability that a student is admitted in the Math Department Major at a college is <math>45 \%</math>. Suppose that this year 100 students will apply for admission into the Math major.
# What is the distribution of the number of students admitted? Use <math>SOCR</math> to graph and print this distribution. What is the shape of this distribution? What is the mean and standard deviation of this distribution?
# Write an expression for the exact probability that among the 100 students at least 55 will be admitted.
# Use SOCR to compute the probability of part (3).
# Use the normal distribution applet in SOCR to approximate the probability of part (3) (do not forget the continuity correction). What is the error of the approximation?

Below you can see the distribution of a normal random variable <math> X </math> with <math> \mu=50, \sigma=5 </math>. In this graph you can also see the probability that <math> X </math> is between 53 and 60.

<center>[[Image: SOCR_Activities_Christou_normal.jpg|600px]]</center>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2011_2012_Stat13_1_Lab4}}

SOCR Courses 2011 2012 Stat13 1 Lab4

2012-02-16T13:05:47Z

SeanWang: /* Normal Probability Distribution Activity */

== [[SOCR_Courses_2011_2012_Stat13_1 | Stats 13.1]] - Laboratory Activity 4==

=== Normal Probability Distribution Activity ===

Note: If at the end of this lab you feel like you could use some more examples, see: [[SOCR EduMaterials Activities Normal Probability examples]]

For the purposes of this lab, we are looking at <math> X \sim N( \mu , 3) </math>.

'''Description''': You can access the [http://www.socr.ucla.edu/htmls/SOCR_Distributions.html SOCR distribution applets here].

====Exercise 1====
Use SOCR to graph and print the distribution of <math> X \sim N(20, 3) </math>. Show on the graph the following points: <math>\mu \pm 1 \sigma, \mu \pm 2 \sigma, \mu \pm 3 \sigma </math>. How many standard deviations from the mean is the value <math> x=27.5 </math>?

====Exercise 2====
Graph the distribution of <math> X \sim N(40, 10)</math>.
# Find <math>P(X>49) </math> Submit a printout.
# Find <math>P(X<22) </math> Submit a printout.
# Find <math>P(12<X<37) </math> Submit a printout.
# Use the mouse or the left cut off or right cut off points to find the <math>8^{th}, 20^{th}, 45^{th}, 55^{th}, 70^{th}, 95^{th} </math> percentiles. After you find these percentiles submit a printout for each one of them.
# Make sure you know how to answer the above questions using the <math>z</math> score <math>z=\frac{x-\mu}{\sigma}</math> and your <math>z</math> table from the handout! You do not need to submit anything here.

====Exercise 3====
The lifetime of tires of brand <math>A</math> follows the normal distribution with mean 40000 miles and standard deviation 4000 miles.
# Use <math>SOCR</math> to find the probability that a tire will last between 40000 and 46000 miles.
# Given that a tire will last more than 46000 miles what is the probability that it will last more than 50000 miles? Submit a printout and explain how you get the answer.
# Given that a tire will last more than 46000 miles what is the probability that it will last less than 50000 miles? Submit a printout and explain how you get the answer.

====Exercise 4====
# The probability that a student is admitted in the Math Department Major at a college is <math>45 \%</math>. Suppose that this year 100 students will apply for admission into the Math major.
# What is the distribution of the number of students admitted? Use <math>SOCR</math> to graph and print this distribution. What is the shape of this distribution? What is the mean and standard deviation of this distribution?
# Write an expression for the exact probability that among the 100 students at least 55 will be admitted.
# Use SOCR to compute the probability of part (3).
# Use the normal distribution applet in SOCR to approximate the probability of part (3) (do not forget the continuity correction). What is the error of the approximation?

Below you can see the distribution of a normal random variable <math> X </math> with <math> \mu=50, \sigma=5 </math>. In this graph you can also see the probability that <math> X </math> is between 53 and 60.

<center>[[Image: SOCR_Activities_Christou_normal.jpg|600px]]</center>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2011_2012_Stat13_1_Lab4}}

SOCR Courses 2011 2012 Stat13 1 Lab4

2012-02-16T13:05:03Z

SeanWang: /* Normal Probability Distribution Activity */

== [[SOCR_Courses_2011_2012_Stat13_1 | Stats 13.1]] - Laboratory Activity 4==

=== Normal Probability Distribution Activity ===

Note: If at the end of this lab you feel like you could use some more examples, see: [[SOCR EduMaterials Activities Normal Probability examples]]

For the purposes of this lab, we are looking at <math> X \sim N( \mu, 3) </math>.

'''Description''': You can access the [http://www.socr.ucla.edu/htmls/SOCR_Distributions.html SOCR distribution applets here].

====Exercise 1====
Use SOCR to graph and print the distribution of <math> X \sim N(20, 3) </math>. Show on the graph the following points: <math>\mu \pm 1 \sigma, \mu \pm 2 \sigma, \mu \pm 3 \sigma </math>. How many standard deviations from the mean is the value <math> x=27.5 </math>?

====Exercise 2====
Graph the distribution of <math> X \sim N(40, 10)</math>.
# Find <math>P(X>49) </math> Submit a printout.
# Find <math>P(X<22) </math> Submit a printout.
# Find <math>P(12<X<37) </math> Submit a printout.
# Use the mouse or the left cut off or right cut off points to find the <math>8^{th}, 20^{th}, 45^{th}, 55^{th}, 70^{th}, 95^{th} </math> percentiles. After you find these percentiles submit a printout for each one of them.
# Make sure you know how to answer the above questions using the <math>z</math> score <math>z=\frac{x-\mu}{\sigma}</math> and your <math>z</math> table from the handout! You do not need to submit anything here.

====Exercise 3====
The lifetime of tires of brand <math>A</math> follows the normal distribution with mean 40000 miles and standard deviation 4000 miles.
# Use <math>SOCR</math> to find the probability that a tire will last between 40000 and 46000 miles.
# Given that a tire will last more than 46000 miles what is the probability that it will last more than 50000 miles? Submit a printout and explain how you get the answer.
# Given that a tire will last more than 46000 miles what is the probability that it will last less than 50000 miles? Submit a printout and explain how you get the answer.

====Exercise 4====
# The probability that a student is admitted in the Math Department Major at a college is <math>45 \%</math>. Suppose that this year 100 students will apply for admission into the Math major.
# What is the distribution of the number of students admitted? Use <math>SOCR</math> to graph and print this distribution. What is the shape of this distribution? What is the mean and standard deviation of this distribution?
# Write an expression for the exact probability that among the 100 students at least 55 will be admitted.
# Use SOCR to compute the probability of part (3).
# Use the normal distribution applet in SOCR to approximate the probability of part (3) (do not forget the continuity correction). What is the error of the approximation?

Below you can see the distribution of a normal random variable <math> X </math> with <math> \mu=50, \sigma=5 </math>. In this graph you can also see the probability that <math> X </math> is between 53 and 60.

<center>[[Image: SOCR_Activities_Christou_normal.jpg|600px]]</center>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2011_2012_Stat13_1_Lab4}}

SOCR Courses 2011 2012 Stat13 1 Lab4

2012-02-16T13:04:55Z

SeanWang: /* Normal Probability Distribution Activity */

== [[SOCR_Courses_2011_2012_Stat13_1 | Stats 13.1]] - Laboratory Activity 4==

=== Normal Probability Distribution Activity ===

Note: If at the end of this lab you feel like you could use some more examples, see: [[SOCR EduMaterials Activities Normal Probability examples]]

For the purposes of this lab, we are looking at <math> X \sim N(\mu, 3) </math>.

'''Description''': You can access the [http://www.socr.ucla.edu/htmls/SOCR_Distributions.html SOCR distribution applets here].

====Exercise 1====
Use SOCR to graph and print the distribution of <math> X \sim N(20, 3) </math>. Show on the graph the following points: <math>\mu \pm 1 \sigma, \mu \pm 2 \sigma, \mu \pm 3 \sigma </math>. How many standard deviations from the mean is the value <math> x=27.5 </math>?

====Exercise 2====
Graph the distribution of <math> X \sim N(40, 10)</math>.
# Find <math>P(X>49) </math> Submit a printout.
# Find <math>P(X<22) </math> Submit a printout.
# Find <math>P(12<X<37) </math> Submit a printout.
# Use the mouse or the left cut off or right cut off points to find the <math>8^{th}, 20^{th}, 45^{th}, 55^{th}, 70^{th}, 95^{th} </math> percentiles. After you find these percentiles submit a printout for each one of them.
# Make sure you know how to answer the above questions using the <math>z</math> score <math>z=\frac{x-\mu}{\sigma}</math> and your <math>z</math> table from the handout! You do not need to submit anything here.

====Exercise 3====
The lifetime of tires of brand <math>A</math> follows the normal distribution with mean 40000 miles and standard deviation 4000 miles.
# Use <math>SOCR</math> to find the probability that a tire will last between 40000 and 46000 miles.
# Given that a tire will last more than 46000 miles what is the probability that it will last more than 50000 miles? Submit a printout and explain how you get the answer.
# Given that a tire will last more than 46000 miles what is the probability that it will last less than 50000 miles? Submit a printout and explain how you get the answer.

====Exercise 4====
# The probability that a student is admitted in the Math Department Major at a college is <math>45 \%</math>. Suppose that this year 100 students will apply for admission into the Math major.
# What is the distribution of the number of students admitted? Use <math>SOCR</math> to graph and print this distribution. What is the shape of this distribution? What is the mean and standard deviation of this distribution?
# Write an expression for the exact probability that among the 100 students at least 55 will be admitted.
# Use SOCR to compute the probability of part (3).
# Use the normal distribution applet in SOCR to approximate the probability of part (3) (do not forget the continuity correction). What is the error of the approximation?

Below you can see the distribution of a normal random variable <math> X </math> with <math> \mu=50, \sigma=5 </math>. In this graph you can also see the probability that <math> X </math> is between 53 and 60.

<center>[[Image: SOCR_Activities_Christou_normal.jpg|600px]]</center>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2011_2012_Stat13_1_Lab4}}

SOCR Courses 2011 2012 Stat13 1 Lab4

2012-02-16T13:04:37Z

SeanWang: /* Normal Probability Distribution Activity */

== [[SOCR_Courses_2011_2012_Stat13_1 | Stats 13.1]] - Laboratory Activity 4==

=== Normal Probability Distribution Activity ===

Note: If at the end of this lab you feel like you could use some more examples, see: [[SOCR EduMaterials Activities Normal Probability examples]]

For the purposes of this lab, we are looking at <math> X \sim N(20, 3) </math>.

'''Description''': You can access the [http://www.socr.ucla.edu/htmls/SOCR_Distributions.html SOCR distribution applets here].

====Exercise 1====
Use SOCR to graph and print the distribution of <math> X \sim N(20, 3) </math>. Show on the graph the following points: <math>\mu \pm 1 \sigma, \mu \pm 2 \sigma, \mu \pm 3 \sigma </math>. How many standard deviations from the mean is the value <math> x=27.5 </math>?

====Exercise 2====
Graph the distribution of <math> X \sim N(40, 10)</math>.
# Find <math>P(X>49) </math> Submit a printout.
# Find <math>P(X<22) </math> Submit a printout.
# Find <math>P(12<X<37) </math> Submit a printout.
# Use the mouse or the left cut off or right cut off points to find the <math>8^{th}, 20^{th}, 45^{th}, 55^{th}, 70^{th}, 95^{th} </math> percentiles. After you find these percentiles submit a printout for each one of them.
# Make sure you know how to answer the above questions using the <math>z</math> score <math>z=\frac{x-\mu}{\sigma}</math> and your <math>z</math> table from the handout! You do not need to submit anything here.

====Exercise 3====
The lifetime of tires of brand <math>A</math> follows the normal distribution with mean 40000 miles and standard deviation 4000 miles.
# Use <math>SOCR</math> to find the probability that a tire will last between 40000 and 46000 miles.
# Given that a tire will last more than 46000 miles what is the probability that it will last more than 50000 miles? Submit a printout and explain how you get the answer.
# Given that a tire will last more than 46000 miles what is the probability that it will last less than 50000 miles? Submit a printout and explain how you get the answer.

====Exercise 4====
# The probability that a student is admitted in the Math Department Major at a college is <math>45 \%</math>. Suppose that this year 100 students will apply for admission into the Math major.
# What is the distribution of the number of students admitted? Use <math>SOCR</math> to graph and print this distribution. What is the shape of this distribution? What is the mean and standard deviation of this distribution?
# Write an expression for the exact probability that among the 100 students at least 55 will be admitted.
# Use SOCR to compute the probability of part (3).
# Use the normal distribution applet in SOCR to approximate the probability of part (3) (do not forget the continuity correction). What is the error of the approximation?

Below you can see the distribution of a normal random variable <math> X </math> with <math> \mu=50, \sigma=5 </math>. In this graph you can also see the probability that <math> X </math> is between 53 and 60.

<center>[[Image: SOCR_Activities_Christou_normal.jpg|600px]]</center>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2011_2012_Stat13_1_Lab4}}

SOCR Courses 2011 2012 Stat13 1 Lab4

2012-02-16T13:03:14Z

SeanWang: /* Normal Probability Distribution Activity */

== [[SOCR_Courses_2011_2012_Stat13_1 | Stats 13.1]] - Laboratory Activity 4==

=== Normal Probability Distribution Activity ===

Note: If at the end of this lab you feel like you could use some more examples, see: [[SOCR EduMaterials Activities Normal Probability examples]]

For the purposes of this lab, we are looking at <math> X \sim N(\mu, \sigma) </math>.

'''Description''': You can access the [http://www.socr.ucla.edu/htmls/SOCR_Distributions.html SOCR distribution applets here].

====Exercise 1====
Use SOCR to graph and print the distribution of <math> X \sim N(20, 3) </math>. Show on the graph the following points: <math>\mu \pm 1 \sigma, \mu \pm 2 \sigma, \mu \pm 3 \sigma </math>. How many standard deviations from the mean is the value <math> x=27.5 </math>?

====Exercise 2====
Graph the distribution of <math> X \sim N(40, 10)</math>.
# Find <math>P(X>49) </math> Submit a printout.
# Find <math>P(X<22) </math> Submit a printout.
# Find <math>P(12<X<37) </math> Submit a printout.
# Use the mouse or the left cut off or right cut off points to find the <math>8^{th}, 20^{th}, 45^{th}, 55^{th}, 70^{th}, 95^{th} </math> percentiles. After you find these percentiles submit a printout for each one of them.
# Make sure you know how to answer the above questions using the <math>z</math> score <math>z=\frac{x-\mu}{\sigma}</math> and your <math>z</math> table from the handout! You do not need to submit anything here.

====Exercise 3====
The lifetime of tires of brand <math>A</math> follows the normal distribution with mean 40000 miles and standard deviation 4000 miles.
# Use <math>SOCR</math> to find the probability that a tire will last between 40000 and 46000 miles.
# Given that a tire will last more than 46000 miles what is the probability that it will last more than 50000 miles? Submit a printout and explain how you get the answer.
# Given that a tire will last more than 46000 miles what is the probability that it will last less than 50000 miles? Submit a printout and explain how you get the answer.

====Exercise 4====
# The probability that a student is admitted in the Math Department Major at a college is <math>45 \%</math>. Suppose that this year 100 students will apply for admission into the Math major.
# What is the distribution of the number of students admitted? Use <math>SOCR</math> to graph and print this distribution. What is the shape of this distribution? What is the mean and standard deviation of this distribution?
# Write an expression for the exact probability that among the 100 students at least 55 will be admitted.
# Use SOCR to compute the probability of part (3).
# Use the normal distribution applet in SOCR to approximate the probability of part (3) (do not forget the continuity correction). What is the error of the approximation?

Below you can see the distribution of a normal random variable <math> X </math> with <math> \mu=50, \sigma=5 </math>. In this graph you can also see the probability that <math> X </math> is between 53 and 60.

<center>[[Image: SOCR_Activities_Christou_normal.jpg|600px]]</center>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

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SOCR Courses 2011 2012 Stat13 1 Lab3

2012-02-01T23:21:53Z

SeanWang: /* Problem 2 */

== [[SOCR_Courses_2011_2012_Stat13_1 | Stats 13.1]] - Laboratory Activity 3==

You can access the applet for any of the [http://www.socr.ucla.edu/htmls/SOCR_Distributions.html SOCR distributions] (http://www.socr.ucla.edu/htmls/SOCR_Distributions.html). Use SOCR to graph the following distributions and answer the questions below.

===[http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial Distribution] Activity ===

==== Problem 1 ====
* X ~ Binom(10, .5)
* Find: P(X = 3), E(X), sd(X) from the SOCR output, and verify them with the formulas discussed in class.

==== Problem 2 ====
* X ~ Binom(10, .1)
* Find P(1 ≤ X ≤ 3) from SOCR, and verify using the binomial formula.

==== Problem 3 ====
* X ~ Binom(10, .9)
* Find and verify:
* P(5 < x < 8)
* P(x < 8)
* P(x ≤ 7)
* P(x ≥ 9)

==== Problem 4 ====
* X ~ Binom(30, .1)
* Find and verify: P(x > 2)

===Distribution Comparison===

* Graph and comment on the shape of the [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html binomial distribution] with n = 30, p = 0.1 and then with n = 20, p = 0.9 (take a snapshot of both).
* Now, keep n = 30 but change p = 0.45. Now, let n = 100, p = 0.1. Take a snapshot of both.
* What changes do you observe in the distribution as the parameters change? Write brief statements about the changes in a chart like this.

{|border="1" cellpadding="20"
|Smaller n
|Larger n
|Smaller p
|Larger p
|-
|

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|}

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2011_2012_Stat13_1_Lab3}}

SOCR Courses 2011 2012 Stat13 1 Lab3

2012-02-01T23:21:14Z

SeanWang: /* Distribution Comparison */

== [[SOCR_Courses_2011_2012_Stat13_1 | Stats 13.1]] - Laboratory Activity 3==

You can access the applet for any of the [http://www.socr.ucla.edu/htmls/SOCR_Distributions.html SOCR distributions] (http://www.socr.ucla.edu/htmls/SOCR_Distributions.html). Use SOCR to graph the following distributions and answer the questions below.

===[http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial Distribution] Activity ===

==== Problem 1 ====
* X ~ Binom(10, .5)
* Find: P(X = 3), E(X), sd(X) from the SOCR output, and verify them with the formulas discussed in class.

==== Problem 2 ====
* X ~ Binom(10, .1)
* Find and verify: P(1 ≤ X ≤ 3)

==== Problem 3 ====
* X ~ Binom(10, .9)
* Find and verify:
* P(5 < x < 8)
* P(x < 8)
* P(x ≤ 7)
* P(x ≥ 9)

==== Problem 4 ====
* X ~ Binom(30, .1)
* Find and verify: P(x > 2)

===Distribution Comparison===

* Graph and comment on the shape of the [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html binomial distribution] with n = 30, p = 0.1 and then with n = 20, p = 0.9 (take a snapshot of both).
* Now, keep n = 30 but change p = 0.45. Now, let n = 100, p = 0.1. Take a snapshot of both.
* What changes do you observe in the distribution as the parameters change? Write brief statements about the changes in a chart like this.

{|border="1" cellpadding="20"
|Smaller n
|Larger n
|Smaller p
|Larger p
|-
|

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|}

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2011_2012_Stat13_1_Lab3}}

SOCR Courses 2011 2012 Stat13 1 Lab3

2012-02-01T23:20:52Z

SeanWang: /* Distribution Comparison */

== [[SOCR_Courses_2011_2012_Stat13_1 | Stats 13.1]] - Laboratory Activity 3==

You can access the applet for any of the [http://www.socr.ucla.edu/htmls/SOCR_Distributions.html SOCR distributions] (http://www.socr.ucla.edu/htmls/SOCR_Distributions.html). Use SOCR to graph the following distributions and answer the questions below.

===[http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial Distribution] Activity ===

==== Problem 1 ====
* X ~ Binom(10, .5)
* Find: P(X = 3), E(X), sd(X) from the SOCR output, and verify them with the formulas discussed in class.

==== Problem 2 ====
* X ~ Binom(10, .1)
* Find and verify: P(1 ≤ X ≤ 3)

==== Problem 3 ====
* X ~ Binom(10, .9)
* Find and verify:
* P(5 < x < 8)
* P(x < 8)
* P(x ≤ 7)
* P(x ≥ 9)

==== Problem 4 ====
* X ~ Binom(30, .1)
* Find and verify: P(x > 2)

===Distribution Comparison===

* Graph and comment on the shape of the [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html binomial distribution] with n = 30, p = 0.1 and then with n = 20, p = 0.9 (take a snapshot of both).
* Now, keep n = 30 but change p = 0.45. Now, let n = 100, p = 0.1. Take a snapshot of both.
* What changes do you observe in the distribution as the parameters change? Write brief statements in this chart.

{|border="1" cellpadding="20"
|Smaller n
|Larger n
|Smaller p
|Larger p
|-
|

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|}

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2011_2012_Stat13_1_Lab3}}

SOCR Courses 2011 2012 Stat13 1 Lab3

2012-02-01T23:19:51Z

SeanWang: /* Distribution Comparison */

== [[SOCR_Courses_2011_2012_Stat13_1 | Stats 13.1]] - Laboratory Activity 3==

You can access the applet for any of the [http://www.socr.ucla.edu/htmls/SOCR_Distributions.html SOCR distributions] (http://www.socr.ucla.edu/htmls/SOCR_Distributions.html). Use SOCR to graph the following distributions and answer the questions below.

===[http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial Distribution] Activity ===

==== Problem 1 ====
* X ~ Binom(10, .5)
* Find: P(X = 3), E(X), sd(X) from the SOCR output, and verify them with the formulas discussed in class.

==== Problem 2 ====
* X ~ Binom(10, .1)
* Find and verify: P(1 ≤ X ≤ 3)

==== Problem 3 ====
* X ~ Binom(10, .9)
* Find and verify:
* P(5 < x < 8)
* P(x < 8)
* P(x ≤ 7)
* P(x ≥ 9)

==== Problem 4 ====
* X ~ Binom(30, .1)
* Find and verify: P(x > 2)

===Distribution Comparison===

* Graph and comment on the shape of the [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html binomial distribution] with n = 30, p = 0.1 and then with n = 20, p = 0.9 (take a snapshot of both).
* Now, keep n = 30 but change p = 0.45. How about when n = 100, p = 0.1? Take another snapshot.
* What changes do you observe in the distribution as the parameters change? Write brief statements in this chart.

{|border="1" cellpadding="20"
|Smaller n
|Larger n
|Smaller p
|Larger p
|-
|

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|}

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2011_2012_Stat13_1_Lab3}}

SOCR Courses 2011 2012 Stat13 1 Lab3

2012-02-01T23:19:30Z

SeanWang: /* Problem 2 */

== [[SOCR_Courses_2011_2012_Stat13_1 | Stats 13.1]] - Laboratory Activity 3==

You can access the applet for any of the [http://www.socr.ucla.edu/htmls/SOCR_Distributions.html SOCR distributions] (http://www.socr.ucla.edu/htmls/SOCR_Distributions.html). Use SOCR to graph the following distributions and answer the questions below.

===[http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial Distribution] Activity ===

==== Problem 1 ====
* X ~ Binom(10, .5)
* Find: P(X = 3), E(X), sd(X) from the SOCR output, and verify them with the formulas discussed in class.

==== Problem 2 ====
* X ~ Binom(10, .1)
* Find and verify: P(1 ≤ X ≤ 3)

==== Problem 3 ====
* X ~ Binom(10, .9)
* Find and verify:
* P(5 < x < 8)
* P(x < 8)
* P(x ≤ 7)
* P(x ≥ 9)

==== Problem 4 ====
* X ~ Binom(30, .1)
* Find and verify: P(x > 2)

===Distribution Comparison===

* Graph and comment on the shape of the [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html binomial distribution] with n = 30, p = 0.1 and then with n = 20,p = 0.9 (take a snapshot of each).
* Now, keep n = 30 but change p = 0.45. How about when n = 100, p = 0.1? Take another snapshot.
* What changes do you observe in the distribution as the parameters change? Write brief statements in this chart.

{|border="1" cellpadding="20"
|Smaller n
|Larger n
|Smaller p
|Larger p
|-
|

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|}

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2011_2012_Stat13_1_Lab3}}

SOCR Courses 2011 2012 Stat13 1 Lab3

2012-02-01T23:19:09Z

SeanWang: /* Problem 2 */

== [[SOCR_Courses_2011_2012_Stat13_1 | Stats 13.1]] - Laboratory Activity 3==

You can access the applet for any of the [http://www.socr.ucla.edu/htmls/SOCR_Distributions.html SOCR distributions] (http://www.socr.ucla.edu/htmls/SOCR_Distributions.html). Use SOCR to graph the following distributions and answer the questions below.

===[http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial Distribution] Activity ===

==== Problem 1 ====
* X ~ Binom(10, .5)
* Find: P(X = 3), E(X), sd(X) from the SOCR output, and verify them with the formulas discussed in class.

==== Problem 2 ====
* X ~ Binom(10, .1)
* Find and verify P(1 ≤ X ≤ 3)

==== Problem 3 ====
* X ~ Binom(10, .9)
* Find and verify:
* P(5 < x < 8)
* P(x < 8)
* P(x ≤ 7)
* P(x ≥ 9)

==== Problem 4 ====
* X ~ Binom(30, .1)
* Find and verify: P(x > 2)

===Distribution Comparison===

* Graph and comment on the shape of the [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html binomial distribution] with n = 30, p = 0.1 and then with n = 20,p = 0.9 (take a snapshot of each).
* Now, keep n = 30 but change p = 0.45. How about when n = 100, p = 0.1? Take another snapshot.
* What changes do you observe in the distribution as the parameters change? Write brief statements in this chart.

{|border="1" cellpadding="20"
|Smaller n
|Larger n
|Smaller p
|Larger p
|-
|

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|
|}

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2011_2012_Stat13_1_Lab3}}

SOCR Courses 2011 2012 Stat13 1 Lab3

2012-02-01T23:18:37Z

SeanWang: /* Problem 2 */

== [[SOCR_Courses_2011_2012_Stat13_1 | Stats 13.1]] - Laboratory Activity 3==

You can access the applet for any of the [http://www.socr.ucla.edu/htmls/SOCR_Distributions.html SOCR distributions] (http://www.socr.ucla.edu/htmls/SOCR_Distributions.html). Use SOCR to graph the following distributions and answer the questions below.

===[http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial Distribution] Activity ===

==== Problem 1 ====
* X ~ Binom(10, .5)
* Find: P(X = 3), E(X), sd(X) from the SOCR output, and verify them with the formulas discussed in class.

==== Problem 2 ====
* X ~ Binom(10, .1)
* Find and verify:
* P(1 ≤ X ≤ 3)

==== Problem 3 ====
* X ~ Binom(10, .9)
* Find and verify:
* P(5 < x < 8)
* P(x < 8)
* P(x ≤ 7)
* P(x ≥ 9)

==== Problem 4 ====
* X ~ Binom(30, .1)
* Find and verify: P(x > 2)

===Distribution Comparison===

* Graph and comment on the shape of the [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html binomial distribution] with n = 30, p = 0.1 and then with n = 20,p = 0.9 (take a snapshot of each).
* Now, keep n = 30 but change p = 0.45. How about when n = 100, p = 0.1? Take another snapshot.
* What changes do you observe in the distribution as the parameters change? Write brief statements in this chart.

{|border="1" cellpadding="20"
|Smaller n
|Larger n
|Smaller p
|Larger p
|-
|

|
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|
|}

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2011_2012_Stat13_1_Lab3}}

SOCR Courses 2011 2012 Stat13 1 Lab3

2012-02-01T23:18:23Z

SeanWang: /* Problem 3 */

== [[SOCR_Courses_2011_2012_Stat13_1 | Stats 13.1]] - Laboratory Activity 3==

You can access the applet for any of the [http://www.socr.ucla.edu/htmls/SOCR_Distributions.html SOCR distributions] (http://www.socr.ucla.edu/htmls/SOCR_Distributions.html). Use SOCR to graph the following distributions and answer the questions below.

===[http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial Distribution] Activity ===

==== Problem 1 ====
* X ~ Binom(10, .5)
* Find: P(X = 3), E(X), sd(X) from the SOCR output, and verify them with the formulas discussed in class.

==== Problem 2 ====
* X ~ Binom(10, .1)
* Find and verify:
* P(1 ≤ X ≤ 3)
* <math>P(1 \leq X \leq 3)</math>

==== Problem 3 ====
* X ~ Binom(10, .9)
* Find and verify:
* P(5 < x < 8)
* P(x < 8)
* P(x ≤ 7)
* P(x ≥ 9)

==== Problem 4 ====
* X ~ Binom(30, .1)
* Find and verify: P(x > 2)

===Distribution Comparison===

* Graph and comment on the shape of the [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html binomial distribution] with n = 30, p = 0.1 and then with n = 20,p = 0.9 (take a snapshot of each).
* Now, keep n = 30 but change p = 0.45. How about when n = 100, p = 0.1? Take another snapshot.
* What changes do you observe in the distribution as the parameters change? Write brief statements in this chart.

{|border="1" cellpadding="20"
|Smaller n
|Larger n
|Smaller p
|Larger p
|-
|

|
|
|
|}

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2011_2012_Stat13_1_Lab3}}

SOCR Courses 2011 2012 Stat13 1 Lab3

2012-02-01T23:18:02Z

SeanWang: /* Problem 2 */

== [[SOCR_Courses_2011_2012_Stat13_1 | Stats 13.1]] - Laboratory Activity 3==

You can access the applet for any of the [http://www.socr.ucla.edu/htmls/SOCR_Distributions.html SOCR distributions] (http://www.socr.ucla.edu/htmls/SOCR_Distributions.html). Use SOCR to graph the following distributions and answer the questions below.

===[http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial Distribution] Activity ===

==== Problem 1 ====
* X ~ Binom(10, .5)
* Find: P(X = 3), E(X), sd(X) from the SOCR output, and verify them with the formulas discussed in class.

==== Problem 2 ====
* X ~ Binom(10, .1)
* Find and verify:
* P(1 ≤ X ≤ 3)
* <math>P(1 \leq X \leq 3)</math>

==== Problem 3 ====
* X ~ Binom(10, .9)
* Find and verify:
* P(5 < x < 8)
* P(x < 8)
* P(x <= 7)
* P(x >= 9)

==== Problem 4 ====
* X ~ Binom(30, .1)
* Find and verify: P(x > 2)

===Distribution Comparison===

* Graph and comment on the shape of the [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html binomial distribution] with n = 30, p = 0.1 and then with n = 20,p = 0.9 (take a snapshot of each).
* Now, keep n = 30 but change p = 0.45. How about when n = 100, p = 0.1? Take another snapshot.
* What changes do you observe in the distribution as the parameters change? Write brief statements in this chart.

{|border="1" cellpadding="20"
|Smaller n
|Larger n
|Smaller p
|Larger p
|-
|

|
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|
|}

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2011_2012_Stat13_1_Lab3}}

SOCR Courses 2011 2012 Stat13 1 Lab3

2012-02-01T23:17:50Z

SeanWang: /* Problem 2 */

== [[SOCR_Courses_2011_2012_Stat13_1 | Stats 13.1]] - Laboratory Activity 3==

You can access the applet for any of the [http://www.socr.ucla.edu/htmls/SOCR_Distributions.html SOCR distributions] (http://www.socr.ucla.edu/htmls/SOCR_Distributions.html). Use SOCR to graph the following distributions and answer the questions below.

===[http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial Distribution] Activity ===

==== Problem 1 ====
* X ~ Binom(10, .5)
* Find: P(X = 3), E(X), sd(X) from the SOCR output, and verify them with the formulas discussed in class.

==== Problem 2 ====
* X ~ Binom(10, .1)
* Find and verify:
* P(1 <= X <= ≤ 3)
* <math>P(1 \leq X \leq 3)</math>

==== Problem 3 ====
* X ~ Binom(10, .9)
* Find and verify:
* P(5 < x < 8)
* P(x < 8)
* P(x <= 7)
* P(x >= 9)

==== Problem 4 ====
* X ~ Binom(30, .1)
* Find and verify: P(x > 2)

===Distribution Comparison===

* Graph and comment on the shape of the [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html binomial distribution] with n = 30, p = 0.1 and then with n = 20,p = 0.9 (take a snapshot of each).
* Now, keep n = 30 but change p = 0.45. How about when n = 100, p = 0.1? Take another snapshot.
* What changes do you observe in the distribution as the parameters change? Write brief statements in this chart.

{|border="1" cellpadding="20"
|Smaller n
|Larger n
|Smaller p
|Larger p
|-
|

|
|
|
|}

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2011_2012_Stat13_1_Lab3}}

SOCR Courses 2011 2012 Stat13 1 Lab3

2012-02-01T23:17:04Z

SeanWang: /* Problem 2 */

== [[SOCR_Courses_2011_2012_Stat13_1 | Stats 13.1]] - Laboratory Activity 3==

You can access the applet for any of the [http://www.socr.ucla.edu/htmls/SOCR_Distributions.html SOCR distributions] (http://www.socr.ucla.edu/htmls/SOCR_Distributions.html). Use SOCR to graph the following distributions and answer the questions below.

===[http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial Distribution] Activity ===

==== Problem 1 ====
* X ~ Binom(10, .5)
* Find: P(X = 3), E(X), sd(X) from the SOCR output, and verify them with the formulas discussed in class.

==== Problem 2 ====
* X ~ Binom(10, .1)
* Find and verify:
* P(1 <= X <= 3)
* <math>P(1 \leq X \leq 3)</math>

==== Problem 3 ====
* X ~ Binom(10, .9)
* Find and verify:
* P(5 < x < 8)
* P(x < 8)
* P(x <= 7)
* P(x >= 9)

==== Problem 4 ====
* X ~ Binom(30, .1)
* Find and verify: P(x > 2)

===Distribution Comparison===

* Graph and comment on the shape of the [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html binomial distribution] with n = 30, p = 0.1 and then with n = 20,p = 0.9 (take a snapshot of each).
* Now, keep n = 30 but change p = 0.45. How about when n = 100, p = 0.1? Take another snapshot.
* What changes do you observe in the distribution as the parameters change? Write brief statements in this chart.

{|border="1" cellpadding="20"
|Smaller n
|Larger n
|Smaller p
|Larger p
|-
|

|
|
|
|}

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2011_2012_Stat13_1_Lab3}}

SOCR Courses 2011 2012 Stat13 1 Lab3

2012-02-01T23:16:38Z

SeanWang: /* Problem 2 */

== [[SOCR_Courses_2011_2012_Stat13_1 | Stats 13.1]] - Laboratory Activity 3==

You can access the applet for any of the [http://www.socr.ucla.edu/htmls/SOCR_Distributions.html SOCR distributions] (http://www.socr.ucla.edu/htmls/SOCR_Distributions.html). Use SOCR to graph the following distributions and answer the questions below.

===[http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial Distribution] Activity ===

==== Problem 1 ====
* X ~ Binom(10, .5)
* Find: P(X = 3), E(X), sd(X) from the SOCR output, and verify them with the formulas discussed in class.

==== Problem 2 ====
* <math>X \sim Binom(10, .1)</math>
* Find and verify:
* P(1 <= X <= 3)
* P(1 <math>\leq</math> X <= 3)
* <math>P(1 \leq X \leq 3)</math>

==== Problem 3 ====
* X ~ Binom(10, .9)
* Find and verify:
* P(5 < x < 8)
* P(x < 8)
* P(x <= 7)
* P(x >= 9)

==== Problem 4 ====
* X ~ Binom(30, .1)
* Find and verify: P(x > 2)

===Distribution Comparison===

* Graph and comment on the shape of the [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html binomial distribution] with n = 30, p = 0.1 and then with n = 20,p = 0.9 (take a snapshot of each).
* Now, keep n = 30 but change p = 0.45. How about when n = 100, p = 0.1? Take another snapshot.
* What changes do you observe in the distribution as the parameters change? Write brief statements in this chart.

{|border="1" cellpadding="20"
|Smaller n
|Larger n
|Smaller p
|Larger p
|-
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|}

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* SOCR Home page: http://www.socr.ucla.edu

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