SOCR - User contributions [en]

SOCR Courses 2008 2009 Stat13 1 Lab6

2008-11-25T07:55:27Z

Somerville: /* Experiment 3 */

== [[SOCR_Courses_2008_2009_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>, 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_2008_2009_Stat13_1_Lab6}}

SOCR Courses 2008 2009 Stat13 1 Lab6

2008-11-24T20:06:59Z

Somerville: /* Experiment 4 */

== [[SOCR_Courses_2008_2009_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>, 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 Normal, 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_2008_2009_Stat13_1_Lab6}}

SOCR Courses 2008 2009 Stat13 1 Lab6

2008-11-24T20:05:03Z

Somerville: /* Experiment 3 */

== [[SOCR_Courses_2008_2009_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>, 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 Normal, 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). This away you can construct contiguous 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_2008_2009_Stat13_1_Lab6}}

SOCR Courses 2008 2009 Stat13 1 Lab6

2008-11-24T20:03:30Z

Somerville: /* Experiment 4 */

== [[SOCR_Courses_2008_2009_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>, 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 Normal, Poisson, Beta, Gamma, Cauchy and other continuous or discrete distributions. 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). This away you can construct contiguous 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_2008_2009_Stat13_1_Lab6}}

SOCR Courses 2008 2009 Stat13 1 Lab6

2008-11-24T20:02:14Z

Somerville: /* Experiment 3 */

== [[SOCR_Courses_2008_2009_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>, 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 Normal, Poisson, Beta, Gamma, Cauchy and other continuous or discrete distributions. 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). This away you can construct contiguous 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.
<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_2008_2009_Stat13_1_Lab6}}

SOCR Courses 2008 2009 Stat13 1 Lab6

2008-11-24T20:00:43Z

Somerville: /* Experiment 3 */

== [[SOCR_Courses_2008_2009_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>, 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 Normal, Poisson, Beta, Gamma, Cauchy and other continuous or discrete distributions. 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
<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). This away you can construct contiguous 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.
<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_2008_2009_Stat13_1_Lab6}}

SOCR Courses 2008 2009 Stat13 1 Lab6

2008-11-24T19:55:25Z

Somerville: /* Experiment 2 */

== [[SOCR_Courses_2008_2009_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>, 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 Normal, Poisson, Beta, Gamma, Cauchy and other continuous or discrete distributions. 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>).
<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). This away you can construct contiguous 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.
<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_2008_2009_Stat13_1_Lab6}}

SOCR Courses 2008 2009 Stat13 1 Lab6

2008-11-24T19:46:06Z

Somerville:

== [[SOCR_Courses_2008_2009_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. '''Step''' through the experiment a few times (by clicking the Run button) and then click '''Refresh Stats Table''' button on the top to see the sample summary statistics. 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.

<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 Normal, Poisson, Beta, Gamma, Cauchy and other continuous or discrete distributions. 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>).
<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). This away you can construct contiguous 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.
<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_2008_2009_Stat13_1_Lab6}}

SOCR Courses 2008 2009 Stat13 1 Lab3b

2008-10-21T15:37:16Z

Somerville: /* Part 1: Histograms */

== [[SOCR_Courses_2008_2009_Stat13_1 | Stats 13.1]] - Laboratory Activity 3==

<center>This Lab is due on '''October 28, 2008'''</center>

===Summary===
This lab is going to serve as a review of what we did in the previous three labs.

====Part 1: Histograms====
Follow these steps:

# Click the Charts tab on the [http://www.socr.ucla.edu SOCR website], and click on the arrow next to Bar Charts then XYPlots. Then click on '''HistogramChartDemo''' and go to the DATA tab.
# Open up another window (or another tab in the same browser window), go to the [http://www.socr.ucla.edu/htmls/SOCR_Modeler.html SOCR Modeler], and click on the '''Data Generation''' button.
# Choose the '''Normal Distribution''' from the drop-down menu. Let’s make the sample size 100 and standard deviation 75. Mean stays 0.
#Check the '''Raw Data''' check-box on the left side – this step is the simplest yet most important.
# Hit the '''Sample''' button and click on the Data tab next to the Graphs tab.
# Select all the data by holding ''Apple+A'' for Macs or ''Ctrl+A'' for PC. Press the COPY button at the top.
# Go back to the window with the '''HistogramChartDemo'''. Click on the '''DATA''' tab and press the '''PASTE''' button on the left hand side of the screen.
# Click on the '''MAPPING''' tab, and click on '''Add''' when C1 is highlighted. Then click on '''UPDATE_CHART''' at the top.
# Click on the '''GRAPH''' tab and play around with the bin size at the bottom. Take a picture of the histogram you created by using the Print Screen command. Comment on the changes you see as you increase/decrease the bin size.

====Part 2: Probability====
# Go to the [http://www.socr.ucla.edu/htmls/SOCR_Experiments.html SOCR Experiments]. We’re going to do the [http://socr.ucla.edu/htmls/exp/Coin_Die_Experiment.html Coin Die Experiment] again, so select it from the drop-down menu.
# Set the distribution of both dice to skewed left (6 should have the most probability).
# Set p = 0.8 and run 100 trials. Count the number of 6’s and take a picture of the graph.
# Repeat Step 3 with p = 0.2. Make sure you take a picture.
# Change the distribution of only the green die to skewed right (6 should have the least probability), and repeat steps 3 and 4. You should comment on changes and have 4 pics.

====Step 3: Binomial Distribution====
# Go to the [http://www.socr.ucla.edu/htmls/SOCR_Experiments.html SOCR Experiments]. Select [http://socr.ucla.edu/htmls/exp/Binomial_Coin_Experiment.html Binomial Coin Experiment].
# Make sure n=20 and p=0.50.
# Choose the Stop 10 option and press fast-forward. This will generate 10 simulations of 20 flips of the coin. Take a picture.
# Choose the Stop 100 option and press fast-forward. This will generate 100 simulations of 20 flips of the coin. Take a picture. Compare the histogram with the histogram from Step 3.
# Compare the histogram with the histogram of normal distribution.

====Step 4: Coin Toss====
# This portion does not require you to do any calculations on SOCR.
# '''Without''' using the equations of binomial distribution, solve the following problem: Let X equal the number of heads when you flip the coin 3 times. Calculate P(X>=2).

<hr>

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SSOCR_Courses_2008_2009_Stat13_1_Lab3b}}

SOCR Courses 2008 2009 Stat13 1 Lab3b

2008-10-21T15:35:27Z

Somerville:

== [[SOCR_Courses_2008_2009_Stat13_1 | Stats 13.1]] - Laboratory Activity 3==

<center>This Lab is due on '''October 28, 2008'''</center>

===Summary===
This lab is going to serve as a review of what we did in the previous three labs.

====Part 1: Histograms====
Follow these steps:

# Click the Charts tab on the [http://www.socr.ucla.edu SOCR website], and click on the arrow next to Bar Charts then XYPlots. Then click on '''HistogramChartDemo''' and go to the DATA tab.
# Open up another window (or another tab in the same browser window), go to the [http://www.socr.ucla.edu/htmls/SOCR_Modeler.html SOCR Modeler], and click on the '''Data Generation''' button.
# Choose the '''Normal Distribution''' from the drop-down menu. Let’s make the sample size 100 and standard deviation 75. Mean stays 0.
#Check the '''Raw Data''' check-box on the left side – this step is the simplest yet most important.
# Hit the '''Sample''' button and click on the Data tab next to the Graphs tab.
# Select all the data by holding ''Apple+A'' for Macs or ''Ctrl+A'' for PC. Press the COPY button at the top.
# Go back to the window with the '''HistogramChartDemo'''. Press the '''PASTE''' button on the left hand side of the screen.
# Click on the '''MAPPING''' tab, and click on '''Add''' when C1 is highlighted. Then click on '''UPDATE_CHART''' at the top.
# Click on the '''GRAPH''' tab and play around with the bin size at the bottom. Take a picture of the histogram you created by using the Print Screen command. Comment on the changes you see as you increase/decrease the bin size.

====Part 2: Probability====
# Go to the [http://www.socr.ucla.edu/htmls/SOCR_Experiments.html SOCR Experiments]. We’re going to do the [http://socr.ucla.edu/htmls/exp/Coin_Die_Experiment.html Coin Die Experiment] again, so select it from the drop-down menu.
# Set the distribution of both dice to skewed left (6 should have the most probability).
# Set p = 0.8 and run 100 trials. Count the number of 6’s and take a picture of the graph.
# Repeat Step 3 with p = 0.2. Make sure you take a picture.
# Change the distribution of only the green die to skewed right (6 should have the least probability), and repeat steps 3 and 4. You should comment on changes and have 4 pics.

====Step 3: Binomial Distribution====
# Go to the [http://www.socr.ucla.edu/htmls/SOCR_Experiments.html SOCR Experiments]. Select [http://socr.ucla.edu/htmls/exp/Binomial_Coin_Experiment.html Binomial Coin Experiment].
# Make sure n=20 and p=0.50.
# Choose the Stop 10 option and press fast-forward. This will generate 10 simulations of 20 flips of the coin. Take a picture.
# Choose the Stop 100 option and press fast-forward. This will generate 100 simulations of 20 flips of the coin. Take a picture. Compare the histogram with the histogram from Step 3.
# Compare the histogram with the histogram of normal distribution.

====Step 4: Coin Toss====
# This portion does not require you to do any calculations on SOCR.
# '''Without''' using the equations of binomial distribution, solve the following problem: Let X equal the number of heads when you flip the coin 3 times. Calculate P(X>=2).

<hr>

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SSOCR_Courses_2008_2009_Stat13_1_Lab3b}}

SOCR Courses 2008 2009 Stat13 1 Lab3b

2008-10-21T01:00:07Z

Somerville:

== '''STATS 13 Review Lab - DUE ON October 28, 2008''' ==

''Summary:'' This lab is going to serve as a review of what we did in the previous three labs.

'''Part 1: Histograms'''

1) Click the Charts tab on the SOCR website: [http://www.socr.ucla.edu[http://www.socr.ucla.edu]], and click on the arrow next to Bar Charts then XYPlots. Then click on HistogramChartDemo and the DATA tab.

2) Open up another window, go to the SOCR website [http://www.socr.ucla.edu[http://www.socr.ucla.edu]], and click the Modeler tab.

3) Click on the Data Generation button, and choose the Normal Distribution from the dropdown menu. Let’s make the sample size 100 and standard deviation 75. Mean stays 0.

4) Check the Raw Data box on the left side – this step is the simplest yet most important.

5) Hit the Sample button and click on the Data tab next to the Graphs tab.

6) Select all the data by holding Apple+A for Macs and Ctrl+A for PC. Press the COPY button at the top.

7) Go back to the window with the HistogramChartDemo. Press the PASTE button on the left hand side of the screen.

8) Click on the GRAPH tab and play around with the bin size at the bottom. Take a picture of the histogram you created by using the Print Screen command. Comment on the changes you see as you increase/decrease the bin size.

'''Part 2: Probability'''

1) Click on the Experiments tab on the SOCR website. We’re going to do the Coin Die Experiment again, so select it from the dropdown menu.

2) Set the distribution of both dice to skewed left (6 should have the most probability).

3) Set p = 0.8 and run 100 trials. Count the number of 6’s and take a picture of the graph.

4) Repeat Step 3 with p = 0.2. Make sure you take a picture.

5) Change the distribution of only the green die to skewed right (6 should have the least probability), and repeat steps 3 and 4. You should comment on changes and have 4 pics.

'''Step 3: Binomial Distribution'''

1) Click on the Experiments tab on the SOCR website. Select Binomial Coin Experiment.

2) Make sure n=20 and p=0.50.

3) Choose the Stop 10 option and press fast-forward. This will generate 10 simulations of 20 flips of the coin. Take a picture.

4) Choose the Stop 100 option and press fast-forward. This will generate 100 simulations of 20 flips of the coin. Take a picture. Compare the histogram with the histogram from Step 3.

5) Compare the histogram with the histogram of normal distribution.

'''Step 4: Coin Toss'''

1) This portion does not require you to do any calculations on SOCR.

2) '''Without''' using the equations of binomial distribution, solve the following problem: Let X equal the number of heads when you flip the coin 3 times. Calculate P(X>=2).

SOCR Courses 2008 2009 Stat13 1 Lab3b

2008-10-21T00:56:19Z

Somerville: /* '''STATS 13 Review Lab - October 21, 2008''' */

== '''STATS 13 Review Lab - October 21, 2008''' ==

''Summary:'' This lab is going to serve as a review of what we did in the previous three labs.

'''Part 1: Histograms'''

1) Click the Charts tab on the SOCR website: [http://www.socr.ucla.edu[http://www.socr.ucla.edu]], and click on the arrow next to Bar Charts then XYPlots. Then click on HistogramChartDemo and the DATA tab.

2) Open up another window, go to the SOCR website [http://www.socr.ucla.edu[http://www.socr.ucla.edu]], and click the Modeler tab.

3) Click on the Data Generation button, and choose the Normal Distribution from the dropdown menu. Let’s make the sample size 100 and standard deviation 75. Mean stays 0.

4) Check the Raw Data box on the left side – this step is the simplest yet most important.

5) Hit the Sample button and click on the Data tab next to the Graphs tab.

6) Select all the data by holding Apple+A for Macs and Ctrl+A for PC. Press the COPY button at the top.

7) Go back to the window with the HistogramChartDemo. Press the PASTE button on the left hand side of the screen.

8) Click on the GRAPH tab and play around with the bin size at the bottom. Take a picture of the histogram you created by using the Print Screen command. Comment on the changes you see as you increase/decrease the bin size.

'''Part 2: Probability'''

1) Click on the Experiments tab on the SOCR website. We’re going to do the Coin Die Experiment again, so select it from the dropdown menu.

2) Set the distribution of both dice to skewed left (6 should have the most probability).

3) Set p = 0.8 and run 100 trials. Count the number of 6’s and take a picture of the graph.

4) Repeat Step 3 with p = 0.2. Make sure you take a picture.

5) Change the distribution of only the green die to skewed right (6 should have the least probability), and repeat steps 3 and 4. You should comment on changes and have 4 pics.

'''Step 3: Binomial Distribution'''

1) Click on the Experiments tab on the SOCR website. Select Binomial Coin Experiment.

2) Make sure n=20 and p=0.50.

3) Choose the Stop 10 option and press fast-forward. This will generate 10 simulations of 20 flips of the coin. Take a picture.

4) Choose the Stop 100 option and press fast-forward. This will generate 100 simulations of 20 flips of the coin. Take a picture. Compare the histogram with the histogram from Step 3.

5) Compare the histogram with the histogram of normal distribution.

'''Step 4: Coin Toss'''

1) This portion does not require you to do any calculations on SOCR.

2) '''Without''' using the equations of binomial distribution, solve the following problem: Let X equal the number of heads when you flip the coin 3 times. Calculate P(X>=2).

SOCR Courses 2008 2009 Stat13 1 Lab3b

2008-10-21T00:51:43Z

Somerville: /* '''STATS 13 Review Lab - October 21, 2008''' */

SOCR Courses 2008 2009 Stat13 1 Lab3b

2008-10-21T00:51:16Z

Somerville: New page: == '''STATS 13 Review Lab - October 21, 2008''' == ''Summary:'' This lab is going to serve as a review of what we did in the previous three labs. '''Part 1: Histograms''' 1) Click the ...

SOCR Courses 2008 2009 Stat13 1

2008-10-21T00:44:26Z

Somerville: /* SOCR 2008-2009 Courses - [http://www.stat.ucla.edu/%7Edinov/courses_students.dir/08/Fall/STAT13.1.dir/STAT13.html Stats 13.1] Laboratory Activities */

== [[SOCR_Courses_2008_2009 | SOCR 2008-2009 Courses]] - [http://www.stat.ucla.edu/%7Edinov/courses_students.dir/08/Fall/STAT13.1.dir/STAT13.html Stats 13.1] Laboratory Activities==

* [[SOCR_Courses_2008_2009_Stat13_1_Lab1 | Lab 1]]

* [[SOCR_Courses_2008_2009_Stat13_1_Lab2 | Lab 2]]

* [[SOCR_Courses_2008_2009_Stat13_1_Lab3 | Lab 3]]

* [[SOCR_Courses_2008_2009_Stat13_1_Lab3b | Lab 3b]]

* [[SOCR_Courses_2008_2009_Stat13_1_Lab4 | Lab 4]]

* [[SOCR_Courses_2008_2009_Stat13_1_Lab5 | Lab 5]]

* [[SOCR_Courses_2008_2009_Stat13_1_Lab6 | Lab 6]]

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

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2008_2009_Stat13_1}}

SOCR Courses 2008 2009 Stat13 1 Lab2

2008-10-06T21:33:21Z

Somerville: /* 9) */

== [[SOCR_Courses_2008_2009_Stat13_1 | Stats 13.1]] - Laboratory Activity 2==

===Coin Die Experiment===

Go to http://www.socr.ucla.edu/htmls/SOCR_Experiments.html and use the scroll bar to find the '''Coin Die Experiment'''. Once you find it, click on the About button and read about the experiment. Answer the following questions with references to graphs where appropriate.

====1)====

There are two random variables (X,Y ), one parameter (p) for the coin, and the probability distribution that governs each die (green or red) involved in this experiment. Describe what each random variable represents.

* Choose fair coin (p = 0.5), and fair dice (green and red).
* Perform 10 runs and take a snapshot
* Perform 100 runs and take a snapshot

====2)====

Construct the probability distribution of Y

====3)====

Comment on the empirical and theoretical distributions how the sample size affects them.

* Reset
* Choose p = 0.8, 1-6 flat for the green die, 3-4 flat for the red die
* Perform 100 runs and take a snapshot

====4)====

Construct the probability distribution of Y

====5)====

Compute the mean of Y

====6)====

Compute the standard deviation of Y

====7)====

Verify that these numbers agree with the ones in the applet.
* Reset
* Choose p = 0.2, green fair die, 2-5 flat for red die
* Perform 100 runs and take a snapshot

====8)====

Construct the probability distribution of Y

====9)====

Comment on the empirical and theoretical distributions. Also, comment on how the coin's parameter (p) affects the theoretical distributions.

===Monte Hall Activity (Let's Make a Deal)===

The Monte Hall Activity can be found here: [[SOCR EduMaterials Activities MontyHall]]

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

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2008_2009_Stat13_1_Lab2}}

SOCR Courses 2008 2009 Stat13 1 Lab2

2008-10-06T21:23:38Z

Somerville: /* 9) */

== [[SOCR_Courses_2008_2009_Stat13_1 | Stats 13.1]] - Laboratory Activity 2==

===Coin Die Experiment===

Go to http://www.socr.ucla.edu/htmls/SOCR_Experiments.html and use the scroll bar to find the '''Coin Die Experiment'''. Once you find it, click on the About button and read about the experiment. Answer the following questions with references to graphs where appropriate.

====1)====

There are two random variables (X,Y ), one parameter (p) for the coin, and the probability distribution that governs each die (green or red) involved in this experiment. Describe what each random variable represents.

* Choose fair coin (p = 0.5), and fair dice (green and red).
* Perform 10 runs and take a snapshot
* Perform 100 runs and take a snapshot

====2)====

Construct the probability distribution of Y

====3)====

Comment on the empirical and theoretical distributions how the sample size affects them.

* Reset
* Choose p = 0.8, 1-6 flat for the green die, 3-4 flat for the red die
* Perform 100 runs and take a snapshot

====4)====

Construct the probability distribution of Y

====5)====

Compute the mean of Y

====6)====

Compute the standard deviation of Y

====7)====

Verify that these numbers agree with the ones in the applet.
* Reset
* Choose p = 0.2, green fair die, 2-5 flat for red die
* Perform 100 runs and take a snapshot

====8)====

Construct the probability distribution of Y

====9)====

Comment on the empirical and theoretical distributions. Also, comment on how probability affects the theoretical distributions.

===Monte Hall Activity (Let's Make a Deal)===

The Monte Hall Activity can be found here: [[SOCR EduMaterials Activities MontyHall]]

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

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2008_2009_Stat13_1_Lab2}}

SOCR Courses 2008 2009 Stat13 1 Lab2

2008-10-06T21:22:43Z

Somerville: /* 1) */

== [[SOCR_Courses_2008_2009_Stat13_1 | Stats 13.1]] - Laboratory Activity 2==

===Coin Die Experiment===

Go to http://www.socr.ucla.edu/htmls/SOCR_Experiments.html and use the scroll bar to find the '''Coin Die Experiment'''. Once you find it, click on the About button and read about the experiment. Answer the following questions with references to graphs where appropriate.

====1)====

There are two random variables (X,Y ), one parameter (p) for the coin, and the probability distribution that governs each die (green or red) involved in this experiment. Describe what each random variable represents.

* Choose fair coin (p = 0.5), and fair dice (green and red).
* Perform 10 runs and take a snapshot
* Perform 100 runs and take a snapshot

====2)====

Construct the probability distribution of Y

====3)====

Comment on the empirical and theoretical distributions how the sample size affects them.

* Reset
* Choose p = 0.8, 1-6 flat for the green die, 3-4 flat for the red die
* Perform 100 runs and take a snapshot

====4)====

Construct the probability distribution of Y

====5)====

Compute the mean of Y

====6)====

Compute the standard deviation of Y

====7)====

Verify that these numbers agree with the ones in the applet.
* Reset
* Choose p = 0.2, green fair die, 2-5 flat for red die
* Perform 100 runs and take a snapshot

====8)====

Construct the probability distribution of Y

====9)====

Comment on the empirical and theoretical distributions.

===Monte Hall Activity (Let's Make a Deal)===

The Monte Hall Activity can be found here: [[SOCR EduMaterials Activities MontyHall]]

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

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2008_2009_Stat13_1_Lab2}}