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SOCR Courses 2008 2009 Stat13 1 Lab5

2008-11-17T19:36:41Z

Rclements17: /* Exercise 2 */

== [[SOCR_Courses_2008_2009_Stat13_1 | Stats 13.1]] - Laboratory Activity 5: Confidence Interval Activity==

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

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

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

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

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

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

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

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

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

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SOCR Courses 2008 2009 Stat13 1 Lab5

2008-11-17T19:31:51Z

Rclements17: /* Exercise 2 */

== [[SOCR_Courses_2008_2009_Stat13_1 | Stats 13.1]] - Laboratory Activity 5: Confidence Interval Activity==

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

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

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

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

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

===Exercise 2===
# Reset and repeat Exercise 1, answering questions 1 and 6, with <math>\alpha=0.01</math>.
# Reset and repeat Exercise 1, answering questions 1 and 6, with sample size now <math>n=80</math>.
# Reset and repeat Exercise 1, answering questions 1 and 6, with sample size <math>n=80</math> and <math>\alpha=1.0E-4</math> (this is <math>10^{-4}</math>).

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

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

<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_Lab5}}

SOCR Courses 2008 2009 Stat13 1 Lab5

2008-11-17T19:30:11Z

Rclements17: /* Exercise 2 */

== [[SOCR_Courses_2008_2009_Stat13_1 | Stats 13.1]] - Laboratory Activity 5: Confidence Interval Activity==

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

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

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

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

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

===Exercise 2===
# Reset and repeat Exercise 1 with <math>\alpha=0.01</math>.
# Reset and repeat Exercise 1 with sample size now <math>n=80</math>.
# Reset and repeat Exercise 1 with sample size <math>n=80</math> and <math>\alpha=1.0E-4</math> (this is <math>10^{-4}</math>).

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

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

<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_Lab5}}

SOCR Courses 2008 2009 Stat13 1 Lab5

2008-11-17T19:25:05Z

Rclements17: /* Exercise 1 */

== [[SOCR_Courses_2008_2009_Stat13_1 | Stats 13.1]] - Laboratory Activity 5: Confidence Interval Activity==

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

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

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

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

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

===Exercise 2===
# Reset and repeat (a) with <math>\alpha=0.01</math>. Take a snapshot and describe what you see.
# Reset and repeat (a) with sample size now <math>n=80</math>. Take a snapshot and describe in detail what you see.
# Reset and repeat (a) with sample size <math>n=80</math> and <math>\alpha=1.0E-4</math> (this is <math>10^{-4}</math>). Take a snapshot and describe in detail what you see.

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

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

<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_Lab5}}

SOCR Courses 2008 2009 Stat13 1 Lab5

2008-11-17T19:21:55Z

Rclements17: /* Description */

== [[SOCR_Courses_2008_2009_Stat13_1 | Stats 13.1]] - Laboratory Activity 5: Confidence Interval Activity==

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

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

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

===Exercise 1===
Using the scroll down button select "Number of Experiments = 100". Select sample size <math>n=20</math>, and choose number of intervals 200. It means: You will select 200 samples and with each sample you will obtain a confidence interval. You will do this 100 times. How many intervals (out of the 200) do you expect to miss the population mean <math>\mu=0</math>? Take a snapshot and describe what you observe.

# What do the number -3, -2, -1, 0, 1, 2, 3 represent?
# What do the blue lines represent?
# How is the confidence interval represented?
# What does the green dot represent?
# Write down the formula on which the confidence intervals are based.

===Exercise 2===
# Reset and repeat (a) with <math>\alpha=0.01</math>. Take a snapshot and describe what you see.
# Reset and repeat (a) with sample size now <math>n=80</math>. Take a snapshot and describe in detail what you see.
# Reset and repeat (a) with sample size <math>n=80</math> and <math>\alpha=1.0E-4</math> (this is <math>10^{-4}</math>). Take a snapshot and describe in detail what you see.

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

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

<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_Lab5}}

SOCR Courses 2008 2009 Stat13 1 Lab4

2008-10-27T19:17:21Z

Rclements17: /* Exercise 2 */

== [[SOCR_Courses_2008_2009_Stat13_1 | Stats 13.1]] - Laboratory Activity 4==

=== Normal Probability Distribution Activity ===

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

'''Description''': You can access the applets for the above distributions at http://www.socr.ucla.edu/htmls/SOCR_Distributions.html .

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

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

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

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

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

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

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

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2008_2009_Stat13_1_Lab4}}

SOCR Courses 2008 2009 Stat13 1 Lab3

2008-10-11T16:51:50Z

Rclements17:

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

You can access the applet for any of the distributions at http://www.socr.ucla.edu under the '''Distributions''' tab. Use SOCR to graph the following distributions and answer the questions below. Also, comment on the shape of each one of these distributions.

===Binomial Distribution Activity ===

==== 1 ====
X~Binom(10,.5)
Find: P(x = 3),E(X), sd(X) and verify them with the formulas discussed in class.
==== 2 ====
X~Binom(10,.1)
Find and verify:

P(1 <= x <= 3)

==== 3 ====
X~Binom(10,.9)
Find and verify:

P(5 < x < 8)

P(x < 8)

P(x <= 7)

P(x >= 9)

==== 4 ====
X~Binom(30,.1)

Find and verify:

P(x > 2)

===Distribution Comparison===

Graph and comment on the shape of binomial with n = 20,p = 0.1 and n = 20,p = 0.9
(take a snapshot of each). Now, keep n = 20 but change p = 0.45. How about when n =
80,p = 0.1? Take another snapshot. What changes do you observe in the distribution as the parameters change? Watch both the change in shape as well as the changing number on the x and y axes.

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

|
|
|
|}

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

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2008_2009_Stat13_1_Lab3}}