SOCR Courses 2011 2012 Stat13 1 Lab5 - Revision history

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IvoDinov: Created page with '== Stats 13.1 - Laboratory Activity 5: Confidence Interval Activity== This is an activity to explore the confidence intervals for the popul…'

2011-11-16T18:32:17Z

Created page with '== Stats 13.1 - Laboratory Activity 5: Confidence Interval Activity== This is an activity to explore the confidence intervals for the popul…'

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

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

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

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

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

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

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

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

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

@@ Line 19: / Line 19: @@
 # How is the confidence interval represented?
 # What does the green dot represent?
-# How many intervals (out of the 200) do you ''expect'' to miss the population mean <math>\mu=0</math>?  How many ''did'' when you ran it?
+# How many intervals (out of the 200) do you ''expect'' to miss the population mean <math>\mu=0</math>?  How many ''did'' when you ran it (in the final experiment)?
 # Write down the formula on which these confidence intervals are based.

@@ Line 26: / Line 26: @@
 #   Reset and repeat Exercise 1, answering questions 5 and 6, with sample size now <math>n=80</math> and <math>\alpha=0.01</math>.  Will the intervals be larger, smaller, or stay the same compared to that of Exercise 1?
 #  Reset and repeat Exercise 1, answering questions 5 and 6, with sample size <math>n=80</math> and <math>\alpha=1.0E-4</math> (this is <math>10^{-4}</math>).  Will the intervals be larger, smaller, or stay the same compared to that of Exercise 1?  Take a snapshot.
-#  What does this tell you about what should happen when you make <math>\alpha</math> smaller or larger?  What about when you make <math>n</math> smaller or larger?
+#  What should happen to the lengths of the intervals as <math>\alpha</math> gets larger?  What about as <math>n</math> gets larger?
 Below you can see a snapshot of the run of 100 intervals with <math> n=36, \ \alpha=0.05 </math>.

@@ Line 23: / Line 23: @@
 ===Exercise 2===
 #  Reset and repeat Exercise 1, answering questions 5 and 6,  with <math>\alpha=0.01</math>.  Will the intervals be larger, smaller, or stay the same compared to that of Exercise 1?
 #   Reset and repeat Exercise 1, answering questions 5 and 6, with sample size now <math>n=80</math> and <math>\alpha=0.01</math>.  Will the intervals be larger, smaller, or stay the same compared to that of Exercise 1?
-#  Reset and repeat Exercise 1, answering questions 5 and 6, with sample size <math>n=80</math> and <math>\alpha=1.0E-4</math> (this is <math>10^{-4}</math>).  Will the intervals be larger, smaller, or stay the same compared to that of Exercise 1?
+#  Reset and repeat Exercise 1, answering questions 5 and 6, with sample size <math>n=80</math> and <math>\alpha=1.0E-4</math> (this is <math>10^{-4}</math>).  Will the intervals be larger, smaller, or stay the same compared to that of Exercise 1?  Take a snapshot.
+#  What does this tell you about what should happen when you make <math>\alpha</math> smaller or larger?  What about when you make <math>n</math> smaller or larger?
 Below you can see a snapshot of the run of 100 intervals with <math> n=36, \ \alpha=0.05 </math>.

@@ Line 26: / Line 26: @@
 #   Reset and repeat Exercise 1, answering questions 5 and 6, with sample size now <math>n=80</math> and <math>\alpha=0.01</math>.  Will the intervals be larger, smaller, or stay the same compared to that of Exercise 1?
 #  Reset and repeat Exercise 1, answering questions 5 and 6, with sample size <math>n=80</math> and <math>\alpha=1.0E-4</math> (this is <math>10^{-4}</math>).  Will the intervals be larger, smaller, or stay the same compared to that of Exercise 1?
 Below you can see a snapshot of the run of 100 intervals with <math> n=36, \ \alpha=0.05 </math>.

@@ Line 15: / Line 15: @@
 Using the scroll down button select "Number of Experiments = 100".  Select sample size <math>n=20</math>, and choose number of intervals 200.  It means:  You will select 200 samples and with each sample you will obtain a confidence interval.  You will do this 100 times.    Take a snapshot and answer the following questions.
 # What do the numbers -3, -2, -1, 0, 1, 2, 3 represent?
 # What do the blue lines represent?
 # How is the confidence interval represented?
 # What does the green dot represent?
 # Write down the formula on which these confidence intervals are based.