SOCR Courses 2008 2009 Stat13 1 Lab5 - Revision history

Rclements17: /* Exercise 2 */

2008-11-17T19:36:41Z

‎Exercise 2

Rclements17: /* Exercise 2 */

2008-11-17T19:31:51Z

‎Exercise 2

Rclements17: /* Exercise 2 */

2008-11-17T19:30:11Z

‎Exercise 2

Rclements17: /* Exercise 1 */

2008-11-17T19:25:05Z

‎Exercise 1

Rclements17: /* Description */

2008-11-17T19:21:55Z

‎Description

IvoDinov at 02:10, 21 October 2008

2008-10-21T02:10:58Z

IvoDinov: New page: == Stats 13.1 - Laboratory Activity 5== ===Confidence Interval Activity=== ==== This is an activity to explore the confidence intervals for the popu...

2008-09-03T18:53:01Z

New page: == Stats 13.1 - Laboratory Activity 5== ===Confidence Interval Activity=== ==== This is an activity to explore the confidence intervals for the popu...

New page

== [[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 "Confidence Interval Experiment".

The confidence interval for the population mean <math>\mu</math> when <math> \sigma </math> is known is given by:
<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.

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

*'''Exercise 2:'''
**'''1.''' Reset and repeat (a) with <math>\alpha=0.01</math>. Take a snapshot and describe what you see.
**'''2.''' Reset and repeat (a) with sample size now <math>n=80</math>. Take a snapshot and describe in detail what you see.
**'''3.''' 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}}

@@ Line 23: / Line 23: @@
 ===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 <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>.
+#   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>).
+#  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>.

@@ Line 23: / Line 23: @@
 ===Exercise 2===
-#  Reset and repeat Exercise 1 with <math>\alpha=0.01</math>.
+#  Reset and repeat Exercise 1, answering questions 1 and 6,  with <math>\alpha=0.01</math>.
-#   Reset and repeat Exercise 1 with sample size now <math>n=80</math>.
+#   Reset and repeat Exercise 1, answering questions 1 and 6, 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>).
+#  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>.

@@ Line 23: / Line 23: @@
 ===Exercise 2===
-#  Reset and repeat (a) with <math>\alpha=0.01</math>.  Take a snapshot and describe what you see.
+#  Reset and repeat Exercise 1 with <math>\alpha=0.01</math>.
-#   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 Exercise 1 with sample size now <math>n=80</math>.
-#  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.
+#  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>.

@@ Line 13: / Line 13: @@
 ===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.
+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 number -3, -2, -1, 0, 1, 2, 3 represent?
 #  What do the blue lines represent?

@@ Line 6: / Line 6: @@
 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:
+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===

@@ Line 1: / Line 1: @@
-== [[SOCR_Courses_2008_2009_Stat13_1 | Stats 13.1]] - Laboratory Activity 5==
+== [[SOCR_Courses_2008_2009_Stat13_1 | Stats 13.1]] - Laboratory Activity 5: Confidence Interval Activity==
-===Confidence Interval Activity===
+This is an activity to explore the confidence intervals for the population mean when the standard deviation is known
-==== 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:
@@ Line 13: / Line 12: @@
 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:'''
+===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.
-**'''1.'''  What do the number -3, -2, -1, 0, 1, 2, 3 represent?
+# What do the number -3, -2, -1, 0, 1, 2, 3 represent?
-**'''2.'''  What do the blue lines represent?
+#  What do the blue lines represent?
-**'''3.'''  How is the confidence interval represented?
+#  How is the confidence interval represented?
-**'''4.'''  What does the green dot represent?
+#  What does the green dot represent?
-**'''5.'''  Write down the formula on which the confidence intervals are based.
+#  Write down the formula on which the confidence intervals are based.
 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>