MAT308: Sampling Distributions of X – Assignment Solution

Sampling Distributions of X

Objectives

In this activity, you will use the applet for investigating the sampling distribution of X.  You will use the applet to explore the shape of the sampling distribution when the population is normally distributed and when the population is not normal.

Materials

Computer with internet access – one per student or pair of students.

Launch the Sampling Distributions applet at http://onlinestatbook.com/stat_sim/sampling_dist/index.html.  When the BEGIN button appears on the left side of the screen, click on it.  You will see a yellow page entitled “Sampling Distributions”.

1. There are choices for the population distribution: Normal, uniform, skewed, and custom.  The default is Normal.  Click the “Animated” button.

• What happens?

• Click the button several more times. What do the black boxes represent?

• What is the blue square that drops down onto the plot below?

• What does the red horizontal band under the population histogram tell us?

Notice the left panel.  Important numbers are displayed there.  Did you notice that the colors of the numbers match up with the objects to the right?  As you make things happen, the numbers change accordingly, like an automatic scorekeeper.

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2. Click on “Clear lower 3” to start clean. Then click on the “10,000” button one time under “Sample;” to simulate taking 10,000 SRSs of size n = 5 from the population.  Answer these questions:

• Does the approximate sampling distribution (blue bars) have a recognizable shape? Click the box next to “Fit normal.”

• Compare the mean of the approximate sampling distribution with the mean of the population.

• How is the standard deviation of the approximate sampling distribution related to the standard deviation of the population?

3. Click “Clear lower 3.” Use the drop-down menus to set up the bottom graph to display the mean for samples for size n = 20.  Then sample 10,000 times.  How do the two distributions of compare:  shape, center, spread?

4. What have you learned about the shape of the sampling distribution of when the population has a Normal shape?

5. Now select “Skewed population. Set the bottom two graphs to display the mean—one for samples of size 2 and the other for samples of size 5.  Click the Animated button a few times to be sure you see what’s happening.  Then “Clear lower 3” and take 10,000 SRSs.  Describe what you see.

6. Change the sample sizes to n = 10 and n = 16 and repeat Step 1. What do you notice?

7. Now change the sample sizes to n = 20 and n = 25 and take 10,000 more samples. Did this confirm what you saw in Step 2?

8. Clear the page and select “Custom” distribution. Click on a point on the population graph to insert a bar of that height.  Or click on a point on the horizontal axis and drag up to define a bar.  Make a distribution that looks as strange as you can. (Note: You can shorten a bar or get rid of it completely by clicking on the top of the bar and dragging down to the axis.)  Then repeat #5 – 7 for your custom distribution.

• Step 1: Set the bottom two graphs to display the mean—one for samples of size 2 and the other for samples of size 5.  Click the Animated button a few times to be sure you see what’s happening.  Then “Clear lower 3” and take 10,000 SRSs.  Describe what you see.

• Step 2: Change the sample sizes to n = 10 and n = 16 and repeat Step 1.  What do you notice?

• Step 3: Now change the sample size to n = 20 and n = 25 and take 10,000 more samples.  Did this confirm what you saw in Step 2?

9. Summarize what you learned about the shape of the sampling distribution of X.