One of the key ideas in Section 2.1 is that the standard deviation of a distribution of sample proportions is estimated by

when the population large enough, where large enough was when the population was more than 20 times the sample size. In Exploration 2.1 you used the words in the Gettysburg Address as the population. There are only 268 words in that speech, so the population is not that large. In this exercise you will use the Sampling Words applet to investigate how population size effects the estimate for the standard deviation of the sample proportions.

a. The proportion of short words in the Gettysburg Address is 0.410. If we take repeated samples of 30 words from the Gettysburg Address, what is the predicted value of the standard deviation of sample proportions? 

b. In the Sampling Words applet, use the pull-down menu to change the Variable to Short. Change the Sample size to 30 and take at least 50,000 samples from the Gettysburg Address. How does the standard deviation of your sample proportions from the applet compare to your answer from part (a)? Why might it be a bit different? 

c. Click on the radio button above the population data that has × 40 next to it. This will make it so you can sample from 40 copies of the Gettysburg Address. Therefore, the population size is now 40 × 268 = 10,720. Now take at least 50,000 samples. How does the standard deviation of your distribution of sample proportions from the applet compare to your answer from part (a)? Did a larger population size help you get a standard deviation closer to what was predicted in part (a)?  

Transcribed Image Text:

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