Question 3.12
3.12 A sampling experiment. Figures 3.1 and 3.2 show how the sample proportion behaves when we take many samples from the same population. You can follow the steps in this process on a small scale.
Figure 3.4 represents a small population. Each circle represents an adult. The white circles are people who favor a constitutional amendment that would define marriage as being between a man and a woman, and the colored circles are people who are opposed. You can check that 50 of the 100 circles are white, so in this population the proportion who favor an amendment is p = 50100 = 0.5.
(a) The circles are labeled 00, 01, . . . , 99. Use line 101 of Table A to draw an SRS of size 4. What is the proportion of the people in your sample who favor a constitutional amendment?
56
(b) Take nine more SRSs of size 4 (10 in all), using lines 102 to 110 of Table A, a different line for each sample. You now have 10 values of the sample proportion . Write down the 10 values you should now have of the sample proportion .
(c) Because your samples have only four people, the only values can take are 04, 14, 24, 34, and 44. That is, is always 0, 0.25, 0.5, 0.75, or 1. Mark these numbers on a line and make a histogram of your 10 results by putting a bar above each number to show how many samples had that outcome.
(d) Taking samples of size 4 from a population of size 100 is not a practical setting, but let’s look at your results anyway. How many of your 10 samples estimated the population proportion p = 0.5 exactly correctly? Is the true value 0.5 in the center of your sample values? Explain why 0.5 would be in the center of the sample values if you took a large number of samples.