Question: 32 We want to know what percent of American adults approve of legal gambling. This population proportion, p is a parameter. To estimate p take an SRS and find the proportion \(\hat{p}\) in the sample who approves of gambling. If we take many SRSs of the same size, the proportion (p-hat) will vary from sample to sample. The distribution of its values in all SRSs is the sampling distribution of this statistic.

Figure 10.10 (below) is a small population. Each circle represents an adult. The colored circles are people who disapprove of legal gambling, and the white circles are people who approve. You can check that 60 of the 100 circles are white, so in this population the proportion who approve of gambling is p = 60/100=0.6

a) The circles are labeled 00, 01,…, 99. Use line 101 of Table B to draw an SRS of size 5. What is the proportion (p-hat) of the people in your sample who approve of gambling?

Line 101 shows the following random digits:

19223, 95034, 05756, 28713, 96409, 12531, 42544, 82853.

b) Take 9 more SRSs of size 5 (10 in all), using lines 102 to 110 (below) of Table B, a different line for each sample. You now have 10 values of the sample proportion (p-hat). What are they?

102: 73676, 47150, 99400, 01927, 27754, 42648, 82425, 36290

103: 45467, 71709, 77558, 00095, 32863, 29485, 82226, 90056

104: 52711, 38889, 93074, 60227, 40011, 85848, 48767, 52573

105: 95592, 94007, 69971, 91481, 60779, 53791, 17297, 59335

106: 68417, 35013, 15529, 72765, 85089, 57067, 50211, 47487

107: 82739, 57890, 20807, 47511, 81676, 55300, 94383, 14893

108: 60940, 72024, 17868, 24943, 61790, 90656, 87964, 18883

109: 36009, 19365, 15412, 39638, 85453, 46816, 83485, 41979

110: 38448, 48789, 18338, 24697, 39364, 42006, 76688, 08708

c) Because your samples have only 5 people, the only values (p-hat) can take are 0/5, 1/5, 2/5, 3/5, 4/5, and 5/5. That is, (p-hat) is always 0, 0.2, 0.4, 0.6, 0.8, 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. (You have begun to construct the sampling distribution of (p-hat), although 10 samples is a small start)

d) How many of your 10 samples estimated the population proportion p = 0.6 exactly correctly? Is the true value 0.6 roughly in the center of your sample values?