3.30 What Percent of the US Population Are Senior Citizens? What Percent of the

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3.30 What Percent of the US Population Are Senior Citizens? What Percent of the US Population Are Senior Citizens? People 65 years and older are the fastest growing segment of the US population, and constituted 13% of the population in 2010.17 Figure 3.9 shows sample proportions from two sampling distributions of the proportion 65 years and older in the US: One shows samples of size 100, and the other shows samples of size 1000.

Figure 3.9 Sampling distributions for n = 100 and n = 1000

What is the center of both distributions?

What is the approximate minimum and maximum of each distribution?

Give a rough estimate of the standard error in each case.

Suppose you take one more sample in each case. Would a sample proportion of 0.17 (that is, 17% senior citizens in the sample) be surprising to see from a sample of size 100? Would it be surprising from a sample of size 1000?

3.40 A Sampling Distribution for Performers in the Rock and Roll Hall of Fame Exercise 3.38 tells us that 206 of the 303 inductees to the Rock and Roll Hall of Fame have been performers. The data are given inRockandRoll. Using all inductees as your population:

Use StatKey or other technology to take many random samples of size n = 10 and compute the sample proportion that are performers. What is the standard error of the sample proportions? What is the value of the sample proportion farthest from the population proportion of p = 0.68? How far away is it?

Repeat part (a) using samples of size n = 20.

Repeat part (b) using samples of size n = 50.

Use your answers to parts (a), (b), and (c) to comment on the effect of increasing the sample size on the accuracy of using a sample proportion to estimate the population proportion.

3.58 Do You Find Solitude Distressing? “For many people, being left alone with their thoughts is a most undesirable activity,” says a psychologist involved in a study examining reactions to solitude.26 In the study, 146 college students were asked to hand over their cell phones and sit alone, thinking, for about 10 minutes. Afterward, 76 of the participants rated the experience as unpleasant. Use this information to estimate the proportion of all college students who would find it unpleasant to sit alone with their thoughts. (This reaction is not limited to college students: in a follow-up study involving adults ages 18 to 77, a similar outcome was reported.)

Give notation for the quantity being estimated, and define any parameters used.

Give notation for the quantity that gives the best estimate, and give its value.

Give a 95% confidence interval for the quantity being estimated, given that the margin of error for the estimate is 8%

Explanation / Answer

3.58)

Notation for the quantity being estimated is p i.e. proportion, p = 76/146 = 0.5205

Margin of error, ME = 0.08

95% CI = (p - ME, p+ME)

= (0.5205 - 0.08, 0.5205 + 0.08)

= (0.4405, 0.6005)

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