Question

In a survey of sleeping habits, 8400 adults were selected randomly and contacted by telephone. Based on a medical association's recommended amount of sleep for adults, respondents were asked, "Do you typically sleep more than 6 hours during the night?" Of those surveyed, only 46% reported that they did. Suppose the researchers that conducted the survey described above decide to conduct it again but will now interview only 5000 people. In the first survey of 8400 people, the observed proportion was 46%. Which of the following is TRUE about the sample proportion for the second survey?

a. By the law of large numbers, it would again be 46%.
b. By the law of large numbers, the smaller (second) survey will certainly produce a sample proportion farther from the true population proportion than the larger (first) survey.
c. The proportion computed from the sample of 5000 people would be more accurate because smaller samples tend to be more homogeneous than larger samples.
d. None of the above.

Answer 1

d. None of the above.

Explanation:

a. By the law of large numbers, it would again be 46%.

FALSE. This proportion (46%) is a sample statistic, that can or can not be repeated in another sample.

b. By the law of large numbers, the smaller (second) survey will certainly produce a sample proportion farther from the true population proportion than the larger (first) survey.

FALSE. Smaller samples will produce wider confidence intervals for the estimation of the population proportion, but larger samples does not necessarily gives us better point estimations of the true proportion. A small sample can be closer to the true proportion than a large sample, although is less probable.

c. The proportion computed from the sample of 5000 people would be more accurate because smaller samples tend to be more homogeneous than larger samples.

FALSE. There is no evidence to claim that smaller samples are more homogeneous.

d. None of the above. TRUE