7. Sampling Distributions & Confidence Intervals: Mean

True or False: The distribution of the sample mean, x̄, will be approximately normally distributed if the sample is obtained from a population that is not normally distributed, regardless of the sample size.

Upper Leg Length The upper leg length of 20- to 29-year-old males is approximately normal with a mean length of 43.7 cm and a standard deviation of 4.2 cm.

Source: “Anthropometric Reference Data for Children and Adults: U.S. Population, 1999–2002”; Volume 361, July 7, 2005.

d. What effect does increasing the sample size have on the probability? Provide an explanation for this result.

Reading A recent Gallup poll asked Americans to disclose the number of books they read during the previous year. Initial survey results indicate that s = 16.6 books.

d. How many subjects are needed to estimate the number of books Americans read the previous year within four books with 99% confidence? Compare this result to part (a). How does increasing the level of confidence in the estimate affect sample size? Why is this reasonable?

Television A researcher wanted to determine the mean number of hours per week (Sunday through Saturday) the typical person watches television. Results from the Sullivan Statistics Survey I indicate that s = 7.5 hours.

d. How many people are needed to estimate the number of hours people watch television per week within 2 hours with 90% confidence? Compare this result to part (a). How does decreasing the level of confidence in the estimate affect sample size? Why is this reasonable?

A simple random sample of size n = 20 is obtained from a population with μ = 64 and σ = 17.

d. Compare the results obtained in parts (b) and (c) with the results obtained in parts (b) and (c) in Problem 17. What effect does increasing the sample size have on the probabilities? Why do you think this is the case?

Television A researcher wanted to determine the mean number of hours per week (Sunday through Saturday) the typical person watches television. Results from the Sullivan Statistics Survey I indicate that s = 7.5 hours.

b. How many people are needed to estimate the number of hours people watch television per week within 1 hour with 95% confidence?

A bookstore is interested in the average length of books in its adult fiction section. Find the minimum sample size required to create a $99\%$ confidence interval with a margin of error no more than $30$ pages if...

(B) The standard deviation is unknown, but the longest book is $697$ pages and the shortest is $298$ pages.

[NW] Determining Sample Size A physical therapist wants to determine the difference in the proportion of men and women who participate in regular, sustained physical activity. What sample size should be obtained if she wishes the estimate to be within 3 percentage points with 95% confidence, assuming that

a. she uses the 1998 estimates of 21.9% male and 19.7% female from the U.S. National Center for Chronic Disease Prevention and Health Promotion?