Question

use the information to construct 90% and 99% confidence intervals for the population mean. interpret the results and compare the widths of the confidence intervals. researchers surveyed a random sample of 32 u.s. workers ages 16 years and over who did not work from home and asked how long (in minutes) it takes them to get from home to work. the responses are listed. 33 49 36 44 27 28 23 40 16 47 17 36 28 39 35 16 76 13 5 52 28 50 32 69 43 27 18 42 35 27 65 86 from past studies, the researchers assume that σ is 18.4 minutes. interpret the results. choose the correct answer below. a. you can be certain that the daily commute time was within the 90% confidence interval for approximately 29 of the 32 workers, and was within the 99% confidence interval for approximately 30 of the 32 workers. b. you can be 90% confident that the population mean commute time is outside the bounds of the 90% confidence interval, and 99% confident for the 99% interval. c. you can be certain that the population mean commute time is either between the lower bounds of the 90% and 99% confidence intervals or the upper bounds of the 90% and 99% confidence intervals. d. you can be 90% confident that the population mean commute time is between the bounds of the 90% confidence interval, and 99% confident for the 99% interval.

Explanation:

• Confidence intervals give a range of values within which the population mean is likely to lie.
• A 90% confidence interval means that if we were to take many samples and construct confidence intervals, about 90% of them would contain the true population mean. Similarly, for a 99% confidence interval, about 99% of such intervals would contain the true population mean.
• Option A is incorrect because confidence intervals are about the population mean, not about individual data points (workers in this case).
• Option B is incorrect as confidence intervals are for the population mean being inside the interval, not outside.
• Option C is incorrect as it misrepresents the concept of confidence intervals. Confidence intervals are single - range estimates for the population mean, not a choice between two ranges.
• Option D correctly interprets the meaning of 90% and 99% confidence intervals.

Answer:

D. You can be 90% confident that the population mean commute time is between the bounds of the 90% confidence interval, and 99% confident for the 99% interval.