Question

in a random sample of 25 people, the mean commute time to work was 32.6 minutes and the standard deviation was 7.1 minutes. assume the population is normally distributed and use a t - distribution to construct a 95% confidence interval for the population mean μ. what is the margin of error of μ? interpret the results.
the margin of error of μ is 2.9.
(round to one decimal place as needed.)
interpret the results.
a. with 95% confidence, it can be said that the commute time is between the bounds of the confidence interval.
b. it can be said that 95% of people have a commute time between the bounds of the confidence interval.
c. if a large sample of people are taken approximately 95% of them will have commute times between the bounds of the confidence interval.
d. with 95% confidence, it can be said that the population mean commute time is between the bounds of the confidence interval.

Explanation:

A confidence interval gives an estimated range of values which is likely to include an unknown population parameter. A 95% confidence interval means that if we were to take many samples and calculate a confidence interval for each sample, approximately 95% of those intervals would contain the true population parameter. Here, the parameter of interest is the population mean \(\mu\) (the mean commute time for all people, not just the sample).

• Option A is incorrect because it refers to the commute time (a single value for an individual) instead of the population mean.
• Option B is wrong as it implies 95% of people (individuals) have commute times in the interval, but confidence intervals are about the population mean, not individual data points.
• Option C is incorrect for similar reasons as B; it's not about the proportion of people in a large sample, but about the population mean.
• Option D is correct because with 95% confidence, we are making a statement about the population mean \(\mu\) (that it lies within the bounds of the confidence interval).

Answer:

D. With 95% confidence, it can be said that the population mean commute time is between the bounds of the confidence interval.