Question

The commute times for workers in a city are normally distributed with an unknown population mean and standard deviation. If a random sample of 20 workers is taken and results in a sample mean of 21 minutes and sample standard deviation of 6 minutes, find a 95% confidence interval estimate for the population mean using the Student's t-distribution.

Answer 1

The 95% confidence interval estimate for the population mean using the Student's t-distribution is between 10.626 minutes and 31.374 minutes.

Explanation:

The first step to solve this problem is finding how many degrees of freedom, we have. This is the sample size subtracted by 1. So

df = 20 - 1 = 19

Now, we have to find a value of T, which is found looking at the t table, with 19 degrees of freedom(y-axis) and a confidence level of 0.99(
`t_(95)`). So we have T = 1.729

The margin of error is:

M = T*s = 1.729*6 = 10,374.

In which s is the standard deviation of teh sample. So

The lower end of the interval is the sample mean subtracted by M. So it is 21 - 10.374 = 10.626 minutes

The upper end of the interval is the sample mean added to M. So it is 21 + 10.374 = 31.374 minutes.

The 95% confidence interval estimate for the population mean using the Student's t-distribution is between 10.626 minutes and 31.374 minutes.