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 37 workers is taken and results in a sample mean of 31 minutes and sample standard deviation of 5 minutes, find a 95% confidence interval estimate for the population mean using the student's t-distribution.

Answer 1

The 95% confidence interval for the population mean commute time is between 29.33 minutes and 32.67 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 = 37 - 1 = 36

95% confidence interval

Now, we have to find a value of T, which is found looking at the t table, with 36 degrees of freedom(y-axis) and a confidence level of
`1 - (1 - 0.95)/(2) = 0.975`. So we have T = 2.0262

The margin of error is:

`M = T(s)/(√(n)) = 2.0262(5)/(√(37)) = 1.67`

In which s is the standard deviation of the sample and n is the size of the sample.

The lower end of the interval is the sample mean subtracted by M. So it is 31 - 1.67 = 29.33 minutes

The upper end of the interval is the sample mean added to M. So it is 31 + 1.67 = 32.67 minutes.

The 95% confidence interval for the population mean commute time is between 29.33 minutes and 32.67 minutes.