Question

The average natural gas bill for a random sample of 21 homes in the 19808 zip code during the month of February was $311.90 with a sample standard deviation of $51.60. Based on the sample, construct a 90% confidence interval for the true mean natural gas bill for homes in the 19808 zip code area in February.

Answer 1

The 90% confidence interval for the true mean natural gas bill for homes in the 19808 zip code area in February is between $292.48 and $331.32

Explanation:

We have the standard deviation for the sample, which means that the t-distribution is used to solve this question.

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 = 21 - 1 = 20

90% confidence interval

Now, we have to find a value of T, which is found looking at the t table, with 20 degrees of freedom(y-axis) and a confidence level of
`1 - (1 - 0.9)/(2) = 0.95`. So we have T = 1.7247

The margin of error is:

`M = T(s)/(√(n)) = 1.7247(51.60)/(√(21)) = 19.42`

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 311.9 - 19.42 = $292.48

The upper end of the interval is the sample mean added to M. So it is 311.9 + 19.42 = $331.32

The 90% confidence interval for the true mean natural gas bill for homes in the 19808 zip code area in February is between $292.48 and $331.32