Answer:
The 90% confidence interval for the mean production rate fro the new method is (75.9, 84.1).
Explanation:
We have to calculate a 90% confidence interval for the mean.
The population standard deviation is not known, so we have to estimate it from the sample standard deviation and use a t-students distribution to calculate the critical value.
The sample mean is M=80.
The sample size is N=18.
When σ is not known, s divided by the square root of N is used as an estimate of σM:

The degrees of freedom for this sample size are:

The t-value for a 90% confidence interval and 17 degrees of freedom is t=1.74.
The margin of error (MOE) can be calculated as:

Then, the lower and upper bounds of the confidence interval are:

The 90% confidence interval for the mean production rate fro the new method is (75.9, 84.1).