The Utah State Tax Commission attempts to set up payroll tax–withholding tables such that by the end of the year, an employee’s income tax withholding is about $100 below his actual income tax owed to the state. The commission director claims that when all the Utah tax returns are in, the average additional payment will be less than $100.

A random sample of 50 accounts revealed an average additional payment of $114 with a sample standard deviation of $50.

a. Testing at a significance level of 0.10, do the sample data refute the director’s claim?

b. Determine the largest sample mean (with the same sample size and standard deviation) that would fail to refute the director’s claim.