A mail-order business prides itself in its ability to fill customers’ orders in six days or less on average.

Periodically, the operations manager selects a random sample of customer orders and determines the number of days required to fill the orders. Based on this sample information, he decides if the desired standard is being met. He will assume that the average number of days to fill customers’ orders is six or less unless the data strongly suggest otherwise.

a. Establish the appropriate null and alternative hypotheses.

b. On one occasion when a sample of 40 customers was selected, the average number of days was 6.65, with a sample standard deviation of 1.5 days. Can the operations manager conclude that his mail-order business is achieving its goal? Use a significance level of 0.025 to answer this question.

c. Calculate the p-value for this test. Conduct the test using this p-value.

d. The operations manager wishes to monitor the efficiency of his mail-order service often. Therefore, he does not wish to repeatedly calculate t-values to conduct the hypothesis tests. Obtain the critical value, xa, so that the manager can simply compare the sample mean to this value to conduct the test.