Recall that the bank manager wants to show that the new system reduces typical customer waiting times to less than six minutes. One way to do this is to demonstrate that the mean of the population of all customer waiting times is less than 6. Letting this mean be μx, in this exercise we wish to investigate whether the sample of 100 waiting times provides evidence to support the claim that μx is less than 6.
For the sake of argument, we will begin by assuming that μx equals 6. and we will then attempt to use the sample to contradict this assumption in favor of the conclusion that μx is less than 6.
Recall that the mean of the sample of 100 waiting times is  = 5.46 and assume that μ, the standard deviation of the population of all customer waiting times, is known to be 2.47.
a. Consider the population of all possible sample means obtained from random samples of 100 waiting times. What is the shape of this population of sample means? That is. what is the shape of the sampling distribution of ? Why is this true?
b. Hind the mean and standard deviation of the population of all possible sample means when we assume that μ equals 6.
c. The sample mean that we have actually observed is = 5.46. Assuming that μ equals 6, find the probability of observing a sample mean that is less than or equal to = 5.46.
d. If μ equals 6, what percentage of all possible sample means are less than or equal to 5.46? Since we have actually observed a sample mean of = 5.46, is it more reasonable to believe that (l) μ equals 6 and we have observed one of the sample means that is less than or equal to 5.46 when μ equals 6, or (2) that we have observed a sample mean less than or equal to 5.46 because μ is less than 6? Explain. What do you conclude about whether the new system has reduced the typical customer waiting time to less than six minutes?