The bank manager wants to show that the new system reduces typical customer waiting times to less than 6 minutes. One wa

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The bank manager wants to show that the new system reducestypical customer waiting times to less than 6 minutes. One way todo this is to demonstrate that the mean of the population of allcustomer waiting times is less than 6. Letting this meanbe µ, in this exercise we wish to investigate whetherthe sample of 106 waiting times provides evidence to support theclaim that µ is less than 6.
For the sake of argument, wewill begin by assuming that µ equals 6, and wewill then attempt to use the sample to contradict this assumptionin favor of the conclusion that µ is less than6. Recall that the mean of the sample of 106 waiting timesis
= 5.37 and assume that σ, the standard deviation ofthe population of all customer waiting times, is known to be2.25.
(a) Consider the population of allpossible sample means obtained from random samples of 106 waitingtimes. What is the shape of this population of sample means? Thatis, what is the shape of the sampling distribution of
?
(b) Find the mean and standard deviationof the population of all possible sample means when we assumethat µ equals 6. (Round your answerto 4 decimal places.)
(c) The sample mean that we have actuallyobserved is
= 5.37. Assuming that µ equals 6,find the probability of observing a sample mean that is less thanor equal to
= 5.37. (Round "z-value" to 2 decimalsand final answer to 4 decimal places.)
(d) If µ equals 6, whatpercentage of all possible sample means are less than or equal to5.37? What do you conclude about whether the new system has reducedthe typical customer waiting time to less than 6minutes? (Round your answer to 2 decimalplaces.)