The bank manager wants to show that the new system reduces typical customer wait

Question

The bank manager wants to show that the new system reduces typical customer waiting times to less than 6 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 U, in this exercise we wish to investigate whether the sample of 95 waiting times provides evidence to support the claim that mu is less than 6. For the sake of argument, we will begin by assuming that mu equals 6, and we will then attempt to use the sample to contradict this assumption in favor of the conclusion that mu is less than 6. Recall that the mean of the sample of 95 waiting times is bar X = 5.47 and assume that sigma, the standard deviation of the population of all customer waiting times, is known to be 2.23. (a) Consider the population of all possible sample means obtained from random samples of 95 waiting times. What is the shape of this population of sample means? That is, what is the shape of the sampling distribution of bar X? Normal because the sample is (Click to select). (b) Find the mean and standard deviation of the population of all possible sample means when we assume that mu equals 6. mu_x = 6. sigma_x = (c) The sample mean that we have actually observed is bar X = 5.47. Assuming that mu equals 6, find the probability of observing a sample mean that is less than or equal to bar x = 5.47. P(bar x lessthanorequalto 5.47) (d) If mu equals 6, what percentage of all possible sample means are less than or equal to 5.47? What do you conclude about whether the new system has reduced the typical customer waiting time to less than 6 minutes? %7 Conclude that mu is than 6.

Explanation / Answer

Part-a

As random sample size n=95>30, so the sampling distribution of sample means xbar is normal

Part-b

Sigma_xbar=sigma/sqrt(n)=2.23/sqrt(95)=0.2288

Part-c

P(xbar<=5.47)=0.0103 using excel function =NORMDIST(5.47,6,0.2288,TRUE)

Part-d

Only 100*0.0103=1.03% of all possible sample means are less than or equal to 5.47

We conclude that mu is less than 6

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