Question

A bank manager has developed a new system to reduce the time customers spend waiting to be served by tellers during peak business hours. The bank manager hopes that the new system will lower the average waiting time to less than six minutes. A one-month trial of the system is conducted and 25 customers have been selected and the waiting times were recorded. The sample resulted in a mean of 5.2 minutes and a standard deviation of 2.4 minutes. a- At 5% significance level, can the manager conclude that the new system is efficient in reducing the average waiting time? Show all steps needed. b- What is the required assumption, if any, of the test used in (a)? c- Another satisfaction suggested to use 10% significance level, what would be the conclusion? d- The manager got puzzled by the results obtained in (a) and (b). Explain to the manager why the conclusions were different and give him/her some insights and recommendations regarding the efficiency of the new system.

Answer 1

(a)

Let the average waiting time be denoted by

Here we are to test

The given data is summarized as

Sample size	n=25
Sample mean	=5.2
Sample SD	s=2.4

The test statistic is given by

The test statistic follows t distribution with df 24. The p-value is obtained as 0.054257

As the p-value is more than the p-value, we fail to reject the null hypothesis at 5% level of significance and hence conclude that the new system has significantly lowered the average waiting time to less than six minutes.

(b)

The assumptions are as follows:

• The sample is drawn from a population that follows Normal distribution
• The observations are independent
• The variance must be homogeneous.

(c)

As the p-value is 0.054257, less than 0.10, we reject the null hypothesis at 5% level of significance and hence conclude that the new system has not significantly lowered the average waiting time to less than six minutes.

(d)

As the significance level changes, the test conclusions may also change based on the p-value thus obtained. Here it can be concluded at 5% level of significance that the new system has significantly lowered the average waiting time to less than six minutes, whereas at 10% level of significance level it can be concluded that the new system has not significantly lowered the average waiting time to less than six minutes.

The higher confidence level indicates lower significance level and thus lower significance level should be used, i.e. the test with 5% level of significance should be used.

Hence the manager can conclude that the new system has significantly lowered the average waiting time to less than six minutes at 5% level of significance.

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