Question

The manager at the Lawrence National Bank is interested in determining whether there is a difference in the mean time that customers send completing their transactions depending on which of the four tellers they use. To conduct the test, the manager has selected simple random samples of 15 customers for each of the tells and has time them (in seconds) from the moment they start the transaction to the time the transaction is complete and they leave the teller station. The manager then asked one of her assistants to reform the appropriate statistical test. The assistant returned with the following partially completed ANOVA table.

Summary

Groups               Count                               Sum                    Average                          Variance

Teller 1               15                                     3,043.9                                          827.4

Teller 2               15                                     3,615.5                                          472.2

Teller 3               15                                     3,427.7                                          445.6

Teller 4               15                                     4,072.4                                          619.4   

ANOVA

Source of Variation        SS          df                       MS                           F-ratio                p-value              F-crit

Between Groups             36,530.6                                                                                 0.0000               2.7694

Within Groups

Total                                 69,633.7 59

1. State the appropriate null and alternative hypotheses.
2. Fill in the missing parts of the ANOVA table and perform the statistical hypothesis test using α=0.05
3. Based on the result of the test in part c, if warranted, use the Tukey-Kramer method with α=0.05 to determine which teller requires the most time on average to complete the customer’s transaction.

Answer 1

a)

Ho: µ1=µ2=µ3=µ4
H1: not all means are equal

b)

AnovaTable
Source of variation	SS	df	MS	F-ratio	P-value	F-crit
Between Groups	36,530.60	3	12176.8667	20.5994162	0	2.7694
Within Groups	33,103.10	56	591.126786
Total	69,633.70	59

c)

Level of significance	0.05
no. of treatments,k	4
DF error =N-k=	56
MSE	591.12679
q-statistic value(α,k,N-k)	3.7448

if absolute difference of means > critical value,means are significnantly different ,otherwise not

critical value = q*√(MSE/2*(1/ni+1/nj)) = 23.51

                      

population mean difference		critical value			result
µ1-µ2	38.11	23.51			means are different
µ1-µ3	25.59	23.51			means are different
µ1-µ4	68.57	23.51			means are different
µ2-µ3	12.52	23.51			means are not different
µ2-µ4	30.46	23.51			means are different
µ3-µ4	42.98	23.51			means are different

.Teller 4  requires the most time on average to complete the customer’s transaction