Question

The manager at the Lawrence National Bank is interested in determining whether there is a difference in the mean time that customers spend completing their transactions depending on which of four tellers the use. To conduct the test, the manager has selected simple random samples of 15 customers for each of the tellers and has timed them (in seconds) from the moment they start their transaction to the time the transaction is completed and they leave the teller station. The manager then asked one of her assistants to perform 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

AnovaTable
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

a. State the appropriate null and alternative hypotheses.

b. Fill in the missing parts of the ANOVA table and perform the statistical hypothesis test using α = 0.05

c. Based on the result of the test in part c, if warranted, use the Turkey-Kramer method with α = 0.05 to determine which teller requires the most time on average to complete a 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	591.1
q-statistic value(α,k,N-k)	3.7448

critical value = q*√(MSE/2*(1/ni+1/nj)) = 23.51
if absolute difference of means > critical value,means are significnantly different ,otherwise not                      
                      

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 a customer’s transaction.