Question

A bank with a branch located in a commercial district of a city has the business objective of improving the process for serving customers during the noon-to-1 pm lunch period. To do so, the waiting time (defined as the number of minutes that elapses from when the customer enters the line until he or she reaches the teller window) needs to be shortened to increase customer satisfaction. A random sample of 15 customers is selected and the waiting times are recorded. Suppose that another branch, located in a residential area, is also concerned with the noon-to-1 pm lunch period. A random sample of 15 customers is selected and the waiting times are recorded.

1. Is there evidence of a difference in the variability of the waiting times between the two branches? (Use ? = 0.05.)

2. Determine the p-value for the test in part (1) above.

3. What assumptions about the distribution of waiting times for each branch is necessary for the test you conducted in part (1) to be valid?

4. Conduct an appropriate test to compare the mean waiting times of the two branches. Report the p-value of this test and state your conclusions.

Customer	Waiting Time	Branch
1	4.21	Residential
2	4.50	Residential
3	5.55	Residential
4	6.10	Residential
5	3.02	Residential
6	0.38	Residential
7	5.13	Residential
8	5.12	Residential
9	4.77	Residential
10	6.46	Residential
11	2.34	Residential
12	6.19	Residential
13	3.54	Residential
14	3.79	Residential
15	3.20	Residential
16	9.66	Commercial
17	10.49	Commercial
18	5.90	Commercial
19	6.68	Commercial
20	8.02	Commercial
21	5.64	Commercial
22	5.79	Commercial
23	4.08	Commercial
24	8.73	Commercial
25	6.17	Commercial
26	3.82	Commercial
27	9.91	Commercial
28	8.01	Commercial
29	5.47	Commercial
30	8.35	Commercial

Answer 1

1. The hypothesis being tested is:

Null hypothesis	H₀: σ₁² / σ₂² = 1
Alternative hypothesis	H₁: σ₁² / σ₂² ≠ 1
Significance level	α = 0.05

Method	Test Statistic	DF1	DF2	P-Value
F	0.62	14	14	0.380

2. The p-value is 0.380.

Since the p-value (0.380) is greater than the significance level (0.05), we fail to reject the null hypothesis.

Therefore, we can conclude that there is a difference in the variability of the waiting times between the two branches.

3. When running a two-sample equal-variance t-test, the basic assumptions are that the distributions of the two populations are normal, and that the variances of the two distributions are the same.

4. The hypothesis being tested is:

H₀: µ₁ = µ₂

H₁: µ₁ ≠ µ₂

Residential	Commercial
4.2867	7.1147	mean
1.6380	2.0822	std. dev.
15	15	n
	28	df
	-2.82800	difference (Residential - Commercial)
	3.50925	pooled variance
	1.87330	pooled std. dev.
	0.68403	standard error of difference
	0	hypothesized difference
	-4.134	t
	.0003	p-value (two-tailed)

The p-value is 0.0003.

Since the p-value (0.0003) is less than the significance level (0.05), we can reject the null hypothesis.

Therefore, we can conclude that there is a difference in the mean waiting times of the two branches.