Question

For each of the following two-samples t-tests (problems 1-6): (a) Determine if a F test for the ratio of two variances is appropriate to calculate for the context. If it is appropriate, conduct the analysis and report the result. Include what statistical conclusion you should draw from the analysis (i.e., whether you should conduct a pooled-variance t-test or an unequal-variances t-test). (b) Identify the most appropriate t-test to conduct for the situation/data given. Don’t forget to consider if the context requires one/two-tail tests. (c) Provide a statistical and practical interpretation of your findings.

1. A bank with a branch located in a commercial district of a city has the business objective of developing an improved process for serving customers during the 12-1pm lunch period. Management decided to first study the waiting time in the current process. The waiting time is defined as the number of minutes that elapses from when the customer enters the line until he or she reaches the teller window. Data are collected from a random sample of 15 customers and stored in Bank1. Suppose that another branch, located in a residential area, is also concerned with improving the process of serving customers in the 12-1pm lunch period. Data are collect from a random sample of 15 customers and stored in Bank2. Can we conclude the wait times are different at the two branches? (Use a 0.05 level of significance)

Bank 1

Waiting Time
7.48
3.86
7.18
6.72
7.18
6.15
8.35
9.46
7.26
5.41
7.86
4.72
9.13
5.46
8.71

Bank 2

Waiting Time
4.56
4.84
6.61
9.95
8.28
5.21
4.5
3.91
5.63
6.77
4.84
3.44
5.19
3.88
3.67

Answer 1

using minitab>stat>basic stat>two variance

we have

Test and CI for Two Variances: Bank 1, bank 2

Method

Null hypothesis σ(Bank 1) / σ(bank 2) = 1
Alternative hypothesis σ(Bank 1) / σ(bank 2) ≠ 1
Significance level α = 0.05

Statistics

95% CI for
Variable N StDev Variance StDevs
Bank 1 15 1.630 2.657 (1.208, 2.530)
bank 2 15 1.812 3.284 (1.051, 3.596)

Ratio of standard deviations = 0.899
Ratio of variances = 0.809

95% Confidence Intervals

CI for
CI for StDev Variance
Method Ratio Ratio
Bonett (0.504, 2.376) (0.255, 5.644)
Levene (0.484, 2.561) (0.234, 6.558)

Tests

Test
Method DF1 DF2 Statistic P-Value
Bonett 1 — 0.10 0.752
Levene 1 28 0.00 0.989

since p value of levenes F stat is less than so we reject Ho and conclude that population variance are not equal

so we will use unequal variance t test

using minitab>stat>basic stat>two sample t

we have

Two-Sample T-Test and CI: Bank 1, bank 2

Two-sample T for Bank 1 vs bank 2

N Mean StDev SE Mean
Bank 1 15 7.00 1.63 0.42
bank 2 15 5.42 1.81 0.47

Difference = μ (Bank 1) - μ (bank 2)
Estimate for difference: 1.577
95% CI for difference: (0.285, 2.868)
T-Test of difference = 0 (vs ≠): T-Value = 2.51 P-Value = 0.019 DF = 27

(b)the most appropriate t test is independent sample t test with unequal variance and test is two tailed

(c) the value of test statistic t is and p value is 0.019 since p value of t test is less than 0.05 so we reject Ho and conclude that the wait times are different at the two branches.