Question

A problem with a phone line that prevents a customer from receiving or making calls is upsetting to both the customer and the telecommunications company. The file Phone contains samples of 20 problems reported to two different offices of a telecommunications company and the time to clear these problems (in minutes) from the customers’ lines: Central Office I Time to Clear Problems (minutes) 1.48 1.75 0.78 2.85 0.52 1.60 4.15 3.97 1.48 3.10 1.02 0.53 0.93 1.60 0.80 1.05 6.32 3.93 5.45 0.97 Central Office II Time to Clear Problems (minutes) 7.55 3.75 0.10 1.10 0.60 0.52 3.30 2.10 0.58 4.02 3.75 0.65 1.92 0.60 1.53 4.23 0.08 1.48 1.65 0.72 Perform a hypothesis test to determine if there’s evidence in this data of a difference in the mean waiting time between the two offices by answering the following questions:

(a) What are the null and alternate hypotheses for this test?

(b) Assuming that the population variances from both offices are equal, what is the value of the test statistic?

(c) Using the critical value approach, is there evidence of a difference in the mean waiting time between the two offices? (Use α = 0.05.) (Answer this using complete sentences and be sure to include what decision rule you used to arrive at your conclusion.)

(d) Find the p-value in (c) and interpret its meaning.

(e) What assumption (other than equal variances) is necessary in (c)?

(f) Assuming that the population variances from both offices are equal, construct and interpret a 95% confidence interval estimate of the difference between the population means in the two offices.

Answer 1

a) NULL HYPOTHESIS H0:

ALTERNATIVE HYPOTHESIS Ha:

b) When variances are equal is

Test statistic is

c) Based on the information provided, the significance level is α=0.05, and the degrees of freedom are df=38. In fact, the degrees of freedom are computed as follows, assuming that the population variances are equal:Hence, it is found that the critical value for this two-tailed test is tc​=2.024, for α=0.05 and df=38.The rejection region for this two-tailed test is R={t:∣t∣>2.024}.

Since it is observed that ∣t∣=0.354≤tc​=2.024, it is then concluded that the null hypothesis is not rejected.

d) The p-value is p=0.725, and since p=0.725≥0.05, it is concluded that the null hypothesis is not rejected.

e) Assumptions: samples are independent to each other and population variances are unknown. Populations from samples are taken are normally distributed.

f)

the 95% confidence interval for the difference between the population means μ1​−μ2​ is −0.954<μ1​−μ2​<1.359, which indicates that we are 95% confident that the true difference between population means is contained by the interval (-0.954, 1.359).