Question

A problem with a telephone line that prevents a customer from receiving or making calls is upsetting to both the customer and the telephone company. Samples of 20 problems reported to two different offices of a telephone company and the time to clear these problems​ (in minutes) from the​ customers' lines are provided. Complete​(a) through​ (d) below.

a. Assuming that the population variances from both offices are​ equal, is there evidence of a difference in the mean waiting times between the two​ offices? (Use alphaαequals=​0.05.) Let mu 1μ1 be the mean waiting time of the first office and mu 2μ2 be the mean waiting time of the second office. Determine the hypotheses. Choose the correct answer below. Determine the test statistic and critical value.

b. Find the p-value and interpret its meaning

c. What other assumption is necessary in a?

d.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.

First office and time data

1.60
1.59
0.89
2.55
0.56
1.71
4.07
3.92
1.47
3.17
1.16
0.49
0.96
1.86
0.99
1.07
6.38
3.91
5.58
0.96

Second office and time data

7.52
3.98
0.24
1.03
0.51
0.48
3.17
2.14
0.59
4.06
3.54
0.74
1.99
0.64
1.55
4.20
0.12
1.66
1.65
0.63

Answer 1

In this problem, we are comparing the mean waiting times between two offices of a telephone company. We need to test if there is evidence of a difference in the means. We can set up the hypotheses, calculate the test statistic and critical value, find the p-value, determine the necessary assumption, and construct and interpret a confidence interval.

Step-by-step explanation:

a. Hypotheses:
Null Hypothesis (H0): μ₁ = μ₂ (there is no difference in the mean waiting times between the two offices)
Alternative Hypothesis (H1): μ₁ ≠ μ₂ (there is a difference in the mean waiting times between the two offices)

Test Statistic and Critical Value:
Since the sample sizes are small and the population variances are assumed to be equal, we can use the t-test for independent samples. The test statistic is given by: t = (x₁ - x₂) / √[(s₁²/n₁) + (s₂²/n₂)], where x₁ and x₂ are the sample means, s₁ and s₂ are the sample standard deviations, and n₁ and n₂ are the sample sizes. The critical value can be determined using a t-distribution table or a t-distribution calculator.

b. P-value and Interpretation:
The p-value is the probability of obtaining a test statistic as extreme as the observed value, assuming the null hypothesis is true. By comparing the p-value to the significance level (α), we can determine if there is sufficient evidence to reject the null hypothesis. If the p-value is less than α, we reject H0. If the p-value is greater than or equal to α, we fail to reject H0. The interpretation of the p-value depends on the chosen significance level.

c. Assumption:
In addition to assuming equal population variances, another assumption necessary in this test is that the two samples are independent of each other. This means that the observations in one sample are not influenced by or related to the observations in the other sample.

d. Confidence Interval:
To construct a confidence interval estimate of the difference between the population means, we can use the formula: (x₁ - x₂) ± t*(s), where (x₁ - x₂) is the sample mean difference, t* is the critical value from the t-distribution corresponding to the chosen confidence level, and s is the standard error of the difference, calculated as √[(s₁²/n₁) + (s₂²/n₂)]. The confidence interval provides a range of plausible values for the true difference between the population means, with the chosen confidence level representing the likelihood of the interval containing the true difference.