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

a)Our hypothesis will be,

Vs.,

To calculate the test statistic, we need to calculate the sample mean and sample sd of both the groups,

Which are, = 2.29 and s₁= 1.70

= 2.022 and s₂= 1.88

Now, the test statistic can be given by,

t = = 0.3923

and df= = 38

Critical t-value from t-table at this df and level of significance is 2.024.

b)P-value = 0.6970

Since the P-value is very high, we fail to reject the null hypothesis. the means differ significantly for both the offices.

Interpretation:- Assuming that both the means are equal, you’d obtain the observed difference or more in 69.7% of data due to random sampling error.

c)We need to assume that both the samples are independent taken and follows approximately the Normal distribution.

d)we need to calculate the pooled standard deviation,

s_p= = 3.2163

and t-value from the t-table is 2.02.

So, CI=

= 2.29-2.022

=0.268 2.055

=[-1.7865, 2.3225]

we are 95% confident that the true diffrence in population mean lies between -1.7865 and 2.3225.