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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 a= 0.05.) 

The following is a sample of 20 problems reported to the first office minutes) to clear those problems.

1.49 1.84 0.78 2.78 0.53 1.56 4.02 3.94 1.66 3.04 1.21 0.33 0.94 1.82 0.92 1.23 6.25 3.83 5.54 0.87
 

The following is a sample of 20 problems reported to the second time (in minutes) to clear those problems.

7.59 3.95 0.11 1.22 0.28 0.64 3.22 2.24 0.56 4.01 3.79 0.74 1.80 0.63 1.63 4.07 0.22 1.64 1.32 0.90

d. What is your statistical decision?


The first step in solving this problem is to calculate the sample statistics, 
The sample mean and the sample standard deviations for the two samples. 

For the first sample, the sample mean is 2.229 and the sample standard deviation is 1.690. 

The second sample has a mean of 2.028, and the standard deviation has a standard deviation is 1.889. 

use the pooled standard deviation because variances are assumed to be equal. 

Formula for the pooled variance is

s^2 = (n1-1*s1^2 + n2-2*s2^2)/(n1 + n2 - 2)

s2^2 = (19 * 1.690^2  + 19*1.889^2)/(20+20-2) =  3.212

Note that the degrees of freedom for this pooled variance t-test is n1 + n2  - 2. 

20 + 20  - 2 = 38. 


t-test = (x_bar1 - x_bar2)/sqrt(s^2 * (1/n1 + 1/n2))

t = (2.229 - 2.028)/sqrt(3.212 * (1/20 + 1/20)) = 0.3547

Note that this is a two-tailed test, so the critical value at the 0.05 level of significance can be calculated. 

Use Excel function to calculate the critical value and the p-value associated with the t-statistic. 

Critical value from Excel, = =T.INV.2T(0.05,38)  = 2.0244

The p -value calculation is Excel, =T.DIST.2T(0.3547,38)  = 0.7248

The process is t reject the null hypothesis if the critical value is greater than the test statistic or if the p value is less than alpha. 

Conclusion: Do not reject the null hypothesis.

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