2.)A sample of 65 observations is selected from one population with a population standard deviation of 0.75. The sample mean is 2.67. A sample of 50 observations is selected from a second population with a population standard deviation of 0.66. The sample mean is 2.59. Conduct the following test of hypothesis using the .08 significance level.

Here the null hypothesis and alternative hypothesis are

H0 : µ1 is less than or equal to µ2
Ha : µ1 is greater than µ2

1.Is this a one-tailed or a two-tailed test?

This is a one tailed test, because we are looking at testing whether µ1 > µ2, and not µ1µ2

2.State the decision rule.

Reject the null hypothesis if the p-value is less than the level of significance, 0.08

3.Compute the value of the test statistic.

The test statistic for testing Ho is

follows Standard Normal distribution.

Here it is given that,

= 2.67, = 2.59, =65, = 50, = 0.75, =0.66

Thus,

= 0.6071

5.What is the p-value?

The p-value of the test is given by,

p-value = P[ Z > 0.6071] = 0.2719

4.What is your decision regarding H0?

Since the p-value is much greater than 0.08, we do not reject the null hypothesis H0.

4.)As part of a study of corporate employees, the director of human resources for PNC, Inc., wants to compare the distance traveled to work by employees at its office in downtown Cincinnati with the distance for those in downtown Pittsburgh. A sample of 35 Cincinnati employees showed they travel a mean of 370 miles per month. A sample of 40 Pittsburgh employees showed they travel a mean of 380 miles per month. The population standard deviation for the Cincinnati and Pittsburgh employees are 30 and 26 miles, respectively. At the .05 significance level, is there a difference in the mean number of miles traveled per month between Cincinnati and Pittsburgh employees?

Let µ1 and µ2 be the mean distance traveled by Cincinnati employees and Pittsburgh employees respectively.

  • Step 1:Formulation of the null and the alternative hypotheses

Here we want to test the null hypothesis H0 : µ1= µ2 against the alternative hypothesis
Ha : µ1≠ µ2

  • Step 2: Specification of the level of significance

Here the level of significance is 0.05

  • Step3:Calculation of the test statistic

The test statistic for testing Ho is

follows Standard Normal distribution.

Here it is given that,

= 370, = 380, = 35, = 40, = 30, = 26

Thus,

= -1.5319

  • Step 4:Determination of the p-value

The p-value of the test is given by,

p-value = P[|Z| > 1.5319] = 0.1256

  • Step 5:Selection of the appropriate hypothesis

Since the p-value is greater than the level of significance, 0.05, we fail to reject the null hypothesis Ho. So at the .05 significance level, there is no difference in the mean number of miles traveled per month between Cincinnati and Pittsburgh employees.