Question

A simple random sample of 100 postal employees is used to test if the average time postal employees have worked for the postal service has changed from the value of 7.5 years recorded 20 years ago. The sample mean was x = 7 years with a standard deviation of s = 2 years. Assume the distribution of the time the employees have worked for the postal service is approximately Normal. The hypotheses being tested are H0: ! = 7.5 versus Ha: ! ≠ 7.5. A one-sample t test will be used.

a. What are the appropriate degrees of freedom for this test?

b. What is the P-value for the one-sample t test? What is your conclusion?

c. What is a 95% confidence interval for !, the population mean time the postal service employees have spent with the postal service? Compare this confidence interval with your conclusion in part (b).

d. Suppose the mean and standard deviation obtained were based on a sample of size n = 25 postal workers rather than 100. What do we know about the value of the P-value?

Answer 1

(a)

Degrees of Freedom = n = 1 = 100 - 1 = 99

(b)

SE = s/

= 2/ = 0.2

Test statistic is:

t = (7 - 7.5)/0.2 = - 2.50

ndf = 99

By Technology, p - value = 0.0141

So, p - value = 0.0141

Since p - value =0.0141 is less than = 0.05, the difference is significant. Reject null hypothesis.

Conclusion:

The average time the postal employees have worked for the postal service has changed from the value of 7.5 years recorded 20 years ago.

(c)

From Table, critical values of t = 1.9842

Confidence Interval:

7 (1.9842 X 0.2)

= 7 0.3968

= ( 6.6032 , 7.3968)

So,

Confidence Interval:

6.6032 < < 7.3968

Since 7.5 is not included in the confidence interval, it confirms the conclusion in part (b) that the average time the postal employees have worked for the postal service has changed from the value of 7.5 years recorded 20 years ago.

(d)

Degrees of Freedom = n = 1 = 25 - 1 = 24

SE = s/

= 2/ = 0.4

Test statistic is:

t = (7 - 7.5)/0.4 = - 1.25

ndf = 24

By Technology, p - value = 0.2234

Since p - value =0.2234 is greater than = 0.05, the difference is not significant. Fail to reject null hypothesis.

Conclusion:

The average time the postal employees have worked for the postal service has not changed from the value of 7.5 years recorded 20 years ago.