Question

You are the manager of a restaurant that delivers pizza to college dormitory rooms. You have just changed your delivery process in an effort to reduce the mean time between the order and completion of delivery from the current 25 minutes. A sample of 36 orders using the new delivery process yields a sample mean of 22.4 minutes and a sample standard deviation of 6 minutes. Perform a hypothesis test to determine if there’s evidence that the population mean delivery time has been reduced below the previous population mean value of 25 minutes by answering the following questions:

(a) What are the null and alternate hypotheses for this test?

(b) What is the value of the test statistic for this test?

(c) Using the critical value approach, at the 0.05 level of significance, what is the decision rule?

(d) What is your conclusion in context of the problem? (Answer this question in a complete sentence(s) and include why, referring to the decision rule.)

(e) Using the p-value approach, at the 0.05 level of significance, what is the decision rule?

(f) What is your conclusion in context of the problem? (Answer this question in a complete sentence(s) and include why, referring to the decision rule.)

(g) Compare your conclusions in (a) and (b). Are they the same? Different? Should this always be the case?

Answer 1

(a)

H0: Null Hypothesis: $\mu \geq$ 25 ( the population mean delivery time has not been reduced below the previous population mean value of 25 minutes )

HA:Alternative Hypothesis: $\mu <$ 25 ( the population mean delivery time has been reduced below the previous population mean value of 25 minutes ) (Claim)

(b)

Test Statistic is given by:

T = - s/n 22.4 - 25 6//36

(c)

$\alpha$ = 0.05

df = 36 - 1 = 35

From Table, critical value of t = - 1.69

Decision Rule:
Reject H0 if t < - 1.69

(d)

Since calculated value of t = - 260 isless than critical value of t =- 1.69, the test statistic is in the critical region.

Reject null hypothesis.

(e)

Using the p-value approach, at the 0.05 level of significance, the decision rule is:

Reject H0 if p - value < $\alpha$ .

(f)

By Technology, p - value = 0.0068.

Since p - value =0.0068 is less than $\alpha$ = 0.05, the test statistic is in the critical region.

Reject null hypothesis.

(g)

The conclusions in (a) and (b) are the same. This should always be the same.