Chapter 13, Problem 13.2.1RE

For Exercises 1 through 13, follow this procedure:

a. State the hypotheses and identify the claim.

b. Find the critical value(s).

c. Compute the test value.

d. Make the decision.

e. Summarize the results.

Use the traditional method of hypothesis testing unless otherwise specified.

1. Price of Pizza A marketing student hypothesized that the median price for a 12-inch pepperoni pizza is $9.00. A random selection of 30 restaurants within a 30-mile radius indicated the following prices for a 12-inch pepperoni pizza. Test the claim that the median price is not over 9.00 at α = 0.05.

To determine

To identify: The claim and state $H_0$ and $H_1$

Answer to Problem 13.2.1RE

The claim is “the median price for a 12-inch pepperoni pizza is $9.00”.

Null hypothesis: $H_0: \text{Median}=9$ (claim)

Alternative hypothesis: $H_1:\text{Median}\ne 9$

Explanation of Solution

Given info:

The data shows the prices for a 12-inch pepperoni pizza.

Justification:

Here, the claim is that the median price for a 12-inch pepperoni pizza is $9.00. This can be written as $\text{Median}=9$. The complement of the claim is $\text{Median}\ne 9$. In the given experiment, the null hypothesis indicates the claim.

To determine

To find: The critical value.

Answer to Problem 13.2.1RE

The critical value is ±1.96.

Explanation of Solution

Calculation:

Condition:

• If $n\le 25,$ the test statistic for a sign test is $x$. Here, x is the smaller number of + or – signs.
• If $n>25$, the test statistic for a sign test is $z=\frac{\left(x+0.5\right)-0.5n}{\frac{\sqrt{n}}{2}}$. Here, x is the smaller number of + or – signs and $n$ is the total number of + and – signs.

Software Procedure:

Step-by-step procedure to obtain the test value using the MINITAB software:

• Choose MegaStat > Nonparametric Tests > Sign Test.
• In Input ranges, select the column of Prices.
• Enter the Hypothesized value as 9.
• Choose not equal in alternative.
• Click OK.

Output using the MINITAB software is given below:

Determine sample size n:

Number of $(-)$ sign = 9

Number of $(+)$ sign = 21

The sample size is,

$\begin{array}{c}n=\text{sum of }(+)\text{ and }(-)\text{ signs}\\ =21+9\\ =30\end{array}$

Here, the sample size is greater than 25. Therefore, the z test is more appropriate.

Critical value:

From the Standard Normal Table, the critical value at α = 0.05 is ±1.96.

To determine

To find: the test statistic.

Answer to Problem 13.2.1RE

The test statistic is –2.01.

Explanation of Solution

The test statistic for sign test is,

$\begin{array}{c}z=\frac{\left(9+0.5\right)-0.5\left(30\right)}{\frac{\sqrt{30}}{2}}\\ =\frac{9.5-15}{2.74}\\ =\frac{-5.5}{2.74}\\ =-2.01\end{array}$

Thus, the test statistic is –2.01.

To determine

To decide: Whether to reject or fail to reject the null hypothesis at a level of significance

$\alpha =0.05$.

Answer to Problem 13.2.1RE

The decision is “reject the null hypothesis”.

Explanation of Solution

Decision rule:

• If $z>z_0\left(=+1.96\right)$, then reject the null hypothesis
• If $z<-z_0\left(=-1.96\right)$, then reject the null hypothesis

Conclusion:

Here, the test statistic is less than the critical value. That is, $z\left(=-2.01\right)<z_0\left(=-1.96\right)$

Thus, the decision is “reject the null hypothesis”.

To determine

To interpret: The decision in the context of the original claim.

Answer to Problem 13.2.1RE

The conclusion is that there is enough evidence to reject the claim that the median price for a 12-inch pepperoni pizza is $9.00.

Explanation of Solution

From part (d), the null hypothesis is rejected. Thus, there is enough evidence to reject the claim that the median price for a 12-inch pepperoni pizza is $9.00 at 5% level of significance.