Question

In Exercises 13–16, refer to the exercise identified and find the value of the test statistic. (Refer to Table 8-2 on page 362 to select the correct expression for evaluating the test statistic.)

Exercise 6 “Cell Phone”

Short answer

The value of the test statistic (z-score) is equal to -12.33.

Step-by-step solution

Given information

Out of 1128 adults, 87% said they have a cell phone.

Hypotheses

It is claimed that less than 95% of adults have a cell phone.

Corresponding to the given claim, the following hypotheses are set up:

Null hypothesis: The proportion of adults who have a cell phone is equal to 0.95.

$H_0:p=0.95$

Alternative hypothesis: The proportion of adults who have a cell phone is less than 0.95.

$H_1:p<0.95$

Test statistic

Since the claim involves testing the equality of the sample proportion with a hypothesized value, the test statistic used will be the z-score.

The value of the sample proportion is computed below:

$$\begin{gathered} \hat{p}=87\% \\ =\frac{87}{100} \\ =0.87 \end{gathered}$$

The given value of the proportion ofadults who have cell phones is supposed to be equal to 0.95.

Thus, p=0.95.

$$\begin{gathered} q=1-p \\ =1-0.95 \\ =0.05 \end{gathered}$$

The value of the test statistic is computed below:

$$\begin{gathered} z=\frac{\hat{p}-p}{\sqrt{\frac{pq}{n}}} \\ =\frac{0.87-0.95}{\sqrt{\frac{0.95\times 0.05}{1128}}} \\ =-12.33 \end{gathered}$$

Thus, the test statistic is equal to -12.33.