Question

Test Statistics. 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 hypothesised value, the test statistic used will be the z-score.

The value of the sample proportion is computed below:

\(\begin{array}{c}\hat p = 87\% \\ = \frac{{87}}{{100}}\\ = 0.87\end{array}\)

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

Thus, p=0.95.

\(\begin{array}{c}q = 1 - p\\ = 1 - 0.95\\ = 0.05\end{array}\)

The value of the test statistic is computed below:

\(\begin{array}{c}z = \frac{{\hat p - p}}{{\sqrt {\frac{{pq}}{n}} }}\\ = \frac{{0.87 - 0.95}}{{\sqrt {\frac{{0.95 \times 0.05}}{{1128}}} }}\\ = - 12.33\end{array}\)

Thus, the test statistic is equal to -12.33.