Q1-13BSC - Elementary Statistics

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 5 “Online Data”

Step-by-Step Solution

Answer

The value of the test statistic (z-score) is equal to 4.28.

Step 1: Given information

Out of 565 randomly selected adults, 59% would erase all of their personal information online if they could.

Step 2: Hypotheses

It is claimed that most adults would erase all of their personal information online if they could.

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

 

Null hypothesis: The proportion of adults who would erase all of their personal information online if they could is equal to 0.5.

\({H_0}:p = 0.5\)

 

Alternative hypothesis: The proportion of adults who would erase all of their personal information online if they could is greater than 0.5.

\({H_1}:p > 0.5\)

Step 3: 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:

 

\(\begin{array}{c}\hat p = 59\% \\ = \frac{{59}}{{100}}\\ = 0.59\end{array}\)

 

The given value of the proportion of adults who would erase all of their personal information online if they could is supposed to be equal to 0.5.

 

Thus, p=0.5.

\(\begin{array}{c}q = 1 - p\\ = 1 - 0.5\\ = 0.5\end{array}\)

 

The value of the test statistic is computed below:

 

\(\begin{array}{c}z = \frac{{\hat p - p}}{{\sqrt {\frac{{pq}}{n}} }}\\ = \frac{{0.59 - 0.50}}{{\sqrt {\frac{{0.5\left( {1 - 0.5} \right)}}{{565}}} }}\\ = 4.2785\\ \approx 4.28\end{array}\)

 

Thus, the test statistic is equal to 4.28.