Question

After once again losing a football game to the archrival, a college’s alumni association conducted a survey to see if alumni were in favor of firing the coach. An SRS of 100 alumni from the population of all living alumni was taken, and 64 of the alumni in the sample were in favor of firing the coach. Suppose you wish to see if a majority of all living alumni is in favor of firing the coach. The appropriate standardized test statistic is

$$\begin{gathered} \left(a\right)z=\frac{0.64-0.5}{\sqrt{{\displaystyle \frac{0.64(0.36)}{100}}}} \\ z=0.64-0.50.64(0.36)100 \end{gathered}$$

role="math" localid="1654432946823" $$\begin{gathered} \left(b\right)t=\frac{0.64-0.5}{\sqrt{{\displaystyle \frac{0.64(0.36)}{100}}}} \\ t=0.64-0.50.64(0.36)100 \end{gathered}$$

$$\begin{gathered} \left(c\right)z=\frac{0.64-0.5}{\sqrt{{\displaystyle \frac{0.5(0.5)}{100}}}} \\ z=0.64-0.50.5(0.5)100 \end{gathered}$$

$$\begin{gathered} \left(d\right)z=\frac{0.64-0.5}{\sqrt{{\displaystyle \frac{0.64(0.36)}{64}}}} \\ z=0.64-0.50.64(0.36)64 \end{gathered}$$

$$\begin{gathered} \left(e\right)z=\frac{0.5-0.64}{\sqrt{{\displaystyle \frac{0.5(0.5)}{100}}}} \\ z=0.5-0.640.5(0.5)100 \end{gathered}$$

Short answer

The required correct option is C.

The appropriate standardized test statistic is$z=\frac{0.64-0.5}{\sqrt{\frac{0.5(0.5)}{100}}}$

Step-by-step solution

Given information

$$\begin{gathered} n=100 \\ x=64 \\ p=50\%=0.50 \end{gathered}$$

Concept used

The following expression is used to compute the test statistic:

$z=\frac{\hat{p}-p}{\sqrt{\frac{p(1-p)}{n}}}$

Where sample proportion $\hat{p}$

Sample size:$n$

The objective is to find out the correct option which shows the appropriate standardized test statistic.

To begin, compute the sample proportion, which is the ratio of the number of successes to the sample size:

$$\begin{gathered} \hat{p}=\frac{x}{n} \\ \Rightarrow \hat{p}=\frac{64}{100} \\ \Rightarrow \hat{p}=0.64 \end{gathered}$$

The following expression is used to compute the test statistic:

$z=\frac{\hat{p}-p}{\sqrt{\frac{p(1-p)}{n}}}$

Fill in the blanks with the values from the given data.

$$\begin{gathered} z=\frac{\hat{p}-p}{\sqrt{\frac{p(1-p)}{n}}} \\ \Rightarrow z=\frac{0.64-0.5}{\sqrt{{\displaystyle \frac{0.5(1-0.5)}{100}}}} \\ \Rightarrow z=\frac{0.64-0.5}{\sqrt{{\displaystyle \frac{0.5(0.5)}{100}}}} \end{gathered}$$

Therefore, the correct option is$\left(c\right)$