To find the value of the z-test statistic using the formula and data provided, we will follow these steps:
1. Identify the given values:
- Population proportion, [tex]\( p = 0.68 \)[/tex]
- Sample size, [tex]\( n = 200 \)[/tex]
- Number of people who drink regularly in the sample, 140
2. Calculate the sample proportion ([tex]\(\hat{p}\)[/tex]):
[tex]\[
\hat{p} = \frac{\text{number of successes in sample}}{\text{sample size}} = \frac{140}{200} = 0.7
\][/tex]
3. Determine [tex]\( q \)[/tex] (the complement of [tex]\( p \)[/tex]):
[tex]\[
q = 1 - p = 1 - 0.68 = 0.32
\][/tex]
4. Calculate the standard error ([tex]\(\text{SE}\)[/tex]):
[tex]\[
\text{SE} = \sqrt{\frac{p \cdot q}{n}} = \sqrt{\frac{0.68 \cdot 0.32}{200}} \approx 0.03298
\][/tex]
5. Calculate the z-test statistic:
[tex]\[
z = \frac{\hat{p} - p}{\text{SE}} = \frac{0.7 - 0.68}{0.03298} \approx 0.61
\][/tex]
Therefore, the value of the z-test statistic is approximately [tex]\( 0.61 \)[/tex].
So, the correct answer is:
b.) 0.61
