A research study claims that [tex]$68\%$[/tex] of adults drink regularly. Edward conducts a random sample of 200 people and finds that 140 people drink regularly.

[tex]z = \frac{\hat{p} - p}{\sqrt{\frac{pq}{n}}}[/tex]

Using the formula and data provided, what is the value of the z-test statistic? Answer choices are rounded to the hundredths place.

A. 0.41
B. 0.61
C. 0.59



Answer :

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