To find the value of the test statistic [tex]\( z \)[/tex] using the given formula [tex]\( z = \frac{\hat{p} - p}{\sqrt{\frac{p q}{n}}} \)[/tex], we need to follow these steps methodically:
1. Identify the given values:
- [tex]\( p \)[/tex] (the population proportion claim) = 0.10
- [tex]\( \hat{p} \)[/tex] (the sample proportion) = 0.15
- [tex]\( n \)[/tex] (the sample size) = 800
2. Calculate [tex]\( q \)[/tex], which is [tex]\( 1 - p \)[/tex]:
[tex]\[
q = 1 - p = 1 - 0.10 = 0.90
\][/tex]
3. Find the standard error of the proportion:
[tex]\[
\text{Standard Error} = \sqrt{\frac{p \cdot q}{n}} = \sqrt{\frac{0.10 \cdot 0.90}{800}}
\][/tex]
Plugging in the values:
[tex]\[
\sqrt{\frac{0.09}{800}} = \sqrt{0.0001125} \approx 0.01061
\][/tex]
4. Calculate the test statistic [tex]\( z \)[/tex]:
[tex]\[
z = \frac{\hat{p} - p}{\text{Standard Error}} = \frac{0.15 - 0.10}{0.01061} \approx \frac{0.05}{0.01061} \approx 4.71
\][/tex]
So, the value of the test statistic [tex]\( z \)[/tex] is approximately [tex]\( 4.71 \)[/tex].
Therefore, the correct answer is:
[tex]\[
\boxed{4.71}
\][/tex]
