Test the hypothesis using the P-value approach. Be sure to verify the requirements of the test.
\[
\begin{array}{l}
H_{0}: p=0.8 \text { versus } H_{1}: p> 0.8 \\
n=200 ; x=175, \alpha=0.05
\end{array}
\]
Is $n p_{0}\left(1-p_{0}\right) \geq 10 ?$
No
Yes
Use technology to find the P-value.
P-value $=0.004$
(Round to three decimal places as needed.)
the null hypothesis, because the P-value is
than $\alpha$.

Step 1 ：Given the null hypothesis \(H_{0}: p=0.8\) and the alternative hypothesis \(H_{1}: p>0.8\). The sample size is \(n=200\), the number of successes is \(x=175\), and the significance level is \(\alpha=0.05\).

Step 2 ：First, we need to check if the sample size is large enough for the normal approximation to the binomial distribution to be valid. This is done by checking if \(np_{0}(1-p_{0})\) is greater than or equal to 10, where \(n\) is the sample size, and \(p_{0}\) is the proportion under the null hypothesis.

Step 3 ：Calculate the test statistic and the P-value. The test statistic is calculated as \((x - np_{0}) / \sqrt{np_{0}(1-p_{0})}\), where \(x\) is the number of successes. The P-value is the probability of observing a test statistic as extreme as the one calculated, assuming the null hypothesis is true.

Step 4 ：Finally, compare the P-value to the significance level. If the P-value is less than the significance level, we reject the null hypothesis. If the P-value is greater than the significance level, we fail to reject the null hypothesis.

Step 5 ：The calculated P-value is approximately 0.004, which is less than the significance level of 0.05. Therefore, we reject the null hypothesis that the proportion \(p\) is equal to 0.8. The evidence suggests that the proportion \(p\) is greater than 0.8.

Step 6 ：Final Answer: We reject the null hypothesis, because the P-value is less than \(\alpha\). The P-value is approximately \(\boxed{0.004}\).