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 $=$ (Round to three decimal places as needed.)

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

Step 3 ：Calculate \(np_{0} = n \times p_{0} = 200 \times 0.8 = 160.0\) and \(np_{0}(1-p_{0}) = 160.0 \times (1-0.8) = 32.0\). Since \(np_{0}(1-p_{0})\) is greater than 10, the sample size is large enough.

Step 4 ：Next, calculate the test statistic as \(z = \frac{x - np_{0}}{\sqrt{np_{0}(1-p_{0})}} = \frac{175 - 160.0}{\sqrt{32.0}} \approx 2.65\).

Step 5 ：Then, calculate the P-value, which is the probability of observing a test statistic as extreme as, or more extreme than, the observed test statistic, under the null hypothesis. The P-value is approximately 0.004.

Step 6 ：Since the P-value is less than the significance level of 0.05, we reject the null hypothesis. This means that we have sufficient evidence to support the alternative hypothesis that the proportion p is greater than 0.8.

Step 7 ：Final Answer: The P-value is approximately \(\boxed{0.004}\).