Step 1 :The problem is asking to test the hypothesis using the P-value approach. The null hypothesis is that the proportion p is equal to 0.8, and the alternative hypothesis is that the proportion p is greater than 0.8. The sample size is 200 and the number of successes is 175. The significance level is 0.05.
Step 2 :Before we can perform the test, we need to check if the sample size is large enough. The condition for this is that both np0 and n(1-p0) are greater than or equal to 10. Here, p0 is the proportion under the null hypothesis, which is 0.8 in this case.
Step 3 :Calculate np0 and n(1-p0): np0 = 200 * 0.8 = 160.0, n(1-p0) = 200 * (1 - 0.8) = 40.0
Step 4 :Both np0 and n(1-p0) are greater than 10. Therefore, the sample size is large enough to perform the test.
Step 5 :\(\boxed{\text{Yes, } n p_{0}(1-p_{0}) \geq 10}\)