For this problem, you would need to conduct a 1-proportion z-test. The key indicator that you would need to do a proportions test is that the problem states that your observed proportion is "25 out of 400," which is 0.0625. The procedure you would follow may differ from mine, but I like to use the acronym, "PHANTOMS" when I conduct hypothesis tests. Define your parameters. State your hypotheses. Assess the conditions. Name the test. Calculate the test statistic. Obtain a P-value. Make a decision. State your conclusion in context.
a.) Since proportion tests are used to generalize results to the entire population, your null hypothesis would be that p = 0.05 ; not p-hat= 0.05. The problem says that the director company claims that more than 5% of all its subscribers would pay for the channel, so, correspondingly, your alternative hypothesis would be p > 0.05.
b.) The problem does not give a specific percentage for the confidence interval, so we cannot make any further calculations regarding the value of z*. However, this is the procedure to find the z-statistic. Since this is a 1-proportion z-test, we would use the formula p̂ - p0 / sqrt [p0(1-p0) / n], where p̂ is your hypothesized proportion (0.05), p0 is your observed proportion (0.0625), and n is your sample size (400). From there, just plug the numbers into the equation to find your z-statistic. Or, if you decide to run the test on your graphing calculator, enter 0.05 for hypothesized proportion, 25 for x, and 400 for n.
c.) After plugging the numbers into the equation or your graphing calculator, you should get a P-value of approximately 0.1257.
d.) Now, we need to make a decision. In this case, since our P-value is greater than alpha (0.1257 > 0.05), we fail to reject the null hypothesis.
e.) Now, you need to write your conclusion in context. For this problem, you should write something like this: At 5% significance, we do not have enough evidence to conclude that the marketing director's claim that more than 5% of all the company's subscribers would pay an extra $10 a month to access the new sports channel.
Hope this helped! :)