For this exercise. We are given the following information. So we have these hypotheses and we're told that the true parameter P is p prime and P prime is less than the null hypothesis proportion. So therefore the alternative hypothesis is actually true now. For part, they were asked to show the following. Where's Ed is the test statistic for a one proportion test. So the one proportion test statistic is given as follows. Now, if we calculate the expectation on Zed, everything inside the brackets is constant except for the sample proportion. So the hypothesized proportion and and are both constant. So the expected value can be rewritten like this. This is because the expected value for a constant is that constant. Now, we also know from Chapter two that the expected value for a sample proportion is the true value. So this becomes and this is what we're trying to show, at least at least with respect to the expected value. So now, for the variants now, in our test statistic again, everything is a constant except for this sample proportion. So that is the only varying parameter. Now I will rewrite this like this, so I'm taking this constant here outside of the variance brackets. So therefore I squared. So that's why these square root symbol has disappeared. And we also know from Chapter two that the variance for a sample proportion is given by the following. So therefore we have the following result and this is what else we were trying to show in part A. So that completes part A. Now in part B, we want to show at the power of the lower tailed test is that is we want to find the probability of getting a test statistic lesson or equal to negative of the critical value when the alternative hypothesis is actually true. Now, if our test statistic is that and we have just found its expected value and it's variants, we would expect that this value is normally distributed as standard normally distributed because we are taking a parameter, we're subtracting its mean from it and then normalizing by its standard deviation. So to find the power of the test, we're looking for the probability that the standard normal distributed parameter is less than negative, the critical value and now just plugging in our values for the standard deviation, which is the square root of the variance of the test statistic and the expected value for the test statistic. And if you look at what's asked for in question be when you were asked to show this right here now for Part C were given this information. So we have some hypotheses and were given the true proportion as 0.8 and we're testing at a significance level of 0.1 This means that are critical. Value is negative, 2.3 to 6, and we are also given n equals 225. And so we know that we know that the alternative hypothesis is actually true and were asked, What is the probability that our test will detect this? What is the probability that our test will result in us rejecting the null hypothesis? In other words, what is the probability of us getting a test statistic less than minus 2.3 to 6? So all we have to do here is plug in all the numbers into this formula, and this gives us a value of 0.978 So the power of this test, the probability of rejecting the null hypothesis when the alternative hypothesis is true, is 0.978