In this problem, we're going to be testing the claim that oxygen treatment is effective for people who have very painful cluster headaches. We have two samples. One sample received oxygen treatment on the other sample received possible treatments, so 150 patients were treated with oxygen, and out of the 150 people, 116 are free from headaches, so the proportion is 116 divided by 115. And for those who are given the possible, you have 29 free from from the headaches 15 minutes of the treatment out off a total of 148 patients. So we're going to test the claim that the oxygen treatment is effective and we're going to use two approaches. The fast approach will be the hypothesis hypothesis test, and the second one will be. The confidence interval meant better now for the hypothesis test method. We're going to have the Nahal hypotheses as P one equals p two, which is to say that the proportions are equal and to prove that them oxygen treatment is effective. It would be that the proportion off those who received oxygen treatment and war free from headaches is greater than the proportion off those who received the placebo. And we're free from, um, headaches. Now we can work out the test statistics by substituting the values we obtained into the formula for that. And when we do so, the calculator value of that is nine point 96 Next, we can get the critical value for this, uh, one tales test at the 0.1 significance level. On the calculated value of that is the critical value of set for that is going to be 2.33 So we can compare the conclusive body of that in the critical money upset. And in this case, we have 2.33 can shape that critical region, and we see that 9.96 is within the critical region. And for that reason we reject the national hypothesis. This'll means that there is sufficient evidence to support the claim that the cure rate with oxygen treatment is higher than the cure it for those given a placebo. So it appears that the oxygen treatment is effective from the hypothesis test. Let's see what happens when we conduct the confidence interval test So for the confidence interval, we need to substitute the value, Uh, the values that we have obtained the into the formula and to get the imagine of error e And when we do so, imagine of error is 0.1101 And when we can also substitute the values, uh, into the expression for the confidence interval you obtained, the interval limits US zero 0.4669 less than p one in a speed too less than 0.6871 And when you look at the intervals at the interval limits, we do not have zero in between. So the internal limits do not include zero, and that means that the two curates are not equal. In other words, we would have to weigh would have to reject the null hypothesis. So this confidence interval test is in agreement with the hypothesis test. And so it appears it appears that the cure it with oxygen treatment is higher than the cure it for those given a placebo, and that means that the oxygen treatment is effective. Next, we're going to to give a reason as's, too, as whether or not the oxygen treatment is effective based on the two tests. So based on the two results the conflict, the hype of the serious test and also the confidence interval test, there's an agreement that there is a difference between those two proportions. So the results. Both results suggest that the oxygen treatment is effective in curing cluster headaches.