In this problem, we're going to be considering sample data and independent. Simple random samples were picked each or two samples each off Science 200. In the first sample, 112 people had the same attributes. In second sample, 88 people had the same attributes. So the a Z we can see here that were given anyone is 200 on X two. X one is 112 and then enter is 200 next to is 88. From thes values, we can obtain three other values, like P one hut, which is 112 divided by 200. So P one hunt equals 0.5 six. And we can obtain que one hot by subtracting 0.1 a 0.56 from one. That would be 0.44 from the second sample. We have p too hot, obtained from 88 divided by 200. On that is 0.4 form and cue. Too hot. He's one man, a 0.4 form, which is your 0.56 You can get PBA from adding the X one and x two, and that will give us 200 divide by the sum of anyone and to which is 400. And that P bob becomes your 4000.5 and Cuba will also be 0.5 because they needed to add up to one. Now, in the first part of the question, we're going to be getting the 95% confidence interval estimate for the difference. P one minus B two So and then we'll tell whether the result suggests what the results suggest about equality off P one and p two. So let's move on to find the 95% confidence interval on the next 5% confidence interval off P one when I speak to is obtained from adding and subtracting the margin of error, so you want to walk out the margin of error E, which is given by the formula, was P upon hot shoes. One hunt over n one speak to hut que to hunt over and to. So what we need to do to get imagine of area is to substitute the values that we have obtained on into the formula, and when we do that correctly, we find the value off. E is your point 0973 Next we add and subtract. Imagine of error from Deb. A P one had means be too hot. In this case, p one heart minus p too hot. He is 0.56 men and 0.44 which is 0.12 So the confidence interval limits are as your 0.1 to minus 2.973 which becomes 0.0 227 So that's the lower limit on for the upper one. We have 0.12 plus 0.973 which he calls zero point 21 73 So therefore the 95% confidence interval estimate will be your brains. You 227 It's less than p one in the spey too, just less than your point 217 We round it off to three significant digits so we can see that zero is not contained in this confidence interval. And for that reason, it appears that P one is equal to P. Two can be rejected, So it tells us that, uh, the equality off the one and P two can be rejected because zero is not contained in the interval. Next we walk out the 95% confidence interval estimate for each off the proportion. So we need to get the margin of error, um, for each proportion, and then get the confidence interval for P one on. Then we get the confidence interval for P two. So let's proceed. We get e the margin of error by working out the following. That is for the first for the first sample. And when we substitute the palace, it is going to be 1.96 Scratch it off P one hunt, which is the 0.56 times the value of Cuban 100 0.44 divided by end one, which is 200. And when we walked this out, we get the value off E zero point as usual. 688 So, for the first, uh, proportion, Imagina vera is 0.66688 and this valley will still be the margin of error for the second sample because we have the same numbers. Um, for P one, have you noticed that the value off P 100 Cuban heart that the pairs are the same 0.560 point 440.440 point 56 and also, the sample size is, uh, the same, and one is 200 extra 200. So the margin of error is identical for both, uh, the fast sample on the second sample. So then, to get the confidence interval for P one who would need to add and subtract the margin of error to the value off people? So the value of P one heart had you obtained it a zero point 56 And so the confidence interval for P one will be 0.56 minus zero point zero, 688 and uh, 0.56 plus 0.688 And when we walked this out to you, obtain zero point for nine want to this less than P one? It's just less than zero point 6 to 88 When you run off to three significant digits, it's going to be 0.491 It's less than P one, which is less than 0.6 to 9. So that's the 95% confidence interval for the first proportion p one now for the second proportion P two had we have 0.4 from, So we're going to address attract 0.688 eso. It's going to be there for the lower interview. It's going to be 0.44 my next year from zero 688 less than P two, just less than 0.44 class zero friends zero 688 And when you walk this out, it's going to be easier point 3712 less than P two, just less than 0.5088 And when you simplify, it's going to be 0.371 is less than P two. That's less than zero point 509 to 3 significant digits. So, according to this 95% confidence interval estimate for each proportion, we noticed that there is an overlap between the confidence interval levels. Interpol um confidence intervals. So, in other words, that the values the P to the confidence interval for P two is contained in the confidence interval for P one. Yes, and that suggests that there's an A P one equals p two cannot be rejected. You know, we could assume that P one and P two are equal according to this comparison off the individual proportions. Next read, go and conduct hypothesis test I used between these two proportions. In other words, we test the non hypotheses. H not p one equals p two on the alternative hypotheses. H one p one is not equal to Peter. So we need to proceed and walk out the test statistic. Zed, which is obtained from a p one hunt minus p to heart all that minus zero divide by people are sorry, this square it off, people Cuba off n one plus people cube off end too. So we had all these figures and we just need to substitute them into the formula. And when we do that correctly, the value off the test statistic is 2.4. And since this is going to be done with the 0.5 level of significance, the critical values that he is plus or minus 1.96 So now we can compare their value of that on the critical value. So in this case, this is a critical region shaded so one negative 1.96 and positive 1.96 And in our case our 2.4 is within the critical region and for that reason would have to make the conclusion to reject the non hypothesis. So I'm by rejecting ANALITICO. This is we we mean that a sufficient evidence to reject the equality off the off the proportions that is rejecting that p one is equal to P two Now, finally, we're going to I used this results to make a conclusion about equality off P one and p two. So the conclusion here is to reject, um, they reject p one he calls Peter. That is, according to them, confidence interval test. How the confidence next five percent confidence interval estimates of P one minute speak to on also from the hypothesis test that he led, uh, the critical the test statistic of 2.4. So meaning that we would have to reject the null hypothesis. And we also supposed to tell which of the three methods is least effective in testing the quality off p one and p two and we can see that the method off the overlapping the conflict individual confidence interval is the least effective method because according to this method, we note that the P two overlaps the confidence interval of P to overlap with the confidence interval of P one, we suggest that the two proportions could be equal, so that's the least effective method in testing the difference between the confidence between the two proportions.