Equivalence of Hypothesis Test and Confidence Interval Two dif...

Question

Equivalence of Hypothesis Test and Confidence Interval Two different simple random samples are drawn from two different populations. The first sample consists of 20 people with 10 having a common attribute. The second sample consists of 2000 people with 1404 of them having the same common attribute. Compare the results from a hypothesis test of p1=p2 (with a 0.05 significance level) and a 95% confidence interval estimate of p1-p2

Accepted Answer

Welcome back, everyone. In this problem, a study investigates the prevalence of a behavior in two populations. In the first sample of 18 people, 7 exhibit the behavior. In the second sample of 950 people, 630 exhibit the behavior. Using a 0.05 significance level, test if the two population proportions are equal and construct a 95% confidence interval for the difference in proportions. What do the results show? Now, let's focus on the first part of this problem. At the 0.05 significance level, we want to test if the two population proportions are equal. Now to test whether they are equal, we'll use a two proportion Z test. Nor in our problem If we let population one, OK. Or if we let group one be the first sample, OK, so actually let me say. Given the first sample, OK. Where X1. Is the number who exhibit the behavior 7. Out of 18 people, so the sample says N1 is 18, then the sample proportion P1 would be equal to 7/18th, which is approximately 0.3889. On the other hand, OK, in the second sample. Notice that 630 people exhibit the behavior, so X2 is 630 out of 950, so N2 is 950, which means our sample proportion will be 630 divided by 950, which is approximately 0.6632. Now, how is this going to help us? Well, like we said, we are testing the hypothesis that the proportions test if the two proportions are equal, sorry, so the null hypothesis is that P1 is equal to P2. Which means that the alternative hypothesis is that P1 is not equal to P2, which again is why we've chosen to use the two proportion Z test. Now, for that test, OK, recall that the Z statistic. is going to be equal to P1 minus P2 for our sample proportions divided by our standard error SE, OK? Where our standard error. is equal to the square root of or pulled proportion P multiplied by 1 minus P multiplied by 1 divided by N1 plus 1 divided by N2. So if we're going to find our test statistic, first we need to find our pooled proportion P. Now here, OK. Since P. Is gonna be equal to X1 + X2 divided by N1 plus N2. That means it will be 7 + 630. Divided by 18 + 950, that's 637 divided by 968, which is approximately 0.6581. I know we can use that to find or a test statistic because no that means. Our standard error, sorry, first is gonna be the square root. Uh 0.6581. Multiplied by 1 minus 0.6581. Multiplied by 1/18 plus 1 divided by 950. Which is approximately equal to 0.1129. And no, this implies then that our test statistic Z is equal to a difference in sample proportions. That's 0.3889 minus 0.6632 all divided by 0.1129, which is approximately negative 2.43. Now, how do we add context to this Z statistic? What, what does this mean? Well, if we were to put this in our normal distribution in. And we're doing a two proportion test. We're basically saying that for our critical Z value, OK, for this test, if the test statistic falls within this region, OK, for this two-tier test, then we would fail to reject. OK, so actually, let me put that uh shaded in black here. So if it falls here, we would fail to reject, OK? But if it falls outside of the critical values, whether to the left or to the right, OK, then we would be able to reject or not hypothesis. So essentially what we're seeing is that. If Our Z value is greater than the critical Z value, then we're able to reject the null hypothesis. In this case we're talking about the absolute values of each, whether it is less than the negative Z value or greater than the positive Z value. No, 4, OK. Uh, let's find our critical value. 4, alpha equals 0.05 for a two-tailed test. OK. A tot test. Then, the critical Z value is equal to plus R minus 1.96. So it's -1.96 to the left and 1.96 to the right. No remember or Z value is -2.43 and that would, that would be less sorry than -1.96. So in other words, here, Our absolute value of Z. Which is equal to 2.43, is greater than the absolute value of our critical Z value, which is 1.96. OK. And because of that, then we can reject each or reject our null hypothesis. So, that would be the result if we were to test if the two population proportions are equal. In other words, by rejecting the null hypothesis, then we can conclude that or there is a significant difference, OK? Therefore, The population proportions. Are not equal. Now, let's move on to the next part of our problem. We were told to construct a 95% confidence interval for the difference in proportions. How do we do that? Well, to do that, OK, recall, you know we're doing it for the confidence interval. Or a confidence interval. And you know what, for a previous statement, we can't. Say that they are not equal, but what we're really saying, let me just fix this here. Um, there is sufficient evidence to suggest that they are not equal. So furthermore, there is sufficient evidence. OK, let me just add that here. My apologies. To suggest that. There's sufficient evidence to suggest that the population proportions are not equal. Let's get back to our confidence interval. For this recall that the formula for the confidence interval. Says that P1, the minus P2, the difference in proportions. OK. Or rather that a confidence interval is the difference in proportions plus R minus the Z value for a two-tail test multiplied by the unpuled standard error, OK? Now, here We already found that this Z value is the critical Z value of 1.96. So we just need to find the difference between proportions and our unpoed standard error. Now, starting with our Anput standard error, OK, recall that it's gonna be equal to the square root. Of P1 multiplied by 1 minus P1. OK. All divided by N1 plus P2 multiplied by 1 minus P2 all divided by N2. Yeah, so no, because we have these values, we can substitute them. So that's really gonna be equal to the square root of 0.3889 multiplied by 0.6111. Divided by 18 plus 0.6632 multiplied by 0.3368 divided by 950. Now, when we calculate, we get that to be approximately 0.11592358, 1952, OK? So now if we next find the difference in our intervals P1 minus P2. Our difference in our proportions, it's gonna be equal to 0.3889 minus 0.6632, which equals negative 0.2743. So now that we have our Z value or difference in proportions and or unput standard error, then we can construct our interval because no, that means it will be equal to. Negative 0.2743 plus or minus, OK, 1.96 multiplied by 0.115923581952. Let me simplify it, that's negative 0.2743 plus minus 0.227210220626, OK? Which means then that the lower bound will be equal to -0.2743 minus our margin of error, which is negative 0.5015. While the upper bound will be equal to or a difference plus or margin of error which is negative 0.0471. Therefore, our interval goes from 0.5015 to 0.0471. Now that we have the interval and we've done the test, what can we conclude? Well, this tells us then. That both the test. The hypothesis test and the confidence interval. OK. Indicate a significant difference. OK. But they both indicate a significant, not significance, sorry, but a significant difference. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Hypothesis Testing

Confidence Intervals

Proportions and Sample Size

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