In Exercises 29 and 30, (a) identify the claim and state Ho an...

Question

In Exercises 29 and 30, (a) identify the claim and state Ho and Ha, (b) find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic z, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the samples are random and independent.

A medical research team conducted a study to test the effect of a drug used to treat a type of inflammation. In the study, 68 subjects took the drug and 68 subjects took a placebo. The results are shown below. At α=0.05, can you reject the claim that the proportion of subjects who had at least 24 weeks of accrued remission is the same for the two groups? (Source: The New England Journal of Medicine)

Accepted Answer

Hello everyone. Glad to have you back. Our next problem says. Researchers studied the effect of a dietary supplement on cholesterol reduction. In the study, 90 participants took the supplement, and 90 took a placebo. Of those who took the supplement, 27 achieved a significant reduction in cholesterol, while 63 did not. Of those who took the placebo, 9 achieved a significant reduction, while 81 did not. At the 0.05 significance level, is there evidence to reject the claim that the proportion of participants achieving significant cholesterol reduction is the same for both groups? A no, do not reject the null hypothesis. The proportions are different. Yes, reject the null hypothesis. The proportions are the same. No, do not reject the null hypothesis. Proportions are the same. Or yes, reject the null hypothesis. The proportions are different. So we're looking at comparing proportions between populations. So, we're going to be looking for the Z test statistic. Jot down Z. Test statistic So, we're going to start as always, with our hypotheses. In this case, the null hypothesis would be that our proportions that achieved reduction is the same between those who took the drug and who took a placebo. So that would be that P1 or proportion 1 is equal to P2 or proportion 2. Since P is the probability of a success. Our alternative hypothesis would then be that P1 does not equal P2. So we're looking at a two-tailed test. So now we need to look at calculating our Z value. So let's look at that formula for Z to see what numbers we need. And when we're assuming the null hypothesis, our equation for our Z test statistic is the sample proportion. Of Group one Minus the sample proportion of group 2. And then that divided by the standard error. And then we have a formula for standard error, but we will get to that. So, what is our proportion of success in sample one, so we say. Sample proportion one. Would be the number of successes in population one or who took the supplement. Divided by the number of total. So, in the ones that took the supplement, 27 achieved reductions, so that's our no brainer. And we know that 90 took the supplement. So it's 27 divided by 90, which is going to equal 0.3. So this is supplement. And then the sample portion too, so peep with that little uh hat on it. Is equal to, we look at the placebo group, then we know that 9 achieved reduction. Out of 90 who took it. So 9 divided by 90, so 0.1, is our sample proportion for the placebo group. So We've got our numerator. What do we need in our denominator? So we need to look at the. Formula for standard error. Which is going to be the square root, and it will be of RP. This is all under the square root. So this is the pooled proportion. For both groups, so Phat, it's a sample proportion, but no uh subscript there because it's the pooled proportion. So again, under a square root, we have Phat, so pooled proportion, Q hat, so the pooled proportion of not not having a success for our failure. And that's going to be multiplied by one. Divided by And one, the number in in group one. And add that to the reciprocal of N2. So then all of that, Phat, Q hat, and our um some there are all under the square root sign. So let's calculate that. Well, we need that pooled proportion. So, the pooled proportion we calculate in a straightforward enough way, so, PHAT is going to equal the total number of successes. So that's 27 + 9 in both groups. Divided by the total number of participants in both groups, so 90 plus 90. And that pulled proportion will then equal 0.2. So, let's go ahead and calculate our standard error. If our full proportion P P hat is, so let's take the square root, and we have underneath the square root sign, 0.2, that's P. Our Q hat would then be 1 minus P, so 0.8. And then that will be multiplied by 1 divided by 90, plus 1 divided by 90. That each group has 90 participants. So the square root of that whole. Expression So you can pause the video for a second and enter those numbers into the calculator. And you should get a value of 0.0597 for that. SE value. So, now we have everything we need for our Z test statistic. So, that's going to be equal to the numerator. Phat sub 1, so our proportion of success in sample one, so it's 0.3, and subtract that proportion for sample two, so minus 0.1. And then that's going to be divided by our standard error, which is 0.0597. So go ahead and calculate that. And we're going to get 3.351. So that is our Z score. I'm gonna go ahead and highlight that in blue, it's. We need that. And now we just need to look up our critical Z value. So Z critical. When alpha is equal to 0.05. It's a two-tailed test, so we're going to have positive and negative values. This is a very small table, easy to look up, it's going to be positive or negative, 1.96. Now, we need to compare that test statistic with our critical Z value. When I like the criticality value as well, so we can pick out what we need. And again, as we're comparing a two-tailed test, we look at our Z equaling 3.351. And we say that the absolute value of 3.351. is greater than 1.96. Again, because it's two-tailed, we can fall on the outside in either direction, positive or negative. So Z critical value. Since our Z is greater than our critical Z value, the end result is we reject. The null hypothesis. Of course, is that the proportions are equal in the two groups. So, back we go to our answer choices, and we're going to get choice D. Yes, reject the null hypothesis. That's because of the comparison between our Z values. But we're rejecting it because the proportions are different. So our no hypothesis was that the proportions would be equal. So, there we have it. We were looking to see if this dietary supplement had an effect on cholesterol reduction, had two groups of 90 participants, one that took the supplement, one with a placebo. So, we looked at the proportion of success in the two groups, and went ahead and calculated a Z value for that, and then compared that to the critical Z value, and found that that Z value is greater than our critical Z value, which causes us to reject the null hypothesis in this case, meaning the proportions are different. See you in the next video.

Key Concepts

Hypothesis Testing

Critical Value and Rejection Region

Standardized Test Statistic (z)

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