In Exercises 11–14, test the claim about the difference betwee...

Question

In Exercises 11–14, test the claim about the difference between two population means and at the level of significance . Assume the samples are random and independent, and the populations are normally distributed.
Claim: μ1=μ2; α=0.1
Population statistics:σ1=3.4 and σ2=1.5
Sample Statistics: x̅1=16, n1=29, x̅2=14, n2=28

Claim: μ1=μ2; α=0.1

Population statistics:σ1=3.4 and σ2=1.5

Sample Statistics: x̅1=16, n1=29, x̅2=14, n2=28

Accepted Answer

Welcome back, everyone. In this problem, a researcher wants to test whether two different diets result in the same average weight loss. The claim is that the population means mu1 is equal to mu 2 at a significant level of alpha equals 0.05. The population standard deviations are sigma 1 equals 2.8 and sigma 2 equals 2.1. Sample statistics are that the sample mean for the first X bar 1 is 8.5, the sample size N1 is 35, and the sample mean X2 is 7.2, while N2 equals 32. Test the claim. A says that there is insufficient evidence to reject the claim that the two different diets result in the same average weight loss and B says there is sufficient evidence to reject the claim that the two different diets result in the same average weight loss. Now in order for us to test this claim, let's define a few things for starters, let's let. New one Represent the mean weight loss for diet one. OK. Which means that sigma one is the population standard deviation for diet one, X bar 1 is the sample mean for diet one, and N1 is the sample size for diet one, OK. And let's let me 2 then represent the mean weight loss for diet 2. Which means the same reasonings apply or the same reasoning applies for their notations. Then our null hypothesis based on our statement. Is that mu1 is equal to mu 2, OK? In other words, there is no difference between both diets. Sorry, there is no difference in average weight loss for both diets. Well, our alternative hypothesis H1, is that there is a difference. In other words, M1 is not equal to mu 2. Now, because the population standard deviations are known, we can use a 2 sample Z test for means, which means then. That we can set our decision rule. To basically say that if our the absolute value of our calculated Z test statistic, OK, is greater than our critical Z value, then we haven't, we can reject our no hypothesis. Otherwise, OK, then we fail to reject the null hypothesis. So this is our decision rule and in that case we'll need to figure out our Z test statistic. We'll need to calculate it using the information given. We need to find our critical Z value for the significance level in a Z table and then. We can compare to make our decision. So let's first start by finding the Zest statistic. What do we know? Well, recall that. The z value or a Z value is going to be equal to the difference between the sample means. OK, minus the difference between the population means. All divided by the square root. Let me make some space here. The square root of sigma 1 squared divided by N1 plus sigma 2 squared divided by N2. Now if we look back in our problem statement we have all of this information, so if we substitute here. For starters, we know that X bar 1 is 8.5, while XB 2 is 7.2. We know that under the assumption of another hypothesis, mu1 equals m2. So that means mu1 minus m2 will be 0, OK. And we also know. That or sorry, sigma one is going to be is, is 2.8 rather from the information given. So that's 2.8 squared divided by 35 and 1 plus sigma 2, 2.1 squared. OK, divided by N2, which is 32. And now, in this case, we should find that we get Z after calculating to be approximately equal to 2.161. So this is our Z test statistic. Now we can use it to compare to our critical value, so we need to find our critical value. Now for a two-tailed, for a two-tailed test. Add a significant level alpha of 0.05, OK? Then the absolute value of our critical value is going to be equal to 1.96. OK? In other words, the critical Z values are plus R minus 1.96. Know that we have our critical value and our Z test statistic. Let's compare. What do you notice? Well, Notice that OK, notice that our Z value, the absolute value of our Z value, that's the absolute value of 2.161. OK, 2.161 is greater than 1.96. Then the test statistic falls into the rejection region and thus we can reject the null hypothesis. That means there is enough evidence to conclude that the null hypothesis is false. In other words, at the 0.05 level of significance, there is sufficient evidence to reject the claim that the two different diets result in the same average weight loss. Therefore, B is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

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