Comparing Two Means Treating the data as samples from larger p...

Question

Comparing Two Means Treating the data as samples from larger populations, test the claim that there is a significant difference between the mean of presidents and the mean of popes.

Accepted Answer

Hi, everyone. Let's take a look at this practice problem. This problem says a nutritionist is comparing the average daily calorie intake of two groups, vegetarian and non-vegetarians. A sample of 15 vegetarians has a mean intake of 2200 calories with a standard deviation of 180 calories. A sample of 20 non-vegetarians has a mean intake of 2500 calories with a standard deviation of 210 calories. At the 0.05 significance level, what is the value of the test statistic for testing whether there is a significant difference in mean daily calorie intake between the vegetarian and non-vegetarian groups? And we're given four possible choices as our answers. For choice A, we have T is equal to -2.54. For choice B, we have T is equal to 4.54. For choice C, we have T is equal to -4.54, and for choice D, we have T is equal to -5.44. So, the first thing that we're going to do is write down the hypotheses that we'll be testing. So, for our null hypothesis, H dot, we're going to have that m1 is going to be equal to U2. Where mu1 is the population mean intake for vegetarians, and mu m2 is the population mean intake for non-vegetarians. And our alternative hypothesis, HA is going to be that m1 is not equal to U2. Now, if you look at the information that we're given in the problem, We see that we're giving information about the sample data, but we don't have any information about the population data, so our population standard deviation are not really known for each group. Also, our sample sizes are relatively small, so we're going to want to use a T distribution rather than a Z distribution. Now, our groups, the vegetarian and non-vegetarians, are independent of each other. So we'll want to use a two-sample T test. We also should not assume equal variances. So we're going to want to use a Welch's tea test, since it's a safer choice when the variance equality is unknown. So, we call our formula for the Welch's T test. We'll have T is equal to the quantity. Of X4 1 minus X2. In quantity divided by The square root Of the quantity Of S1 squared divided by N1. Plus S2 squad divided by N2 in quantity. Where X bar 1 is going to be the main intake for the vegetarian group, XB 2 is going to be the mean intake for the non-vegetarian group. And S1 and S2 are going to be the standard deviations for the vegetarian and non-vegetarian groups respectively, as well as N1 and N2 are going to be the sample sizes for each of those groups, respectively. And we were actually given all of this information and the problems. So that means that T is going to be equal to the quantity for X 1, we have 2200, then we'll have minus X2, which is gonna be 2500. In quantity divided by The square root Of the quantity, or S1, we have 180. But we're gonna need to square that, and then divide that by N1, which is 15. Then we'll have plus, for S2, we're gonna have a value of 210. And that value is going to need to be squared and divided by in 2, which in this case is 20. And when we evaluate this expression, this is going to be equal to. -4.54. So here we just round it to two decimal places. And so that is actually the answer that we are looking for for this problem. And if we compare that to our answer choices, we see that the correct answer choice corresponds to answer choice C. So, I hope that this explanation has been useful, and I'll see you in the next video.

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