 two sample t-tests for population means using Excel. We have some data about the price of houses with and without fireplace, and we have a pretty good bit of data about 2,000 records here that we want to analyze. Now in this problem, we're not given the population standard deviation of the home's with or without the fireplace. So without the population standard deviations that should tell you that we need to run a t-test, and we're going to run a t-test for the difference in the two means. One way of running this test is use the Excel data data analysis. Look down to we find the two sample t-test, and you have to pick there whether it's going to be equal variances or unequal variances. For us, we're not going to get into how you tell whether or not the variances are equal or not. The problems will tell you to assume equal variances or to assume unequal variances. We're going to use the unequal variance here and click OK. And it brings up a dialog box, and you need to enter the variable ranges here, which you can see the variable range 1, which is the fireplace, and variable range 2, which is the no fireplace. Problem with this data analytics pack is that in some cases you're given the summary data. You're given the sample means and sample standard deviations, and there's no way to run that test using dead analytics. But here we are going to check that the labels are there, which they are put in our alpha, put in our hypothesized main difference, which is zero, and we're going to output to a new worksheet. I'm just going to label one and click OK. And we get the information we need, except what I found is students get confused. It gives you a one-tail probability and a one-tail statistics, and it doesn't really give students an understanding is this the right tail or the left tail. They have to think it through. It's not labeled. And so they make a lot of mistakes. What I'm going to recommend that you do is to use the calculator that's in your package. And I'm going to show you how to do that now. This is the calculator. There are two versions, one for raw data that you're seeing here because we have raw data in this example, but we also have a summary data version that gets over the problem that I talked about with the data analytics package. Let's go back here to summary data. We enter our information in the blue cells, and here we know that the main difference is zero, and we're going to use an alpha 0.05. And then we enter our data over in these two columns, J and K in the blue cells. So let's do that. Here's our data. I'm going to click in that first cell, the no fireplace data, and use the control shift down error because we've got a lot of data, 741 records, then control C to copy it and control up arrow to go back up. And we've got that data ready to copy. And I'm going to go here into that first blue cell, right click, and then I want to paste special just the values because I want to get that header in there that tells me it's the no fireplace. Now I've pasted in the data for the fireplace just as I did for the no fireplace here. And you can see that over here in ourselves it's calculated the mean for the no fireplace, the mean for the fireplace, standard deviation for each of the two samples, the counts of the two samples, and then down below it's starting to calculate the answers for us. We need to select either equal variances on this drop down or not equal. Looks like equal. You can see all the answers changed there. I'm going to select not equal since that's what we're told for this particular problem. You're given the test statistic, the standardized test statistic, and then you're given the upper and lower critical values for this value of alpha, the two-tail, left-tail, and right-tail probabilities, and using that information you should be able to draw your conclusions.