A well-known automotive magazine took three top-of-the-line...

A well-known automotive magazine took three top-of-the-line midsize automobiles manufactured in the United States, test-drove them, and compared them on a variety of criteria. In the area of gasoline mileage performance, five automobiles of each brand were each test-driven 500 miles; the miles per gallon data obtained follow. Use $\alpha=.05$ to test whether there is a significant difference in the mean number of miles per gallon for the three types of automobiles. $$\begin{array}{ccc} & \text { Automobile } & \\ \mathbf{A} & \mathbf{B} & \mathbf{C} \\ 19 & 19 & 24 \\ 21 & 20 & 26 \\ 20 & 22 & 23 \\ 19 & 21 & 25 \\ 21 & 23 & 27 \end{array}$$

Transcript

0:00 All right.

00:01 So for this problem, the way that we'll be performing this test for a significant difference is by doing one factor anova.

00:09 Now, i'm going to be doing the calculations here in excel, but i won't be using excel sort of automation of this.

00:15 I will show the step -by -step procedure that you use for doing a problem like this.

00:20 The first step is going to be to find the averages for each one of our treatments.

00:29 Average, which we can find simply by summing up the individual measurements, then dividing that by the number of measurements.

00:38 You can see that it's five for each, or you can simply do average if you're using excel or a graphing calculator.

00:48 So we can see the average for a is 20, for b is 21, and for c is 25.

00:54 Then what we want to do is create a table of the squared deviations.

01:01 Where the idea is in each cell, we'll have the difference between the particular measurement and the average for that group.

01:11 I'll say x bar per group, and then we square that value.

01:17 So, for instance, for a, i'll do this in similar columns as above, but for this first one, well, that's 19.

01:26 We know the average for group a is 20.

01:29 So what i'm going to do, just for the sake of using excel here, i put b dollar sign 7.

01:37 So that basically means that when i apply the formula to the right, it'll move across the same row.

01:44 But it's not, or pardon me, it's going to move across.

01:47 Yeah, it's going to move across the same row, but it won't change which column it is.

01:54 Anyways, so we do b2 minus b dollar sign 7 squared.

01:59 Then apply this downwards.

02:06 So if we look, okay, we're then doing 21 minus 20 squared.

02:11 And then we'd have 20 minus 20 squared and so on.

02:15 And then just to make sure, okay, that went one too many.

02:19 All right.

02:20 So then we can drag this across and apply the same ideas.

02:25 So we have 19 minus 21 squared, then 20 minus 21 squared and so on.

02:30 So now we have the individual squared deviations, then we want to find the sum of all of these squared deviations.

02:40 That is going to provide us with the, pardon me, i just realized i have this mixed around.

02:48 When we sum up all of those squared deviations, that is going to give us our within sum of squares.

02:56 So we just do equals sum of all of these...

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