In conjunction with the housing foreclosure crisis of 2009, many economists expr

Question

In conjunction with the housing foreclosure crisis of 2009, many economists expressed increasing concern about the level of credit card debt and efforts of banks to raise interest rates on these cards. The banks claimed the increases were justified. A Senate sub-committee decided to determine if the average credit card balance depends on the type of credit card used. The cards under consideration are Visa, MasterCard, Discover, and American Express. The sample sizes to be used for each level are 25, 25, 26, and 24, respectively.

Explanation / Answer

the other variations are much harder to find. In fact, the total variation wasn't all that easy to find because I would have had to group all the numbers together. That's pretty easy on a spreadsheet, but with the calculator, it would have meant entering all the numbers once for each list and then again to find the total. Besides that, since there are 156 numbers, and a list can only hold 99 numbers, we would have problems. Below, in the more general explanation, I will go into greater depth about how to find the numbers. Basically, unless you have reason to do it by hand, use a calculator or computer to find them for you.

df = Degrees of Freedom

Now, for the next easy part of the table, the degrees of freedom. We already know the total degrees of freedom, N-1 = 155.

How many groups were there in this problem? Eight - one for each exam. So when we are comparing between the groups, there are 7 degrees of freedom. In general, that is one less than the number of groups, since k represents the number of groups, that would be k-1.

How many degrees of freedom were there within the groups. Well, if there are 155 degrees of freedom altogether, and 7 of them were between the groups, then 155-7 = 148 of them are within the groups. Isn't math great? In general terms, that would be (N-1) - (k-1) = N-1-k+1=N-k.

Oooh, but the excitement doesn't stop there. One of our assumptions was that the population variances were equal. Think back to hypothesis testing where we were testing two independent means with small sample sizes. There were two cases. Case 1 was where the population variances were unknown but unequal. In that case, the degrees of freedom was the smaller of the two degrees of freedom. Case 2 was where the population variances were unknown, but assumed equal. This is the case we have here. The degrees of freedom in that case were found by adding the degrees of freedom together. That's exactly what we'll do here.

Since the first group had n=24, there would be df=23. Similarly, the second group had n=23, so df=22. If you add all the degrees of freedom together, you get 23 + 22 + 21 + 18 + 16 + 15 + 15 + 18. Guess what that equals? You got it ... 148. Isn't this great?

Mean Squares = Variances

The variances are found by dividing the variations by the degrees of freedom, so divide the SS(between) = 2417.49 by the df(between) = 7 to get the MS (between) = 345.356 and divide the SS(within) = 38143.35 by the df(within) = 148 to get the MS(within) = 257.725.

There is no total variance. Well, there is, but no one cares what it is, and it isn't put into the table.

F

Once you have the variances, you divide them to find the F test statistic. Back in the chapter where the F distribution was first introduced, we decided that we could always make it into a right tail test by putting the larger variance on top.

In this case, we will always take the between variance divided by the within variance and it will be a right tail test.

So, divide MS(between) = 345.356 by MS(within) = 257.725 to get F = 1.3400

The rest of the table is blank. There is never a F test statistic for the within or total rows. When we move on to a two-way analysis of variance, the same will be true. There will be F test statistics for the other rows, but not the error or total rows.

Finishing the Test

Well, we have all these wonderful numbers in a table, but what do we do with them?

We have a F test statistic and we know that it is a right tail test. We need a critical value to compare the test statistic to. The question is, which critical F value should we use?

Are you ready for some more really beautiful stuff?

What two number were divided to find the F test statistic? The variance for the between group and the variance for the within group. Notice that the between group is on top and the within group is on bottom, and that's the way we divided. Also notice that there were 7 df on top and 148 df on bottom. There we go. We look up a critical F value with 7 numerator df and 148 denominator df.

Since no level of significance was given, we'll use alpha = 0.05. We have two choices for the denominator df; either 120 or infinity. The critical F value for F(7,120) = 2.0868 and the critical F value for F(7,infinity) = 2.0096. The critical F value with 120 df is larger and therefore less likely to reject the null hypothesis in error, so it's the one we should use. Actually, in this case, it won't matter as both critical F values are larger than the test statistic of F = 1.3400, and so we will fail to reject the null hypothesis with either one.

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