00:01
All right, so we have a clinical psychologist who's interested in the effectiveness of cognitive behavioral therapy for depression.
00:06
And he gives patients the beck depression inventory three times.
00:12
He gives the pre -therapy, post -therapy, and then six months later.
00:16
And then lower scores on these, on this depression inventory indicate less depression.
00:26
So here pre -therapy, the numbers are higher.
00:28
Six months later, the scores are lower, indicating less depression.
00:33
And what we wanna know is whether or not these differences are significant, the difference in these groups, pre -therapy, post -therapy, and six months later.
00:44
And so let's state our hypotheses.
00:47
Our hypotheses are as follows.
00:49
Well, the null hypothesis, h -naught, will be, let me do it up here actually, h -naught.
01:05
The mean scores, the mean beck scores, are the same, or are equal for each group.
01:30
The alternative hypothesis is that the mean scores are not all equal, or not.
02:02
All right, so this, and we're gonna test this set of hypotheses at the alpha of 0 .01 level of significance.
02:11
And what that means is we're gonna reject the null hypothesis if this p value, if the p value we get from our f statistic, from our anova, is less than this alpha.
02:22
And we can also use a critical value approach here.
02:28
And the way we get that is we need to look at, it's called the degrees of freedom.
02:33
So we're gonna start filling in this table down here.
02:35
We'll get our statistic.
02:42
So there's a critical value, and the critical value, using the f statistic world, as i have p value, we'll do both, p value and f statistic.
02:55
If the f statistic, if our f calculated statistic is greater than 8 .02, we're going to go ahead and reject the null hypothesis.
03:07
Reject if f is bigger than this, or if our p value is less than 0 .0.
03:18
And the way you get this f statistic, the critical value here, is by looking at the degrees of freedom of this table.
03:24
So the degrees of freedom is calculated as taking k minus one where k is the number of groups.
03:29
So three groups, pre, post, and six months later.
03:32
So that is going to be, oops, k minus one.
03:35
It's gonna be two.
03:36
The within sum of squares is taken as big n minus k.
03:39
Big n is the total number of data points in the sample, or in our study, and there are 12.
03:45
Three groups, four subject per group.
03:47
So 12, 12 minus three is nine.
03:51
And the total degrees of freedom is n minus one, n is 12, so 11.
03:58
Now this is where we have to do a little bit of work.
04:01
So we need these sum of squares values.
04:04
So the sum of squares, the sum of squares of the within, we have the within, the between, and the total.
04:15
We're gonna do the within and the total because the between is equal to the sum of squares of the total minus the squares of the within.
04:22
So once we get the total and the within, we'll be good to go.
04:26
So the within sum of squares, what we do is we take the sum of the x squared values.
04:32
This is for each of the three groups.
04:39
The sum of the x squared values minus n times x bar squared.
04:46
So, and this is gonna be for each group.
04:49
So i guess you'd have two summations here.
04:57
So this would be for a single group, and then this would be the same of all three groups.
05:03
So the x squared values are given to us right here.
05:07
We have there, but then we also need the mean, and we're given the sums of the x values.
05:11
So the way we get the mean is we just take that sum divided by how many there are.
05:15
In this case, there are four.
05:16
So the means for these groups, so x bar, the way we do that is we do 21 divided by four, right? so that's gonna be 2 .75, 14 divided by four, right, we're just taking these five by four, and that's gonna give us not 3 .5, sorry about that, 3 .5.
05:40
And then seven divided by four is gonna give us 0 .75.
05:47
So what we do, so we're gonna sum up for each of the groups...