00:03
We have a car insurance company, company a, and they claim that its customers pay less for car insurance on average than customers of its competitor, b.
00:12
And so that means our hypothesis will be that the mean of company a and the mean of company b, and that's the mean that the customers pay, are equal, that the mean of a equals the mean of b.
00:28
Whereas the alternative, h1, is that the mean of a is less than the mean of b, because that's what company a is saying.
00:39
Their customers pay less than company b.
00:42
So we'd expect the means of a to be less than that of b.
00:46
And so this is a two -sample t -test, two independent sample t -test, because we're told the populations are approximately normal, so we can assume they're normally distributed.
01:00
And then we're also told that the population variances are equal.
01:05
So this is a equal variances.
01:08
We're going to assume the variances are equal.
01:15
And so that tells us, because there's two different formulas to use for a two -sample t -test, and so we're going to use the one for equal variances.
01:24
And we're testing this claim at the alpha of 0 .10 level of significance.
01:31
And we're going to call a group 1, b group 2.
01:38
And so we're going to, there's two ways we're going to make a decision.
01:45
The first will be the alpha, the critical value.
01:48
So we're going to reject h0 if, there's two ways and it's good to go over both, if the absolute value of our test statistic is greater than our critical value.
02:03
There's that.
02:03
We could also reject if our p -value that we're going to find is less than the alpha of 0 .10.
02:11
And this is a one -tailed test, meaning all this alpha is in one tail.
02:15
And so since it's normal, so essentially what we're looking for, here's our normal distribution, approximately normal.
02:20
That's why the student's t -distribution is approximately normal.
02:23
Here's our zero, our t -square of zero.
02:25
And since we're looking strictly less than, we're going to have some negative t -value here.
02:31
That's where we're looking at the absolute value here.
02:32
I guess you could say the absolute value of that critical value score here.
02:36
And then the area here to the left of it is 0 .10.
02:41
And if our test statistic falls in here somewhere, then we're going to reject h0.
02:47
All right, let's go ahead and do our calculations.
02:49
So we're told the sample sizes are nine and eight for a and b respectively.
02:53
The means are 151, 159, and the sample standard deviations are 10.
02:58
So the test statistic is calculated as x bar one minus x bar two divided by the square root of the pooled variances.
03:08
That's what the equal variances allows us to do.
03:12
We pool the variances and then we take that pooled variance multiplied by one over the sample size of one plus one over the sample size of two.
03:19
Now we need a formula for the pooled variance.
03:21
That's equal to n1 minus one multiplied by the sample variance of the first group plus n2 minus one times the sample variance of the second group, all divided by n1 plus n2 minus two.
03:45
Good.
03:47
So let's go ahead and, oh, something to note.
03:50
We need to note the degrees of freedom and actually the degrees of freedom is right here.
03:54
N1 plus n2 minus two.
03:55
So the degrees of freedom is that same thing.
03:58
N1 plus n2 minus two.
04:02
And this is for the equal variances to sample t -test...