00:01
So, for this problem, i'll go into the full detail for part a.
00:04
For the remaining portions, i'll use some technology to speed up the process.
00:09
So for part a, we're asked, is higher education associated with an increase in income for black men? so that would mean that the null hypothesis is that the two mean values are the same.
00:19
We're testing it against the alternative hypothesis that mu1 is lower than mu2, where mu2 is going to be the mean for black men with college education or higher.
00:32
Now the test statistic, in this case, is going to be a t -statistic, because we have a fairly small sample size.
00:43
The t -statistic is going to be calculated by taking sample mean 1 minus sample mean 2, divided by the square root of sample variance 1 over sample size 1, plus sample variance 2 over sample size 2.
01:01
And i'll note, i use some shorthand here, i call variance 1 over n1 a, and variance 2 over n2 b, because that is very useful for expressing our degrees of freedom.
01:12
We have that the degrees of freedom is going to be equal to a plus b, all squared, divided by a squared over n1 minus 1, plus b squared over n2 minus 1.
01:26
So finding our degrees of freedom, pardon me, first thing that i'll do is calculate out a and b, so that's variance over sample size, so it would be 362 .5, oops, 362, 2 .5 squared, divided by 53 for a, so 2479 .36, and for b that's going to be 687 .15 squared over 21, oops, i need to actually assign that to b, so that's 22 ,484 .5, roughly, if i can actually get my software to behave, there we go.
02:03
And the degrees of freedom will then be a plus b squared, divided by a squared over n1 minus 1, so that's 53, or pardon me, 52, plus b squared over n2 minus 1, so that's 20.
02:25
So we get 24 .54 degrees of freedom, roughly.
02:32
Since we're doing a left tail test here, our rejection region is going to be the set of t values less than the critical, or the negative t value for a one tail proportion of alpha equals 0 .05.
02:47
So we'll just find that using my software here, and we'll do inverse cdf, student t distribution, 24 .53 degrees of freedom, 0 .05 to the left.
02:57
So we reject if our observed test statistic is less than negative 1 .7094.
03:05
Then we'll plug our values into the formula for the test statistic and evaluate.
03:10
So sample mean 1 is 636 .53, sample mean 2 is 1245 .13, then we divide that by, well, just using my shorthand, square root of a plus b.
03:26
So we can see that our t statistic is roughly negative 3 .85, which is less than the critical value.
03:36
It is in the rejection region.
03:41
Therefore, we will reject the null hypothesis and support the claim that higher education is associated with an increase in income for black men.
03:52
Now as i said, for the other parts of this, basically we're just doing the same thing over and over again.
03:58
So i'll focus on putting down the results here.
04:01
So for part b, we have basically the exact same hypotheses, and i'll show you how to use a ti -84 here to find the, or to basically quickly run our tests.
04:15
So we can, actually i'll explain what i'm doing precisely here.
04:21
So we hit stat, go across two tests, and we want a two sample t test.
04:30
We input the stats.
04:34
We have x bar 1 is, now this is for the white men, that's 855 .23.
04:40
Sx 1 is 522 .68.
04:48
Sample size 1 is 588...