The following is a solution to # 12. Let me change that real quick number 12. And we're using a hypothesis test, we're testing that the population mean difference is either negative four or less than negative four. And we're testing at the 5% level of significance. And we have this these summary stats here, so 40 25 80 87 5 and five for the sample size sample mean and sample standard deviation.
Now we're gonna use a two sample T test. The reason why we have to use the T test is because one of these sample size is not at least 30. So because of that, we have to use to sample T. Test even though this 40 is bigger, but it's 25 is not. So here, the to form is going to use T equals pretty self explanatory.
It's very, very typical. The only thing is that's a little weird is this pooled variance here, so S A p squared is in one minus one times the sample variance plus en tu minus one times that sample variance divided by the degrees of freedom. And we're going to use in one plus into a minus two as the degrees of freedom. So the two sample sizes added together uh minus two. So those are degrees of freedom to keep that back your head here.
So, so let's find the uh let me change colors first. So let's find the S. A P squared first. And the way we do that again, we take the sample size minus 1 to 39 times this variant. So that would be five squared Plus this minus ones.
That's 24 times of five squared And that's divided by 40 plus 25 minus two. So 40 plus 25 minus two. That 63. So we have 63 degrees of freedom here. And whenever you plug that in the calculator I got 25.
25. Is the the variance there. So so we're gonna use that for the T. Value for the test statistic. And the way we find that is we take the two sample means Well I already have formula.
So I'm just gonna use the numbers so 80 minus 87. So that's the two sample means subtracted and then minus the hypothesized subtraction or hypothesized difference and that's divided by square root of 25 Times one over N one so 1/40 Plus one over into so 1/25. Okay and whenever you plug that in the calculator you should get negative two point 353. Okay so negative two point 353 is my test statistic. And so now I need to find the p value.
So the p values the probability on left tail test. It's the probability that I get a sample statistic less than the actual value that I got. So if I go to second and you can get this at the table but I like to use the calculator, second vars and go to this T CDF. And the lower bound since this is the left tail test is gonna be negative infinity. Um so I'm gonna use negatively 99 because it's the largest negative number I can get.
And then the upper bound will be that test value that I have some negative 2.353. Okay. And like I said, there are 63° of freedom here. So then whenever I paste I get .011 whenever you're around. So .011 is my p value.
And the alpha value. If you look back what we do with that p values we explicitly compared to the alpha. And so we're comparing it to point oh five and it is in fact less than point oh five. And any time you p values less than the alpha value the significance level we reject H nine. So there is enough evidence to suggest that that difference is less than -4.
Okay, so number uh I guess letter B or part B same thing. But this time we have a two tailed tests. So we're testing that it's 21 or not equal to 21 at the 1% level of significance. So we start this out the same way. Really only difference here is on that p value you need to multiply by two to account for both tails but I still I'm going to find the pooled variance And that's going to be 14, which is that 15 -1 Times this 12 squared plus 33 Which is 34 -1 times this squared eight squared.
And then that's divided by the 15-plus 34. But then -2. Okay, so whenever you put that in the calculator you should get about 87.83. Okay, so 87.83. That's the pooled variance.
I use that to get the test statistics. So t equals the mean differences of 1 92 minus 1, 80 minus the hypothesized difference which was remember 21 minus 21. And then that's divided by the square root of S. And P square which is at age seven 0.83 Times 1/15. So one over the sample sizes and then 1/34.
Okay. And then when we plugged that in you should get negative 3.098. So that's my test value. So then I need to find the probability that we get a sample statistic less than that negative 3.098. But also greater than positive 3.098.
So what I need to do here is I just need to find one of the tails. And what I'm gonna do is find the probability using that T. C. D. F.
Again. So the 6th option that I get less than negative 3.098. And this was 47° of freedom. So 15 plus 34 -2 is 47. And I get this.
But then I just have to remember to multiply by two to account for that other side. So at times too, and I get .003. Now in the end it doesn't matter. I'm gonna reject anyway, But to get a perfect p value, it should be .003 and that is less than alpha. So once again we reject that an old hypothesis, meaning there is enough evidence to suggest that the The mean difference, the population mean differences, not 21..
