 Hi, Eileen. This is a problem that's similar to the one that you were talking about where you're having a hard time coming up with the values for D bar and the standard deviation of D and Going through it the long way manually and using all the formulas That you need to solve for these things is doable, but it takes time and I'm what I'm making a suggestion to you that I would go for getting as many points as I can accurately in a short amount of time and Not satisfying for perfection. The reason I'm saying that is that Stat crunch does not give you the standard deviation of the D values Directly and you can calculate it using the data compute Function But essentially what you're doing you're just doing those long equations Using stat crunch and candidly you can do those long equations faster and excel than you can in stat crunch So I'll be looking at that. I want to get as many of these answers right in the shortest period of time as I have Your final exam and your quizzes are timed and so I would make that trade-off But let me show you The first thing you have to do of course is identify the claim and the null and the alternative the claim is That the scores improve the second time they took the test Okay When we're doing a paired sample or really any two sample tests, we're always subtracting the two means and we are Looking to to say for the null that there is no difference in the the means and The alternative is either that there is a difference or That mean one is greater than mean two or that mean two is greater than the mean one in this case Because we're saying the second test they got better scores that would mean that mean two is Greater than mean one which would mean the mean difference in use of d would be less than or equal to zero and the Null has to be the complement of that would be greater than equal to zero So the claim is that the scores improve the second time First thing you need to do is get the critical value and I want to just go ahead and click on the little icon there Open up that data and stat crunch So we've got our data But for right now we're just going to go and get the calculator We're using the t distribution. We always use the t for paired samples We need our degrees of freedom And we've got 14 pairs So the degrees of freedom would be the number of pairs minus one would be 13 our alpha is 0.01 because we've got a a left tail test remember our Alternative the operator is a less than symbol points to the left. That means it's a lower tail one tail test So we put the entire alpha in there of 0.01. I think that's 0.01. Yes and I click on compute and we get a critical t value of minus 2.650 Which is what they get and our Rejection region is that red area there anything um, any any t statistic standardized t statistics is greater than Excuse me less than minus 2.65 is in the rejection region and we would reject the null hypothesis So i'm going to close that for right now And what I want to suggest to you that would save so much time Is that instead of going through these these long calculations we need to get the test statistic and the D bar and we can get that using the stat t stats paired sample Paired sample unfortunately, and I told you always comes up with raw data We click on it Our first data is the first test Second test we can ignore the where and group by We can go ahead and save the differences Just in case we have time. I'll show you something We're going to do the hypothesis test Remember that when we're doing the hypothesis test, it doesn't matter if we use just the equal because if we're We're Significant for the equal then either the greater than side is also equal or if we're doing a A uh upper tail test the less than side is always significant What we do need to do is to get our alternative Correct and our mu sub d the difference Is less than zero because we're saying mu Two is greater than mu is one so we click on compute And we get this data And we get a couple of things there We get a value that's labeled the mean And that is the mean of the differences d bar So just remember that this mean of the differences is your d bar 39.786 and if we look over here in my stat lab minus 39.786 I'm not sure I said minus there But that mean that the hypothesis test gives us Is our d bar We also get our standardized test statistic minus 3.182 And there in our Uh question is the standardized test statistic correct minus 3.182 So we've gotten a lot right very quickly We don't have the standardized. I'm excuse me the standard deviation of the differences But um I would recommend pressing ahead we've got a p value That is a lot less than 0.01 So that tells us to reject the null hypothesis If we go down here in The problem It asks us to Determine do we reject or fail to reject The standardized test statistic minus 3.182 Falls below it's less than the critical area Critical value of minus 2.650 therefore it's in the critical Excuse me the rejection zone and that tells us also to reject the null hypothesis Then we've got to interpret it Choose the correct answer And the correct answer is at the 1 percent significance level 0.01 There is enough evidence that the critical reading scores improved the second time they took the test Now why is that we're rejecting the null The null Said that there Was no difference or in this case That the second Scores were not better than the first scores The claim is that the second scores are better So we go down here at the 1 percent significance level There is enough evidence that the scores improved the second time The reason this one is wrong Because it just says there's evidence the critic reading scores got worse second time which we prove the opposite And you can see those others So that's how you can solve everything except the standard deviation And but you can get the majority of the answers and get a pretty doggone good score in a much faster time I'll show you how to get the standardized Standard deviation in a shorter video