So we we would be assuming that the mean of the diet coke is equal to the mean weight of the regular coke. And alternately we think that the diet coke is actually less than the regular coke. And we have our data that's given to us in the example of two sample sizes of 36. So really we're assuming that the difference between the two is zero, so that the mean of diet coke minus the mean of regular coke is zero. And then we're going to get a difference that's actually negative. That's what this is our actual difference. And we want to find what this likelihood is and this will be our p value. So let's find our test statistic, our test statistic which we're going to use the conservative estimate and find that 35 degrees of freedom. So we need to know what this test statistic is for for this specific value. So we're going to take our mean of the two, which is that 20.78479 minus the mean of the other, which is that 0.81682 And then we're going to divide that by the square root of And we're gonna take our standard deviation 439 squared divided by the sample size. And again this standard deviation 0.751 square divided by the sample size. And this gives us a test statistic that is negative 22.9 And this is very, very uh Mhm incorrect. As far as where I place that difference. That difference is like way down here in this distribution. So the likelihood of getting this type of test statistic for that difference, Mhm is approximately zero. So we have strong evidence to reject the null. We have strong evidence to reject the null and claimed that the diet coke does have a lesser weight, lesser weight than regular coke. So we have strong evidence of that. That doesn't matter what significance level you use. But we were supposed to use 5% and then it asks us in part B will give us the appropriate confidence interval. If we're going to use this idea. And if we were doing a confidence interval, we would need to use 5% of the lower tail, 5% of the upper tail. We'd actually need a 90% confidence interval to look at that. Not a 95% because we would want 5% of the air at the left tail, 5% at the other tail. And so on my table, I looked at my table and mind skips from 32 to 40. I don't have 35 degrees of freedom. And so I looked for the confidence interval number. So I would have 5% in the upper tail or you would have down at the bottom, it would say a 90% confidence interval and I have this value comes out to be 1.697 Now your book might be more accurate than mine. So if you can get more accurate you would and so we would take the difference that we got and so we would have the point 78479 minus the 0.81682 plus or minus. And then we would use our T star value for that. Now, if we use software will get a different value and then we still use that same standard deviation of the 0.439 squared over the sample size plus 0.751 squared over the sample size. And I'm going to leave that for you to do the calculation of finding what these two numbers are. But we'll find that it basically does not include zero. So these numbers are going to be different now. Part C. Asked why why would this happen? Well, the diet coke uses artificial sweetener versus the regular cope. He uses sugar and this must be more dense. So it's not that you're getting a lesser volume, you're just getting a lesser mass because of the sugar being dancer. Mhm. Okay.