The following is a solution to number nine and were given that there was a random sample of 40 noticed they didn't tell you what the population distribution was, because it doesn't matter because that that uh sample size of 40 is big enough, where the samples of the population doesn't matter, The distribution of the population doesn't matter. So, um if If it had been like 25, for example, though you would need to have a population that's normally distributed, but in this case 40 is big enough, so we don't care about that. And uh were also given the The sample mean of 108.5 and a sample standard deviation of 17.9. And we're supposed to test at the 5% level of significance If the population mean is greater than 100. All right, So, first off, let's say what kind of tests for using? We're going to use the T test because we are given the sample standard deviation, not the sigma. So, if you're given the sigma, if you're given the population standard deviation, then you can use the Z test. But since you're not given the population standard deviation, you're only given the sample standard deviation, you have to use that T test. So we're gonna do a five step process ah five step hypothesis test. Where the first step, you state your hypothesis and the null hypothesis always has something to do with uh equality. So we're saying that the mean is equal to 100, so it's always the uh the parameter equaling whatever they say it is, And then the alternative is you kind of have to do a little bit of reading here, but it says, is there enough evidence at the 5% level of significance that the mean is in fact greater than 100. The population mean is greater than one. That's why we said great of him. Steps two and three are basically the same thing. Well, not the same thing, but you find them the same way whenever you have technology. So I'm going to do these kind of together the T stands for the test statistic. And there is a formula in your book. You can certainly use, I'm gonna use technology because it goes by much more quickly and it's easier to see. And then uh p value is it stands for probability. And that's also found with technology. Usually you can use a chart as well, but I think technology is a little bit easier. So I'm gonna use the t. because I think it works quite nicely. And if you go to stat an air over two tests, we're going to go to the second option here. The T test. So it's a T. Test. And summary stats needs to be highlighted. The stats is there and then you just start filling in your stuff. So um you not, that's the hypothesized value, that's your null hypothesis, and that was 100. The X. Bar. Is your sample mean? And remember that was one of 8.5 S sub x stands for the sample standard deviation set 17.9 in this case, And then your sample sizes the ends, that's 40. And then you get to this alternative hypothesis here and we need to switch this over to greater than mu not so it's greater than the null hypothesis. And then whenever we calculate that gives us basically everything we need. So there's a lot of good stuff here, but the T. And the P. Or what I really care about. So the t. value is about 3.003, and we have a really small p value of .002. So let's write those down and then we'll talk about them. So the T. Value Is 3.003 And then the P value is .002. So what we do in step for because the p value really is the most important thing. We explicitly compare the p value with the alpha value. So I have a P value of 0.2 and I'm comparing that 2.5 and that is less than five. So I have a P value that's less than alpha. And any time the p values less than alpha, you reject the null hypothesis. So we're rejecting H. Not now had the p value been greater than alpha than we would fail to reject H. Not so there's not enough evidence to say that me was greater than 100. But since the P values less than alpha, we are rejecting this null hypothesis, meaning we're accepting this alternative hypothesis as true. So a step five we basically just summarize that. So we're gonna say there is because we're rejecting there is sufficient evidence to suggest That the population mean I'll just say μ is in fact greater than 100. So since that we rejecting the null hypothesis, there is sufficient evidence to suggest that μ is greater than 100. That's the five step hypothesis test for a T. Test whenever sigma is unknown.