The following is a solution to number four. And this says that a random sample of 19 people, I guess uh from a normal distribution. So that's important. The population distribution is normal. That makes that sample size okay. If it weren't normal, then we would need a sample size of at least 30. So the 19 is okay. And the sample mean of that particular uh Uh there is no context of this problem. So the sample mean of this sample is .8 and then the sample standard deviation is .4, and we're supposed to test that alpha equals .01 if the mean is less than one. Um so first off, because we don't know the population standard deviation, we were never given sigma, were only given the ass, we need to use the T test. Alright, We cannot use the Z test because we don't know sigma's we have to use the T test and it's a five step process, like I said. So first off, the uh the first step is to state our Nolan are alternative hypotheses. So the null hypothesis h not always has some sort of equality and we're testing a population means. So you're always gonna use greek symbols here. So mu is 1.0. And then we're testing if it's less than 1.0. So the alternative is that mu is less than 1.0. The 2nd and 3rd step, I'm actually going to write together because we're going to use technology to get it. So the second step is to find the test statistic and you can use a formula for that. But if you have a calculator, if you have, you know, Excel or some sort of software, um it does a lot of the work for you. So that's what I'm going to use just to save some time. So t is the test statistic and then the p value is also going to be given to me. So let's go to the calculator. Now I'm on ATI 84. And if you go to stat on the T- 84 and go down to tests, it's the second option here. The T test. And we need the summary stats here, so make sure summary stats is highlighted And the mute not, that's the hypothesized value. So in this case it's 1.0 or just one. So it's the null hypothesis. The X. bar is your sample mean, and that was given as .8 SX is the sample standard deviation, which was given us four, And then in is the sample size and that was given as 19. And then you get down here, and this is the alternative hypothesis. So it's either not equal to less than or greater than and if you look back that alternative is less than one. So that's all good. And now we can calculate and that gives you some stuff here. But really this is the T. And the P. Is what I need. So the T. Is negative 2.179. So let's go and write that down -2.179. and then the P value Is .02 0.2 And the p value really is probably the most important thing on that what you do. And step forward as you explicitly compare the P value with the alpha value. And remember the alpha value here is 1.2 is greater than one. And whenever the p value is greater than alpha than we fail to reject the null hypothesis. So failed to reject H. Not had that P value been less than alpha than you would reject H not, but since it's greater than alpha than you failed to reject, so any time you failed to reject your conclusion, which is the last step step is going to be something along this line, there is not sufficient evidence to suggest that The population means I'll just use mu is less than 1.0, so there is not sufficient evidence to say that the alternative hypothesis is true, so therefore we're for lack of a better phrase, were accepting that the null hypothesis is true.