 Hello, this is a video about one sample hypothesis testing, testing a mean. Test to claim that mean GPA of night students is significantly different than 3 at the .02 significance level. Based on a sample of 65 students, the sample mean GPA was 2.95 with a standard deviation of .05. So we're going to go through and run the hypothesis test here. First we'll state the hypotheses. In this case, we do know that we are dealing with a mean, mu. The null hypothesis will always be equal to, and the number I'm dealing with is 3. So mu equal to 3 is equal, is the null hypothesis. The alternative, I'm dealing with significantly different, which means not equal to. So my claim is the null hypothesis here. It's important to note that we are dealing with a two-tailed test. That's actually the answer to my next question. We are two-tailed. Then I want to know the test statistic. Well, you have to note that you are dealing with a t-distribution here, and why is that? That's because the population standard deviation sigma is not known. You know the sample standard deviation, but not the population standard deviation. So for the test statistic, you can either use the test statistic formula and calculate it by hand, or you could use Google Sheets. Either way, it's perfectly fine. Now, some information you need to know for Google Sheets is first X bar. The sample mean is equal to 2.95. The sample standard deviation is equal to 2, sorry, is equal to 0.05. And is equal to 65, that's my sample size. The population mean value in question here is 3. And then the type of test I have is two-tailed. So the alternative hypothesis has not equal to. This is the information to put in the Google Sheets to get the test statistic. So in Google Sheets, you will go to the data list tab, and you're going to go to this one variable CLP value t-distribution, you have to make sure you use the one that says t-distribution. So it turns out that our sample mean is 2.95, sample standard deviation 0.05, sample size is 65, mu naught is 3. And it looks like the sign of my null hypothesis is not equal to. This is all the information I need to put in. Once you do that, you'll notice that your test statistic is actually going to be negative 8.06. You have your p-value, too, but the question didn't ask for that. So negative 8.06, the two decimal places. All right, so test statistic t is equal to negative 8.06. And keep that in mind. So the next part of the question is they want you to find the positive critical value. Well, here's the issue. What we just did in Google Sheets only tells us the test statistic and the p-value. It does not tell you the critical value. We have to go to a different area to find the critical value. So we must go to a different tab in Google Sheets. So remember that this is a two-tailed test. This test is two-tailed. So that means in my Bell curve, I have two tails. And you have two critical values as a result. And you're interested in the one that is positive. Now remember when finding critical values, so I have a positive one and I have a negative one. They are t-values because I'm using a t-distribution here. Alpha, which is 0.02, is the area, the total area of the tails. So since I have two tails, I need to split up 0.02 and divide it by 2. To get 0.01. So my first, my right-hand tail has the area of 0.01. Left-hand tail has the area of 0.01 as well. I need the area to the left to find my positive critical value. So area to the left. My area to the left is going to be 1 minus, area to the right, 1 minus 0.01. That would be 0.99. So I know when I go to Google Sheets, my area to the left is 0.99. Since I'm dealing with a t-distribution, it's important that I know my degrees of freedom. What you need to know about degrees of freedom in this case is that it's always a sample size minus 1. So in this case, that would be 65 minus 1, which is 64. So in Google Sheets, we'll go to the Compute tab. This is the only way you can find critical values for hypothesis testing is under the Compute tab. The only thing you have to put in is degrees of freedom in left-tail area. That's all that's required for you to put in here. So your degrees of freedom in this case are going to be 64. And then your left-tailed area is actually going to be 0.99. And this is going to give me a critical value of about 2.39. 2.39 is my critical value. And my negative critical value would actually be negative 2.39. So now, based on this, I can actually come to a conclusion. I can actually come to a conclusion. Rather than taking the p-value and comparing it to alpha, I can actually take the critical value and compare it to the test statistic. So my critical value is 2.39. The tail located beside 2.39, this is called my rejection region. If my test statistic is in the rejection region, then I reject the null hypothesis. In this case, I have a two-tailed test, so I have two rejection regions. I have 2.39, and what's to the right? And then negative 2.39, and what's to the left? So what was the test statistic in this case? Well, recall that our test statistic was negative 8.06. Negative 8.06. So negative 8.06 would be somewhere over here in this shaded region on the left tail. Well, guess what? Since the test statistic lies in the rejection region, we reject the null hypothesis. We reject H0. That's because once again, the test statistic did lie within this rejection region here, the critical region as it's called. All right, so we rejected H0. So reject H0. I remember our claim was down here in the alternative hypothesis. This was our claim. So since I rejected H0 and the claim does not include equality, that puts me here in this first row. The conclusion is there is sufficient evidence to support the claim that and then whatever your claim is. So in this case, there is sufficient evidence to support the claim that the mean GPA of night students is significantly different than 3. So that's how to conduct the hypothesis test when it requires you to find a critical value. Thanks for watching.