Transcript:
Hello, this is Dr. Forever For Econ 4400 In this video, we are going to look at p-values and how to find p-values for hypothesis test on a slope parameter of a linear regression. We define the p-values as the smallest alpha alpha being the significance level of the test, so P-value is the smallest alpha for which we can reject a h0 so we look at the HTML of our test, the smallest offer for which we can reject Our hypothesis is called the p-value, so p-value is a probability like significance level of the test. This is also a probability, so we are usually faced with three options either. Our test is a right-tail test or a left-tail test or a two to a two-sided test For each of these three cases. We will look at how to find the p-value. Our first example is p-value for a right-tail test for a right-tailed test. The P-value is the area under the sampling distribution to the right of the test statistic of the test. So in order to memorize if it’s a right-tail test, p-value is to the right of the test statistic. So this is important here here. Our important value is the test statistic. They have to have the test statistic for a test in order to be able to find the p-value test statistic to remind you was. T which is beta hat minus. The hypothesized value of beta hat divided by standard error of it hat. So we need this number first. In order to find the p-value, then we go to the distribution sampling distribution curve and look at the area to the right of this value in order to get the p-value. If we are looking at the left tail test, the p-value is the area to the left of the test statistic, so we calculate the test statistic. The area to the left of that number would be the p-value foreign left tail test and for a two-tailed test, Both of those areas we need to consider both so the area to the right of the positive, absolute value of the test statistic and the area to the left of the negative absolute value of the test statistic together, those two areas will be the p-value for a test. So let me show this to you in a couple of examples. The first example is for a right tailed test and we are testing for the effect of education. Our model is a model for returns to education, so our model would look like this. We have the wage, which is equal to Beta 0 plus beta of education times education, plus other factors in the model. Now we are testing to see if the effect of Education on wage is larger than 45 or smaller, equal than 45 This is a right tailed test because our alternative is giving values to the right side of 45 Let’s do the test at fat 5% significance level. This is the estimated value for our model. X1 is the education here. These are the standard errors for us and the T ratio in order to keep the T ratio. We are looking at beta hat for the education. So X 1 here was the education variable. So this is the beta hat. 45 is the hypothesized value the value that we used in our HTML. NH alternative and standard error of Beta Hat for US is 8.2 to 2 If you calculate this ratio, this will be approximately point to 0-2 so I have the T ratio always remember. T ratio is essential to find a p-value for a test. Now go back to the p value. This is a right-tail test. P-value would be the area to the right of this number. So if this is my graph point, 202 will be somewhere here close to zero to the right of this value. Boy, look at the area. This whole area would be the p-value for this test now. Let’s look at the left tail test. I change the hypothesis so that it represents a left tailed test. You can see that. H Null is now beta, smaller than 45 I keep the rest of the information. The same. The T ratio would be the same because we have the same value for beta hat. The same value for hypothesized value of beta and standard error is the same so point zero 202 That’s the same T ratio now. This is a left-tail test on looking for the area to the left of the test statistic so on the graph again, point to 0-2 would be here now. The p-value would be the area to the left of this. This area now is the p-value to the left because our test was a left-tail test for a two-tailed test. Same example, the test is changed. The hypothesis is changed so that it is a two sided hypothesis. Now our alternative has Matthews to both side of the hypothesized value. The rest of the information is the same. We have the same. T ratio of point, two zero two p-value is the area to the right of positive 202 and to the left of negative 202 so we are looking at this area and also to the left, so both of these shaded areas together will constitute your p-value. So these two together are the p-value for this test. Now how you calculate the area? That’s a different story. We need solvers to calculate the areas for us, but if we want to show that on a graph. This is how we get the p-values.