 Hello, this is a video about linear regression or correlation performing a hypothesis test. A sample of 20 children was asked to draw a nickel. The diameter of each nickel was recorded as well as each child's family income. Incomes and thousands of dollars and a nickel and nickel diameters are given in the table. Test the claim of significant correlation at the 0.05 significance level. So first thing we're going to start off with is we're going to state our hypotheses here. So for any hypothesis test that deals with correlation, the null hypothesis is always rho. This is our population correlation coefficient is equal to 0. And alternative hypothesis is rho is not equal to 0. What this top hypothesis is saying there is no correlation and what this alternative hypothesis is saying is there is correlation, so keep that in mind. All right, let's calculate the correlation coefficient. And we do that using Google Sheets. You do not want to do it by hand. So in Google Sheets, we'll go to the regression tab and we're going to clear out any data that's already there. And then we need to type in the data. Now you can either try to copy the data over from the actual question itself. If that doesn't work out very well, though, I recommend taking the homework question data, pasting it in the Excel. I do know for a fact that will work. And then copy the data over column by column. So depending on your computer, it may be a little bit more difficult to do this than you think. But if you paste it in Excel, move it over column by column. I know for a fact that it will work for you. All right, so let's copy over our second column, paste it into B2. Make sure you leave A1 and B1 as X and Y. Those are your header columns, your headings for your columns. All right, so it looks like our correlation coefficient, R, is negative 0.297. Negative 0.297, that is R. Now I need to find my critical value. The best way to find your critical value is by looking at your critical value table for linear correlation. So I had 20 pieces of data. And the degrees of freedom would be n minus 2 or 20 minus 2. So let me write that down real fast. For this type of test, degrees of freedom is n minus 2, 20 minus 2, which is 18. So in our critical value table, we just send the data down to 18. And lucky there we have 0.444, 0.444, 0.444, 0.444, 0.444, 0.444, 0.444, 0.444, 0.444, 0.444. Next. Now, anytime you conduct this hypothesis test for correlation, your null is always rho equals 0. There is no linear correlation. Or rho is not equal to 0 is the alternative hypothesis, meaning there is linear correlation. So you can use your p-value alpha method if the p-value is less than alpha, we reject H naught. Or you could compare the correlation coefficient to the critical value. Let's first do the p-value alpha method. Well, I know alpha is 0.05. Now the p-value is, let me give this to you in your Google Sheets. So it's not that hard to find. p-value is about 0.2035, 0.2035. And it's clear to me that the p-value is definitely greater than alpha. So since it's greater than alpha, we will fail to reject the null hypothesis. We'll see what happens when we compare the correlation coefficient, the absolute value of it to the critical value. Well, first off, the absolute value of the correlation coefficient is the absolute value of negative 0.297, which is positive 0.297. So what I want to do is I want to compare 0.297 to 0.444. All right, so I reject the null hypothesis if my correlation coefficient is greater than the critical value. In this case, though, it looks like we are less than the critical value, which means we cannot reject. We fail to reject H0. So regardless, you get the same result no matter what method you use. So fail to reject H0. So that means all eyes are pointing to the fact that there is no linear correlation in this case. So our claim was that there was linear correlation, which means that there is not sufficient evidence to support the claim in this case. Now let's find the regression equation for this data. You get y hat equals AX plus B. That is the general format for a linear regression equation. A and B are given to you in Google Sheets. A is about 24.69. B is about negative 0.06. So let's just fill that in. Y hat equals negative 0.06X plus 24.69. To predict what diameter a child would draw a nickel given a family income of $35,000, it would be appropriate to use what? So in other words, can I just plug in this income of $35,000 into the equation and figure out how big the child would draw the nickel? And the answer is there is no linear correlation based on our test we just conducted. So we cannot use the regression equations, the equation that we just found. We cannot use it because there is no linear correlation. There's no evidence for it. So you cannot use the equation to make predictions. Instead, the best estimate anytime linear correlation doesn't hold true is to use the mean y value. In this case, that would be the mean coin size. That would be the best estimate for how big a child would draw the diameter of a nickel given a family income of $35,000. So that's how you conduct a hypothesis test for linear correlation.