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2 5. You will be prompted for the following information: Xlist: Enter the name of the list containing the predictor (x) variable. For this example, type OLYR and press e. Ylist: Enter the name of the list containing the response (y) variable. For this example, type JUMP and press e. β & ρ: Select the sign that appears in your alternative hypothesis. (Consult your lecture notes for more details regarding the hypothesis testing procedure for linear correlation.) We are only concerned with ρ (rho), which is the population linear correlation coefficient. For the purposes of this course, we will always be using ρ 0 as our alternative hypothesis when testing whether there a significant linear correlation between x and y. Select 0 and press e. RegEQ: Here you can specify one of the built-in Y= functions as a place to store the regression equation that is generated. Doing so will allow you to add the regression line to your scatterplot later. (Note that this same line will be drawn on all subsequent plots, too, unless you press! and C to clear the equation from memory when you are finished working with this data set.) I generally store my regression equations as Y1. Press v and scroll right to Y-VARS and press e to select 1:Function. Then, press e again to select 1:Y1. These keystrokes will enter Y1 into the RegEQ field. Highlight Calculate and press e. 6. Your calculator will return two screens full of output; use the ; and : keys to scroll through all of the output. Let s start by finding the linear correlation coefficient (r) for our data. You will need to scroll down to the bottom of the second screen to find r. For our example, r = (Always round r to 3 decimal places.) What does r tell us? First of all, its sign tells us that there likely is a positive correlation between Olympic year and the winning men s high jump height. The slope of the regression line will also be positive. Second, because r is very close to 1, we can expect that there is a near-perfect positive correlation. 7. Is there a statistically significant linear correlation? That is, can we conclude that the population linear correlation coefficient (ρ) is not equal to 0? The linear regression t test addresses this question. Given how large r was for our data, it is a safe bet that we have a significant linear correlation between OLYR and JUMP. You would report the results of the t test for this example as t 23 = , P = 6.47 x (two-tailed). Note that I reported the degrees of freedom as a subscript (df = n - 2). Round the t-test statistic to 4 decimal places and the P-value to 3 significant figures. Given that the P-value is less than the significance level of α =.05, we can conclude that there is a significant linear correlation between the Copyright 2007 by Laura Schultz. All rights reserved. Page 2 of 5

3 Olympic year and the winning men s high jump height. (I present the formal hypothesis test at the end of this handout.) 8. The coefficient of determination (r 2 ) tells us how much of the variability in y can be explained by the linear correlation between x and y. By convention, r 2 is reported as a percentage. For our example, r 2 = 94.4%. Round r 2 to 3 decimal places. What does this mean? 94.4% of the variation in men s winning high jump heights can be attributed to the linear relationship between the Olympic year and the winning high jump height. 9. Given that we had a significant linear correlation, it is appropriate to construct the linear regression equation for our data. There are two methods for doing so. First, note that the previous calculator displays indicate that y = a + bx. Your calculator reports values for both a (the y-intercept) and b (the slope). The second method is to press! and record the equation given for Y1. In either case, round the y- intercept and slope values to 3 significant figures each when you report the linear regression equation. The linear regression equation for our sample data is ŷ = x. In situations where there is not a significant linear correlation, do not bother constructing a linear regression equation. (Without a significant linear correlation coefficient, we cannot make predictions from a regression equation. The best estimate of ŷ for any corresponding x value is simply ȳ if there is no linear relationship between x and y.) 10. Let s plot the data again and see what it looks like with the regression line. Press #9. Your calculator will return the scatterplot with the regression line in place. Note how well the regression line fits our data. The stronger the linear correlation, the closer the data points will cluster along the regression line. 11. What is the marginal change in men s high jump performance per year? Marginal change is simply the slope of the regression line. Hence, the marginal change for our example is inches/year. In other words, men s high jump performance improves by inches per year. 12. Making predictions from a regression equation. Let s use the regression equation to predict what the winning high jump height would have been in 1940 had the Olympics been held that year. Once again, there are several approaches you can use. Let s start by working off the scatterplot display. Press \$ and then ; to hop from the data points onto the regression line. Type in the x value you want to plug into the regression equation and press e. For this example, recall that the Olympic year is coded as the number of years since Therefore, type 40 and press e. Your calculator will return the display shown to the right. The predicted y value for an x value of 40 is reported on the lower right-hand corner of the display. Round your predictions to 3 significant figures. Hence, we predict that the Copyright 2007 by Laura Schultz. All rights reserved. Page 3 of 5

4 winning men s high jump would have been 80.7 inches had the Olympics been held in Another approach is to work from the home screen. Press `M to exit the plot display. Then, press v>11 to paste Y1 to the home screen and type (40)e. Doing so will tell your calculator to plug x = 40 into the regression equation stored for Y1. Your calculator will return the output screen shown to the right. Once again, you would round to 3 significant figures and report that you predict that the winning Olympic men s high jump height in 1940 would have been 80.7 inches. 14. The final approach is to write down the equation given for Y1 and plug 40 into the equation manually. If you choose to the third approach, make sure you use all the given digits and round only at the end. I strongly advise that you adopt one of the first two approaches for making predictions. 15. When you are finished working with this data set, don t forget to press! and C to clear the regression equation for Y1 from memory. Forgetting to do so will cause you problems the next time you try to generate a stat plot. Here is the formal hypothesis test for this example: Claim: There is a linear correlation between the Olympic year and the winning men s high jump height. Let ρ = the population linear correlation coefficient H 0 : ρ = 0 (There is no linear correlation between the Olympic year and the winning men s high jump height.) H 1 : ρ 0 (There is a linear correlation between the Olympic year and the winning men s high jump height.) Conduct a two-tailed linear regression t test with a significance level of α =.05 Reject H 0 because our P-value is less than α (6.47 x <<<<.05) Conclusion: There is a significant linear correlation between the Olympic year and the winning men s high jump height (t 23 = , P = 6.47 x 10-16, two-tailed). Copyright 2007 by Laura Schultz. All rights reserved. Page 4 of 5

5 Appendix: Data Sets The Olympic year has been coded as the number of years since Men s winning high-jump heights are given in inches. Heights are given in inches. Women s shoe sizes have been converted to men s sizes by subtracting 1.5 from the reported shoe sizes. Copyright 2007 by Laura Schultz. All rights reserved. Page 5 of 5

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