PSY 216. Assignment 15 Answers. What information is provided by the sign (+ or -) of the Pearson correlation?

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1 PSY 216 Assignment 15 Answers 1. Problem 1 from the text What information is provided by the sign (+ or -) of the Pearson correlation? The sign of Pearson s correlation tells us the direction of the relation between the variables. When the sign is positive, as the value of one variable increases, the value of the other variable tends to increase as well. When the sign is negative, as the value of one variable increases, the value of the other variable tends to decrease. 2. Problem 2 from the text What information is provided by the numerical value of the Pearson correlation? The numerical value, or magnitude of Pearson s correlation tells us the strength of the relation between the two variables. When the magnitude is close to 0, either the two variables are not related to each other or the assumptions of Pearson s correlation have been violated. As the magnitude approaches one, the ability to predict the value of one variable from the value of the other variable increases that is, the amount of error, on average, in the prediction decreases as the magnitude increases. 3. Problem 6 from the text For the following scores, X Y XY a. Sketch a scatter plot showing the data points ΣX = 10 M = 2 SSX = 8 ΣY = 15 M = 3 SSY = 18 ΣXY = b. Just looking at the scatter plot, estimate the value of the Pearson

2 correlation. Negative, large magnitude (around -.7) c. Compute the Pearson correlation. SP = ΣXY (ΣX ΣY) / N = 20 (10 * 15) / 5 = -10 r = SP / (SSX X SSY) = -10 / (8 X 18) = -10 / 12 = Problem 9 from the text With a small sample, a single point can have a large effect on the magnitude of the correlation. To create the following data, we started with the scores from problem 8 and changed the first X value from X = 1 to X = 6. (Problem 9 from the text) X Y XY a. Sketch a scatter plot showing the data points ΣX = 15 M = 3 SSX = 18 ΣY = 15 M = 3 SSY = 18 ΣXY = b. Just looking at the scatter plot, estimate the value of the Pearson correlation. Positive, medium magnitude (around.3) c. Compute the Pearson correlation. SP = ΣXY (ΣX ΣY) / N = 50 (15 * 15) / 5 = 5 r = SP / (SSX X SSY) = 5 / (18 X 18) = 5 / 18 = Problem 1 from the text Identifying individuals with a high risk of Alzheimer s disease usually involves a long series of cognitive tests. However, researchers have developed a Minute Screen, which is a quick and easy way to accomplish the same goal. The question is whether the 7-Minute Screen is as effective as the

3 complete series of tests. To address this question, Ijuin et al. (2008) administered both tests to a group of patients and compared the results. The following data represent results similar to those obtained in the study. (Problem 1 from the text) Patient 7-Minute Screen Cognitive Series 7-Minute Screen X Cognitive Series A B C D E F G H I ΣX = 63 M = 7 SSX = 98 ΣX = 153 M = 17 SSX = 236 ΣXY = 1198 a. Compute the Pearson correlation to measure the degree of relationship between the two test scores. SP = ΣXY (ΣX ΣY) / N = 1198 (63 * 153) / 9 = 127 r = SP / (SSX X SSY) = 127 / (98 X 236) = 127 / =.835 b. Is the correlation statistically significant? Use a two-tailed test with α =.01. Step 1: Specify the hypotheses and the α level H0: ρ = 0 H1: ρ 0 α =.05 Step 2: Determine the critical region df = n 2 = 9 2 = 7 From a table of critical values of t, find the two-tailed t with 7 degrees of freedom and α =.05. t = ± Step 3: Calculate the test statistic t = r2 df 1 r 2 = = =.015 Step : Make a decision Because the calculated t is in the tails cut off by the critical t, we can reject H0 and conclude

4 that the 7-Minute Screen and the Cognitive Series are likely correlated with each other. c. What percentage of variance for the cognitive scores is predicted from the 7-Minute Screen scores? (Compute the value of r 2.) r 2 =.835 *.835 =.697 Because r 2 is above 0.25, this is a large effect. 6. Use SPSS with the class data set to answer the following question: Is there a positive correlation between self-reported GPA and the number of points predicted in the class? Give H0, H1 and α. Is this a one-tailed or two-tailed hypothesis? Create a scatterplot. From the SPSS output, report the following: H0: ρgpa, predicted points 0 H1: ρgpa, predicted points > 0 α =.05 One tailed In SPSS, click on Graphs Chart Builder If a dialog box appears warning you to set the measurement level for each variable, click on OK. In the list of chart types, click on Scatter/Dot In the gallery of Scatter/Dot graphs, double click on the graph in the upper left corner (hovering the mouse pointer over it will give a tool tip of Simple Scatter ) Drag the variable you are trying to predict (say Points expected)) to the Y Axis? Box Drag the other variable (GPA) to the X Axis? Box Click OK

5 In SPSS, click on Analyze Correlate Bivariate Move the two variables (GPA and Points predicted) into the Variables box. Click the Options button Check the checkbox to the left of Means and standard deviations Click Continue In the Test of Significance section, select One-tailed Click OK Statistic Value

6 MGPA Mpredicted points r.065 p (In one tailed tests, remember to divide the twotailed p value by 2).38 / 2 =.17 gpa points gpa points ΣX ΣX SS SP = * 297 / 11 = r = SP / (SSX SSY) = / ( * ) =

7 7. Write a sentence or two in APA format that summarizes the results of the analysis. A correlation for the data revealed that the GPA and the number of points expected in a statistics class were not reliably related, r =.065, n = 38, p =.17, one tail.