5. Calculating the Pearson correlation and the coefficient of determination Aa A

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5. Calculating the Pearson correlation and the coefficient of determination Aa Aa Suppose you are interested in the role of exercise in maintaining cardiac health among young adult women between the ages of 25 and 35. You ask 50 patients to record the number of days they exercise in a month to see whether frequency of exercise correlates with systolic blood pressure You decide to use the computational formula to calculate the Pearson correlation between the number of days of exercise in a month and the systolic blood pressure. To do so, you call the number of days of exercise in a month X and the systolic blood pressure Y. Then, you add up your data values (EX and Y), add up the squares of your data values (EX2 and 2), and add up the products of your data values (EXY). The following table summarizes your results × 300 Y 6,000 2X2 2,170 Y2 728338 XY 35,693 , The sum of squares for the number of days of exercise in a month is SSx - The sum of squares for the systolic blood pressure is ssy- The sum of products for the number of days of exercise in a month and the systolic blood pressure is SP = The Pearson correlation coefficient is r= Suppose you want to predict the systolic blood pressure from the number of days of exercise in a month among young women. The coefficient of determination is r2 systolic blood pressure can be explained by the relationship between the systolic blood pressure and the number of days of exercise in a month , indicating that of the variability in the When doing your analysis, suppose that, in addition to having data for the number of days of exercise in a month for these young women, you have data for the number of days they did not exercise in a month. You'd expect the correlation between the number of days they did not exercise in a month and the number of days of exercise in a month to be systolic blood pressure to be and the correlation between the number of days they did not exercise in a month and the

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Answers

Preparatory work

Xbar = (?x)/n = 300/50 = 6; Ybar = (?y)/n = 6000/50 = 120

1. Sum of squares for x: Sxx = ?x2 – (n.Xbar2) = 2170 – (50 x 6 x 6) = 370 ANSWER 1

2. Sum of squares for y: Syy = ?y2 – (n.Ybar2) = 728338 – (50 x 120 x 120) = 8338 ANSWER 2

3. Sum of products for x and y: Sxy = ?xy - (n.Xbar.Ybar) = 35693 – (50 x 6 x 120) = - 307 ANSWER 3

4. Pearson correlation coefficient, r = Sxy/?(Sxx.Syy) = - 0.1738 ANSWER 4

5. coefficient of determination = r2 = 0.0306, indicating that 3.06% of the variability in systolic blood pressure can be explained by the relationship between systolic blood pressure and the number of days of exercise in a month.

6. Since number of days in a month is constant, as the number of days of no exercise in a month increases, the number of days of exercise in the month decreases. Hence, correlation coefficient between the number of days of no exercise in a month and the number of days of exercise in the month is expected to be negative.

7. Further, since correlation coefficient between systolic blood pressure and the number of days of exercise in a month is negative and correlation coefficient between the number of days of no exercise in a month and the number of days of exercise in the month is expected to be negative, it is to be expected that correlation coefficient between systolic blood pressure and the number of days of no exercise in a month is positive.

DONE

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