A researcher gathered data on psychology students’ ratings of their likelihood of attending graduate school and the numbers of credits they had completed in their psychology major (Rajecki, Lauer, & Metzner, 1998). Imagine that each of the following numbers represents the Pearson correlation coefficient that quantifies the relation between these two variables. From each coefficient, what do we know about the relation between the two variables?
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Is age associated with how much people study? How can we calculate the Pearson correlation coefficient for the accompanying data (taken from students in some of our statistics classes)?
Student | Age | Number of Hours Studied Per Week |
---|---|---|
1 | 19 | 5 |
2 | 20 | 20 |
3 | 20 | 8 |
4 | 21 | 12 |
5 | 21 | 18 |
6 | 23 | 25 |
7 | 22 | 15 |
8 | 20 | 10 |
9 | 19 | 14 |
10 | 25 | 15 |
We see from the scatterplot that the data, overall, have a pattern through which we could imagine drawing a straight line. So, it is safe to calculate the Pearson correlation coefficient.
Age (X) | (X − MX) | Hours Studied (Y) | (Y − MY) | (X − MX)(Y − MY) |
---|---|---|---|---|
19 | −2 | 5 | −9.2 | 18.4 |
20 | −1 | 20 | 5.8 | −5.8 |
20 | −1 | 8 | −6.2 | 6.2 |
21 | 0 | 12 | −2.2 | 0 |
21 | 0 | 18 | 3.8 | 0 |
23 | 2 | 25 | 10.8 | 21.6 |
22 | 1 | 15 | 0.8 | 0.8 |
20 | −1 | 10 | −4.2 | 4.2 |
19 | −2 | 14 | −0.2 | 0.4 |
25 | 4 | 15 | 0.8 | 3.2 |
MX = −1 | MY = 14.2 | ∑ [(X − MX)(Y − MY)]= 49 |
The numerator is 49.
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Age (X) | (X − MX) | (X − MX) | Hours Studied (Y) | (Y − MY) | (Y − MY)2 |
---|---|---|---|---|---|
19 | −2 | 4 | 5 | −9.2 | 84.64 |
20 | −1 | 1 | 20 | 5.8 | 33.64 |
20 | − 1 | 1 | 8 | − 6.2 | 38.44 |
21 | 0 | 0 | 12 | − 2.2 | 4.84 |
21 | 0 | 0 | 18 | 3.8 | 14.44 |
23 | 2 | 4 | 25 | 10.8 | 116.64 |
22 | 1 | 1 | 15 | 0.8 | 0.64 |
20 | −1 | 1 | 10 | − 4.2 | 17.64 |
19 | −2 | 4 | 14 | − 0.2 | 0.04 |
25 | 4 | 16 | 15 | 0.8 | 0.64 |
MX =21 | ∑(X − MX)2 = 32 | MY = 14.2 | ∑ (Y − MY)2 = 311.6 |
We now multiply the two sums of squares, then take the square root of the product of the sums of squares.