Homework

Homework Assignment 3 Correlation

Complete the following Pearson correlation problem. Hints on the next page, formula below. A researcher wants to know if there is a linear relationship between body image and exercise habits. Data were collected on 10 college-age females and are given below. Calculate the Pearson r for the data. To get there, calculate SSxy, SSx, and SSy. Then find the coefficient of determination (r

2 ).

The body image score is a subscale score from a validated personality inventory. A score of 1 relates to a body image perception of ectomorphism (tall and skinny); a score of 10 relates to a body image perception of endomorphism (short and fat). The exercise habit score is the total minutes of aerobic exercise achieve per average week by the individual. BI EXER 3 44 9 36 6 37 7 30 2 69 5 30 1 80 4 65 10 0 8 20

  

     

  

  



  

  







 

n

y

n

x yx

n

yx xy

r 22

22

  SSySSx

SSxy r 

How to do a correlation (Pearson r)- 1. Put the data in columns (already done here). 2. Sum each column (add them up). 3. Create two new columns; one for X and one for Y. Label them X

2 and Y

2 .

4. Square each X and Y and write them in the new columns. 5. Create a fifth column called XY. 6. Multiply each X times its corresponding Y. Write these in the XY column. 7. Sum the X

2 , Y

2 , and XY columns.

8. Plug the numbers in the formula above and calculate the r. 9. Always calculate r

2 as well (by squaring the number you got for r).

Here’s a worked example (when you work yours, use the numbers on the first page, not these from the example!)-

X Y X 2 Y

2 XY x = 15

y = 14

1 2 1 4 2 x 2 = 55

2 1 4 1 2 y 2 = 48

3 3 9 9 9 xy = 50

4 3 16 9 12

5 5 25 25 25

= 15 14 55 48 50

First, calculate the numerator, which is called the ‘sums of squares of the cross- products’, or SSxy: Then calculate the left half of the denominator, which is called the ‘sums of squares for X’: Then calculate the right half of the denominator, which is called the ‘sums of squares for Y’: Finally, put them together in the formula and calculate r:

   n

yx xySSxy

  

   5

1415 50 SSxy = 8

  n

x xSSx

2

2   

  5

15 2

55SSx = 10

  n

y ySSy

2

2   

  5

14 2

48SSy = 8.8

SO,

  SSySSx

SSxy r 

  8.810

8 r = 0.85

If calculated the long way, it looks like this:

  

     

  

  



  

  







 

n

y

n

x yx

n

yx xy

r 22

22

  

      5

14

5

15

5

1415

22

4855

50



 r

=   8.80.10

8 =

38.9

8 = 0.85

Then we find r 2 by squaring r:

r 2 = 0.72