Correlation Homework

Question

A researcher is interested in how exercise is related to sleep. She asks people how many hours they exercise per day and how long they sleep per night. The data is as follows (the numbers represent hours for both X and Y):

Participant	Hours exercise per day	Hours sleep per night
A	2.5	7
B	1	5
C	2	8
D	1.5	8
E	2	9
F	2	8
G	3	10
H	1	9
I	0	6
J	2	8

a. Who are the groups being tested?

The groups being tested are

b. What are the null and research hypotheses?

c. What is the mean of X?

What is the mean of Y?

Find the deviances for both groups. (X – My) and (Y – My)

Multiply the deviances like in the lecture

Sum these.

What number do you get? ___________________________

This is the numerator

Square the deviances in X and Y

What is the Sum of Squares of X?

What is the Sum of Squares of Y?

What do you get when you multiply SSx and SSy? _______________________________

Take the square root of that ____________________________

This is your denominator

d. What is r? ______________________

e. What is the critical value? ___________________________________

f. What is your decision? ______________________________

Answer 1

This one walks you through building the Pearson correlation by hand, so let's follow the worksheet's own steps.

b. Hypotheses. Null: there is no relationship between hours of exercise and hours of sleep in the population (rho = 0). Research: there is a relationship (rho does not equal 0, two-tailed).

c. Means. Mean of X (exercise) = 17/10 = 1.7 hours. Mean of Y (sleep) = 78/10 = 7.8 hours.

Deviances and their products. For each participant, compute (X - 1.7) and (Y - 7.8), then multiply the pair. Summing those products gives SP = 6.9. That's your numerator.

Sums of squares. Square each X deviance and sum: SSx = 6.6. Same for Y: SSy = 19.6. Multiply them: 6.6 x 19.6 = 129.36. Square root: 11.37. That's your denominator.

d. r = 6.9 / 11.37 = .61 (rounded from .6067).

e. Critical value. Degrees of freedom = n - 2 = 8. For a two-tailed test at alpha = .05, the critical value from the Pearson r table is .632.

f. Decision. Our r of .61 does not exceed .632, so we fail to reject the null hypothesis. The correlation is moderately positive but not statistically significant with only 10 participants.

Here's the teaching point worth sitting with: the relationship looks fairly strong descriptively, yet we can't call it significant. Small samples demand bigger correlations before we can rule out chance, which is exactly why sample size matters so much in psychology research.