Interpretation of Pearson's r

Question 1

What are the 5 steps to NHST with Pearson’s r?

Answer 1

1) Create H0 and H1
- e.g. H0 = There will be no relationship between duration of illness and wellbeing score
H1 = There will be a negative correlation between the duration of illness and wellbeing score

2) Calculate Pearson’s r
- e.g. r = sample covariance / total variance

3) Obtain p-value using table
- e.g. Conditional probability of obtaining a value of r as low or lower than our value assuming the NULL is true

4) Reject or fail to reject H0
- e.g. The conditional probability of obtaining an r value as low or lower than ours assuming H0 is true is sufficiently small (p < 0.05) for us to reject the null hypothesis

5) Evaluate
- e.g. Our data suggests that there is evidence for a weak to moderate negative correlation (r = -0.358, p < 0.05) between illness duration and well-being. As illness duration increases, wellbeing decreases

Question 2

Every visiting alien from Ziltiodia 10 is hosted by an earth family. There is a theory that Ziltoidian health and wellbeing depends upon being in a happy environment. To examine this theory you conduct a correlational research study, hypothesising that the ZHI (Ziltoidian Health Index - a measure of Ziltoidian health and wellbeing involving both physical and mental health measures) will correlate with average score on the well-established Household Happiness Ratings (HHR) questionnaire. You collect data from 20 Ziltoidians and their host families. The standard deviations for ZHI and HHR scores are 4.81 and 6.85 respectively. The sample covariance between ZHI and HHR scores is 23.77.

a) This is a (…….)-tailed hypothesis

b) The Pearson’s r correlation coefficient based on this data is:

c) The correct row in the table to consult is marked:

d) The p-value associated with the Pearson’s r statistic from part b) is less than:

e) Based on this data you should (reject or fail to reject) the null hypothesis

Answer 2

a) 2
b) 0.721
c) 20
d) 0.01
e) Reject

Question 3

Every visiting alien from Ziltiodia 10 is hosted by an earth family. There is a theory that Ziltoidian health and wellbeing depends upon the average salary of the host family. To examine this theory you conduct a correlational research study, hypothesising that the ZHI (Ziltoidian Health Index - a measure of Ziltoidian health and wellbeing involving both physical and mental health measures) will correlate positively with average household earnings. You collect data from 23 Ziltoidians and their host families. The standard deviations for ZHI score and earnings are 3.52 and £10924.37 respectively. The sample covariance between ZHI score and earnings is 14503.97.

a) This is a (…….)-tailed hypothesis

b) The Pearson’s r correlation coefficient based on this data is:

c) The correct row in the table to consult is marked:

d) The p-value associated with the Pearson’s r statistic from part b) is less than:

e) Based on this data you should (reject or fail to reject) the null hypothesis

Answer 3

a) 1
b) 0.377
c) 23
d) 0.05
e) Reject

Question 4

What does Pearson’s r tell us about the relationship between variables?

Answer 4

Pearson’s r tells us the strength/direction of the relationship between variables

Question 5

What does the p-value tell us about the probability of Pearson’s r?

Answer 5

The p-value tells us the probability that this correlation coefficient could arise by chance assuming the null hypothesis is true

(i.e. when sampling from uncorrelated variables)

Question 6

True or False?

The p-value tells you the strength of the correlation

Answer 6

False

The p-value DOES NOT tell you the strength of the correlation

Question 7

True or False?

An extreme Pearson’s r tells you that there is necessarily a correlation between your variables

Answer 7

False

An extreme Pearson’s r DOES NOT tell you that there is necessarily a correlation between your variables

Question 8

True or False?

Finding a very large Pearson’s r does not mean there is evidence for a correlation

Answer 8

True

Question 9

You test a 2 tailed hypothesis for correlation with N = 8 & find Pearson’s r = -0.7, p > 0.05

What is the correlation suggested by r?

Is it sufficient evidence to reject the null?

Answer 9

The correlation suggested by r is strong

However, there is not sufficient evidence to reject the null

i.e. a value of r as extreme (or more extreme) is quite likely (p > 0.05) to have arisen by chance when sampling from 2 uncorrelated variables

Question 10

You test a 2 tailed hypothesis for correlation with N = 100 and find Pearson’s r = 0.27, p <0.01

What is the correlation suggested by r?

Answer 10

There is strong statistical evidence for a correlation

However, the correlation is still weak

Question 11

What is a weak correlation?

Answer 11

Even though it looks like the variables are correlated, the amount of share variance is relatively small

Even though there is some shared variance, there are potentially other variables than affect the variability of the initial variables

Question 12

What is the formula for Pearson’s r?

Answer 12

r = shared variance / total variance

Question 13

What does it mean when r=0?

Answer 13

No variance shared (zero correlation)

Question 14

What does it mean when 0.1 < r < 0.3?

Answer 14

Little variance shared (weak correlation)

Question 15

What does it mean when 0.4 < r < 0.6?

Answer 15

Some variance shared (moderate correlation)

Question 16

What does it mean when 0.7 < r < 0.9?

Answer 16

Most variance shared (strong correlation)

Question 17

What does it mean when r = 1?

Answer 17

All variance shared (perfect correlation)

Question 18

What is the r value when all variance is shared (perfect correlation)?

Answer 18

r = 1

Question 19

What is the r value when most variance shared (strong correlation)?

Answer 19

0.7 < r < 0.9

Question 20

What is the r value when some variance is shared (moderate correlation)?

Answer 20

0.4 < r < 0.6

Question 21

What is the r value when little variance is shared (weak correlation)?

Answer 21

0.1 < r < 0.3

Question 22

What is the r value when there is no variance shared (zero correlation)?

Answer 22

r = 0

Question 23

How do you calculate the variance shared between 2 variables?

Answer 23

r^2

Question 24

12.82% of the variance in scores on our 2 variables is shared

What does this imply about the variance between the 2 variables?

Answer 24

12.82% of the variance in variable x can be explained by variation in variable y (and vice versa)

87.18% is left unexplained

Simply = Only 12.82% of the variance of variables x and y are due to their influence on themselves.

The other 87.18% of the variance is due to other possible factors

Question 25

What is a measure of shared variance between two variables?

Answer 25

Covariance

Question 26

What does Pearson’s r depend on?

Answer 26

Covariance

Question 27

You carry out a correlational research design involving a sample of 20 participants. You are testing a 2-tailed research hypothesis and obtaining a Pearson’s r value of 0.41. You can report that you have evidence for a moderate correlation between your variables.

a) True
b) False

Answer 27

b) False

Question 28

You carry out a correlational research design involving a sample of 60 participants. You are testing a 1-tailed research hypothesis and obtaining a Pearson’s r value of 0.41. Given that your p-value is < 0.005 you can report that you have evidence for a strong correlation between your variables.

a) True
b) False

Answer 28

b) False