6. Describing Relationships: Correlation

Question 1

Positive Relationship

Answer 1

Occurs insofar as pairs of scores tend to occupy similar relative positions (high with high and low with low) in their respective distributions.

Question 2

Negative Relationship

Answer 2

Occurs insofar as pairs of scores tend to occupy dissimilar relative positions (high with low and vice versa) in their respective distributions.

Question 3

Indicate whether the following statement suggests a positive or negative relationship:

More densely populated areas have higher crime rates.

Answer 3

Positive. The crime rate is higher, square mile by square mile, in densely populated cities than in sparsely populated rural areas.

Question 4

Indicate whether the following statement suggests a positive or negative relationship:

Schoolchildren who often watch TV perform more poorly on academic achievement tests.

Answer 4

Negative. As TV viewing increases, performance on academic achievement tests tends to decline.

Question 5

Indicate whether the following statement suggests a positive or negative relationship:

Heavier automobiles yield poorer gas mileage

Answer 5

Negative. Increases in car weight are accompanied by decreases in miles per gallon.

Question 6

Indicate whether the following statement suggests a positive or negative relationship:

Better-educated people have higher incomes.

Answer 6

Positive. Increases in educational level - grade school, high school, college - tend to be associated with increases in income.

Question 7

Indicate whether the following statement suggests a positive or negative relationship:

More anxious people voluntarily spend more time performing a simple repetitive task.

Answer 7

Positive. Highly anxious people willingly spend more time performing a simple repetitive task than do less anxious people.

Question 8

Scatterplot

Answer 8

A graph containing a cluster of dots that represents all pairs of scores.

Question 9

Linear Relationship

Answer 9

A relationship that can be described best with a straight line.

Question 10

Curvilinear Relationship

Answer 10

A relationship that can be described best with a curved line.

Question 11

Correlation Coefficient

Answer 11

A number between -1 and 1 that describes the relationship between pairs of variables.

Question 12

Pearson Correlation Coefficient (r)

Answer 12

A number between -1.00 and +1.00 that describes the linear relationship between pairs of quantitative variables.

Question 13

Supply a verbal description for the following correlations.

an r of –.84 between total mileage and automobile resale value

Answer 13

Cars with more miles tend to have lower resale values.

Question 14

Supply a verbal description for the following correlations.

an r of –.35 between the number of days absent from school and performance on a math achievement test

Answer 14

Students with more absences from school tend to score lower on math achievement tests.

Question 15

Supply a verbal description for the following correlations.

an r of .03 between anxiety level and college GPA

Answer 15

Little or no relationship between anxiety level and college GPA.

Question 16

Supply a verbal description for the following correlations.

an r of .56 between age of schoolchildren and reading comprehension

Answer 16

Older schoolchildren tend to have better reading comprehension.

Question 17

Speculate on whether the following correlation reflects simple cause-effect relationships or more complex states of affairs.

caloric intake and body weight

Answer 17

Simple causal-effect

Hint: A cause-effect relationship implies that, if all else remains the same, any change in the causal variable should always produce a predictable change in the other variable.

Question 18

Speculate on whether the following correlation reflects simple cause-effect relationships or more complex states of affairs.

height and weight

Answer 18

complex

Hint: A cause-effect relationship implies that, if all else remains the same, any change in the causal variable should always produce a predictable change in the other variable.

Question 19

Speculate on whether the following correlation reflects simple cause-effect relationships or more complex states of affairs.

SAT math score and score on a calculus test

Answer 19

complex

Hint: A cause-effect relationship implies that, if all else remains the same, any change in the causal variable should always produce a predictable change in the other variable.

Question 20

Speculate on whether the following correlation reflects simple cause-effect relationships or more complex states of affairs.

poverty and crime

Answer 20

complex

Hint: A cause-effect relationship implies that, if all else remains the same, any change in the causal variable should always produce a predictable change in the other variable.

Question 21

Correlation Coefficient (Computation Formula)

Answer 21

r = (SPxy) / √(SSx SSy)

Where

SSx = ∑X² – [(∑X)²) / n]

SSy = ∑Y² – [(∑Y)²) / n]

SPxy = ∑XY – [(∑X)(∑Y) / n]

Question 22

Sum of Products (Definition and Computation Formulas)

Answer 22

SPxy = ∑(X – ̅X) (Y – ̅Y)

SPxy = ∑XY – [(∑X)(∑Y) / n]

Question 23

Couples who attend a clinic for first pregnancies are asked to estimate (independently of each other) the ideal number of children. Given that X and Y represent the estimates of females and males, respectively, the results are as follows.

Couple	X	Y
A	        1	2
B	        3	4
C	        2	3
D	        3	2
E	        1	0
F	        2	3

Calculate a value for r, using the computation formula.

Answer 23

SSx = ∑X² – [(∑X)²) / n]
=> 28 – 144/6 = 4

SSy = ∑Y² – [(∑Y)²) / n]
=> 42 – 196/6 = 9.33

SPxy = ∑XY – [(∑X)(∑Y) / n]
=> 32 – 168/6 = 4

r = (SPxy) / √(SSx SSy)
=> 4 / (2√9.33) = 0.65

Question 24

Give the 11 steps of the computational sequence for the calculation of r

Answer 24

1. Assign a value to n, representing the number of pairs of scores.
2. Sum all the scores for X.
3. Sum all the scores for Y.
4. Find the product of each pair of X and Y scores, one at a time.
5. Then add all of these products.
6. Square each X score, one at a time.
7. Then add all squared X scores.
8. Square each Y score, one at a time.
9. Then add all squared Y scores.
10. Substitute numbers into formulas and solve for SPxy, SSx, and SSy.
11. Substitute into formula and solve for r

Question 25

Correlation Matrix

Answer 25

Table showing correlations for all possible pairs of variables.