
AGE 
GPA 

AGE 
Pearson Correlation: 
1.000 
.031 
GPA 
Pearson Correlation: 
.031 
1.000 
a) It is close to zero. If
you plotted the results on a scatter diagram, you would not
see any relationship.
b) Nothing. If you correlating 50 pairs of random numbers,
almost 83% of the time you will get correlation coefficients
that are further from zero than .031. This is the sort of
result you would expect to get if the true correlation is
zero.
c) A negative sign indicates an inverse relationship. Higher
age means smaller GPA. But see a and b above.
d)
5. A researcher interested in what characteristics were correlated with student use of computerassisted lessons found a correlation of .4153 between the number of lessons a student took and the student's highschool gradepoint average.
a) Students who had high
highschool grades tended to use the computerassisted
lessons more and students with low highschool grades tend
to use the lessons less, but there were many exceptions.
b) The result does not appear to be random chance. It
appears that there is a real relationship here. However,
there is no information about what is cause and what is
effect.
6. We expect that as cars get older and have more miles, their value will decrease. We can check this expectation in a preliminary way with correlation. Taking a sample of Cadillacs listed for sale in the want ads of a large city paper, I fed their prices, ages, and mileage into a statistical computer program and got the following results:
PRICE 
MILES 
AGE 
YEAR 

PRICE 
Pearson Correlation: 
1.000 
.777 
.870 
870 
MILES 
Pearson Correlation: 
.777 
1.000 
.593 
.593 
AGE 
Pearson Correlation: 
.870 
.593 
1.000 
1.000 
YEAR 
Pearson Correlation: 
870 
.593 
1.000 
1.000 
a) 34. N is the number of
cars included in the study.
b) Because Age is found by subtracting the year of the car
from the current year. As the year of the car goes up, the
age goes down, a negative relationship.
c) Older cars tend to have more miles on them, a positive
relationship. As cars get older, their value decreases, a
negative relationship.
d) The results are not random. Cars really do lose value as
they get old. And newer cars really do have fewer miles on
them than older cars. Correlation shows something that was
obvious all along.
7. Suppose we are interested in the correlation between last year's income and this year's income for a group of people.



A 
$26,000 
$26,000 
B 
$40,000 
$20,000 
C 
$12,000 
$36,000 
D 
$20,000 
$20,000 
E 
$14,000 
$27,000 
a) No. If everyone earns 90%
of what they earned last year, the correlation will still be
one.
b) If the correlation is zero, there is no relationship
between what people earned last year and what they earned
this year.
c) It will be negative because the two who earned little
last year earned more than average this year, and the one
person who had a high income last year has one of the lowest
incomes this year.
d) If most people repeat their income from one year to the
next with a few exceptions, the correlation will be
positive, but the exceptions will drive us away from one. So
somewhere between .3 and .8.
8. Otto Mobile collects data from 25 cars. He then computes a correlation between weight of the car and its gas mileage. The answer he arrives at is 1.4. Interpret.
He made a mistake. Correlation cannot be less than 1.
9. Brandon Cumber takes 50 numbers from a table of random numbers. He calls the first 25 the X variables and the second 25 the Y variables and computes a correlation. Computing a tvalue from the formula t=(r0)/(standard error of p) he gets 2.102 with 23 degrees of freedom. He therefore claims that he has conclusively proven that either the procedure does not work as advertised or that the table of numbers is not random. Evaluate his claim.
About 5% of the time you will get a tvalue either greater than 2 or less than 2. By random chance he seems to have gotten one of those results that will happen one time in twenty. And in this case we can be sure it is random chance because he was using random numbers.
10. Doyle Zirkle is certain that there is a relationship between the kind of clothes people wear and their personalities. He has heard that correlation is a way of measuring how strong relationships are so he asks you to help compute a correlation to prove that he is correct. Assuming you want to help him, what should you say or do?
To compute a correlation coefficient, you have to be able to measure what it is that you are studying. How do you measure personality or clothing style? You might be able to classify them, which would allow you to use some other statistical procedure, such as Chisquare.
11. There is a relationship between X and Y on the graph below. However, the correlation coefficient will be close to zero. Explain why and illustrate on the graph.
It is important to always be aware that correlation measures linear relationships. There is a relationship here, but it is not linear.