math

1. A researcher carefully computes
the correlation coefficient between
two variables and gets r = 1.23.
What does this value mean?

2. A error was made! All correlation
coefficients: −1 ≤ r ≤ 1.

3. It has been noted that there is a positive
correlation between the U.S. economy
and the height of women's hemlines
(distance from the floor of the bottom of a
skirt or dress) with shorter skirts
corresponding to economic growth and
lower hemlines to periods of economic
recession. Comment on the conclusion
that economic factors cause hemlines to
rise and fall.

4. A positive correlation does exist;
however, correlation does not imply
causation.

5. Given a set of paired data (X,Y)
a. if Y is independent of X, then
what value of a correlation
coefficient would you expect?
b. if Y is linearly dependent on X,
then what value of a correlation
coefficient would you expect?

6. a. r = 0.
b. r ≈ 1 or r ≈ −1 (these two
are same as |r| ≈ 1).

7. A researcher has a large number of
data pairs (age, height) of humans
from birth to 70 years. He computes
a correlation coefficient.
a. Would you expect it to be
positive or negative? Why?
b. What would you suggest to be a
major problem with this approach?

8. a. Positive since in general people grow
in height increasing with age.
b. The underlying data is not linear.
During the first few years of life, height
increases rapidly and irregularly. After
teenage years height is essentially
constant. A correlation coefficient is a
measure of the scatter about a straight
line. A better plan would be to restrict
the data set to children only.

9. A researcher wishes to test the idea that show size
and mathematical ability are correlated; that is,
people with larger feet have higher mathematical
skills. To test this he conducts a study of an entire
town of 2000 persons measuring their shoe size
and administering a math test.
He finds that there is a significant correlation
between shoe size and math skills with people with
larger feet having higher math skills.
What might an important problem with this
approach?

10. The problem is the "entire town."
This includes infants, children, as well
as adults. Clearly small children have
smaller feet and have not yet learned
as many math skills.
A more appropriate study would be
to include only adults or persons in a
specified age group; as an example,
25 to 65.

11. How does the correlation
coefficient relate to the slope
of the regression line?

12. The sign of the correlation coefficient
is the same as the sign of the slope,
but the magnitude of the correlation
coefficient is a measure of scatter
about the slope.