Stat401E
Fall 2010
Lab 10
1. Demand theory states that demand for a product depends
(among other things) on the price of the product and on
consumers' incomes. As the price of a product increases, demand for it
will decrease. As consumers' incomes increase, the demand for the
product will increase. You have data on the cigarette purchases and
the incomes of a random sample of 78 U.S. smokers of Marlboro
cigarettes. Your data were collected at random times over the last 30
years--years during which (even after adjusting for inflation) the
price of a pack of Marlboro cigarettes has varied considerably.
Your data set has the following three variables:
DEMAND
the average number of cigarette packages purchased daily
INCOME
annual income (adjusted for inflation)
PRICE
price of a package of Marlboro cigarettes (adjusted for
inflation)
Correlations among these three variables are as follows:
DEMAND
INCOME
PRICE
DEMAND
1.00
.45
-.35
INCOME
.45
1.00
.15
-.35
.15
1.00
PRICE
(Make sure you read these data correctly. For example, the zero-order
correlation between DEMAND and INCOME is .45 .)
a. What proportion of the variance in DEMAND is explained by the
linear effects of INCOME and PRICE? (Hint: The explained proportion
should include both variance explained separately by INCOME and
PRICE and variance explained in common by INCOME and PRICE.)
b. Of "the variance in DEMAND that is not explained by INCOME," what
proportion of this unexplained variance is explained by PRICE?
c. Find a 95% confidence interval for the first-order partial
correlation between DEMAND and PRICE while controlling for INCOME.
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2. Let
X = j + kW ,
where j and k are positive constants.
a. Show algebraically that the correlation between Y and X
equals the correlation between Y and W. (Hints: Begin
with the formula for r XY [Lecture Notes, p. 144].
Substitute "j + k W " for X and "j + k W " for X in this
formula. Then simplify.)
b. Show algebraically that when k is negative, r XY = – r WY . (Hints:
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See hints for part a, plus remember k  k , but instead k = k .)
c. Finally, show that r XY = r Z Z , where Z X and Z Y are standardized
X Y
forms of X and Y (respectively). (Hints: Begin with the formula for
. Then substitute Z X and Z Y in this formula with each's
expression in terms of an unstandardized variable, its mean, and
rZ
XZ Y
its standard deviation. [See page 137 in the Lecture Notes for such
an expression.]
Then simplify.)
d. Extra Credit: If the regression equation for the regression of Y
ˆ = â + bˆ W , derive the regression coefficients for the
on W equals Y
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regression of Y on X in terms of j, k, â 1 , and bˆ 1 . (Hints: Begin
with formulas for â 2 and bˆ 2 from the regression of Y on X.
Substitute "j + k W " for X and "j + k W " for X in these formulas.
Simplify into expressions including only j, k, â 1 , and bˆ 1 .)
3. The warden of a prison believes that his rehabilitation program does
not work, since there is a zero-order correlation of r = .71 between
how well a prisoner does in the program and the number of crimes
committed after a prisoner leaves prison. However, you discover that
the first-order partial correlation between the two variables equals
-.71 when you adjust for the effects of race (0 = African-American
and 1 = white). Explain how this is possible in the following ways:
a. First of all, show that it is mathematically possible to have a
zero-order correlation of .71 and a first-order partial correlation
as small as -.71 . You can do this by choosing two zero-order
correlations (namely, between race and subsequent crime and between
race and performance in program) such that the partial correlation
is smaller than -.71 . (Hint: In doing this problem use the
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computation formula for a first-order partial
correlation on page 173 of the Lecture Notes.)
b. Next, make a rough sketch of how the relation between
(1) prisoners' performance in the program and (2) their
later criminal record may differ according to (3) race.
(Hints: Instead of drawing a sketch by hand, you may prefer
to obtain a plot and data set using the following URL:
http://www.public.iastate.edu/~carlos/401/lectures/applets/partial.html
Your browser’s print capability can be used to print the 2nd part of
this web page. If you print this part in black-and-white, please
indicate on the sketch which points are for African Americans and
which are for whites. In working on this problem, one strategy is
to place points on the graph [no more than 5 data points per race
and keeping data points at least a quarter-inch apart, please] by
trial and error until you get a positive zero-order correlation
between performance and crimes, but a negative partial correlation
between the two variables when controlling for race. Then erase a
point. If the former correlation becomes larger and the latter
becomes smaller, leave it erased, otherwise replace it. Then add a
point. If the former correlation becomes larger and the latter
becomes smaller, leave it on the plot, otherwise erase it. As the
former correlation approaches 1 and the latter approaches -1, a
clear pattern will emerge.)
c. Finally explain in words a possible (that is, a theoretical)
reason why African-American and white prisoners may react
differently to the rehabilitation program. Be sure to use an
explanation that is consistent with the sketch obtained in part b.
(Hints: Your explanation should account for the association
represented by each correlation. First, why do you theorize that
performance and race are associated as they are in the sketch?
Second, why do you theorize that crimes and race are associated as
they are in the sketch? Third, assuming that the sketch and these
two theoretical arguments are accurate, how could the rehabilitation
program be effective despite the fact that those who perform well in
it are more likely to commit crimes after leaving prison than those
who performed poorly in it?)
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