Review Unit 4 ANS.
AP Statistics
Name:
I. 1.
Sarah’s parents are concerned that she seems short for her age. Their pediatrician has the
following record of Sarah’s height:
Age (in months)
Height (cm)
36
86
48
90
51
91
57
94
60
95
A scatterplot showed a strong positive association between age and height, and a least-squares
regression line was found to be HEIGHT = 71.95 + 0.3833 AGE. The correlation was r = 0.994. The
doctor wants to predict Sarah’s height in middle age (50 years old) if there is no intervention (growth
hormone), and he uses the regression line for this prediction. Check the doctor’s prediction, and then
comment on this procedure
50 years = 600 months
HEIGHT=71.95+0.3833 AGE Sarah's height would be 30
=71.95+0.3833(600)
=301.9 cm
meters, according to the model. Clearly this is a case of extrapolation and leads to an unreasonable
conclusion.
2. There is a strong positive correlation between years of schooling completed (x) and lifetime earnings
(y) for American men. One possible reason for this association is causation: more education leads to
higher-paying jobs. But lurking variables may explain some of the correlation. Suggest some lurking
variables that would explain why men with more education earn more.
Socioeconomic status may well have an impact here. Students from wealthier families likely have
greater access to education and to higher paying jobs. They may be more likely to be part of a family
business or have resources to start their own company.
3. The National Halothane Study was a major investigation of the safety of the anesthetics used in
surgery. Records of over 850,000 operations performed in 34 major hospitals showed the following
death rates for four common anesthetics x.
Anesthetic
Death rate
A
1.7%
B
1.7%
C
3.4%
D__
1.9%
Do these data prove that anesthetic C is causing more deaths than the others, or there another possible
explanation? Explain in a short narrative OR draw an appropriate diagram (and label variables
appropriately). Use an additional sheet if necessary.
Anesthetic C could be used for surgeries that have a higher risk of complications, leading falsely to
conclude that the anesthetic caused the deaths, when the true cause was related to the type of illness.
The severity of the patients' illnesses is also likely a lurking variable. Different anesthesiologists may
prefer to use different anesthetics so there may be differences due to the surgeons.
4.
A student experimenting with a pendulum counted the number of full swings the
pendulum made in 20 seconds for various lengths of string. Her data are shown
below:
Length
(in)
# of swings
6
9
12
15
18
21
24
27
30
33
23
20
17
16
14
13
13
12
11
10
Propose a reasonable model for the number of swings of the pendulum (in 20 seconds) for various string
lengths. Show all work, Use your model to predict the number of swings in 20 seconds for a 42 inch long
string.
A scatterplot shows a strong positive nonlinear
relationship between string length in pendulums and
the number of swings in 20 seconds.
Taking the logs of both x and y and performing linear
regression, we get log ŷ = 1.74296 − .47182log x
Graphing the residuals with lengths, we get
ŷ = 101.74296 x −.47182 where x is length and y is number
of swings.
The patternless residual plot confirms the model, so I
propose a power model:
ŷ = 101.74296 x −.47182
#
S
w
i
n
g
s
l
n
(
S
w
i
n
g
s
)
21
18
15
12
7.5
15.0
22.5
30.0
1.2
1.4
Length
1.3
1.2
1.1
1.0
0.8
1.0
ln(Length)
r 0.0125
e
s 0.0000
i -0.0125
d -0.0250
u
a
0.8 1.0 1.2
l
ln(Length)
1.4
II. A researcher suspected a relationship between people’s preferences in movies and preference in pizza.
A random sample of 100 people produced the following two-way table:
Jurrasic Park
20
5
10
Lethal Weapon (I)
8
15
12
Gone With the Wind
15
2
13
____________________________________________________________
1. Enter the overall (marginal) distributions on the table.
Favorite
Movie
Jurrasic
Park
Lethal
Weapon (I)
Gone With
the Wind
Totals
Pepperoni
Ground Beef
Mushrooms
Totals
20
5
10
35
8
15
12
35
15
2
13
30
43
22
35
100
2. Compute (in percents) the conditional distribution of favorite movie among those who prefer ground
beef topping.
Favorite
Ground Beef
Movie
Jurrasic
3. Briefly describe your finding in words.
5
≈ 22.7%
Park
22
Of people who prefer ground beef topping on pizza, the
Lethal
15
= 68%
majority (68%) prefer the movie Lethal Weapon (I) , while
Weapon (I)
22
23% prefer Jurrasic Park and fewest (9%) prefer Gone With
Gone With
2
the WInd.
≈ 9.1%
the Wind
22
Totals
22 or 100%
4. Compute (in percents) the conditional distribution of
favorite pizza topping among those who chose Gone With the Wind as their favorite movie. Show the
distribution in a table.
Favorite
Movie
Gone With
the Wind
Pepperoni
Ground Beef
Mushrooms
Totals
15
2
13
30
5. Briefly describe your finding in words.
Of those who prefer Gone with the wind, 50% (15/30) prefer pepperoni, 43% (13/30) prefer mushrooms,
and 7% (2/30) prefer ground beef.
6. Of those who prefer pepperoni, what proportion prefer Jurrasic Park?
20
≈ 46.5%
43
7. Of those who prefer Lethal Weapon (I), what proportion prefer ground beef?
15
≈ 42.9%
35