Download Name Stat 2411

Name _______________________ Stat 2411
Spring 2003
Final Exam
For problems involving computations, show how you arrive at your final answer. For
problems that may be left as unsimplified numerical expressions, you may leave any
answers with expressions such as
1000 


 2 
10
C5
30
P4
(1) (58 points) The following are the log(BMI) for 120 fathead minnows in an
experiment conducted by the EPA. BMI is the Body Mass Index (weight/length^2). The
values range from -3.85 to -2.37.
Stem
-23
-24
-25
-26
-27
-28
-29
-30
-31
-32
-33
-34
-35
-36
-37
-38
Leaf
7
2
766
987753333
9998888652221000
9888887764433332200000
888876665443332222100
97766554222222210000
766554432
86621
954410
988
3
21
5
----+----+----+----+-Multiply Stem.Leaf by 10**-1
#
1
1
3
9
16
22
21
20
9
5
6
3
1
2
1
(a) Without punching these numbers into a calculator to find the sample mean and
sample standard deviation, give reasonable guesses at the mean and standard deviation of
these values. Explain how you arrived at your answers. Any reasonable answer is fine. I
just want to know what you would guess for the mean and standard deviation so you
would know if results you get from a calculator or computer make sense.
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(b) If we are going to choose 20 of these 120 fish for histology, how many possible ways
are there for us to choose the 20 fish for histology? (You may leave your answer as an
unsimplified numerical expression.)
(c) There are, of course, 60 fish below the median BMI and 60 fish above the median
BMI. If we want to choose 10 fish from below the median BMI for histology and 10 fish
from above the median BMI for histology, how many ways can we choose the 20 fish for
histology? (You may leave your answer as an unsimplified numerical expression.)
(d) Choosing 10 random fish from above the median BMI and 10 random fish from
below the median BMI is called a ___________________________________ sample.
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(e) Suppose 6 of the 60 fish below the median BMI have tumors and 9 of the 60 fish
above the median have a tumor.
Below median BMI
Above median BMI
Number
of fish
60
60
Number
of fish with tumor
6
9
We choose 10 fish randomly from above the median BMI and independently choose 10
fish randomly from below the median BMI for histology. What is the probability that we
find that 2 of the 10 chosen fish below the median have tumors and 3 of the 10 chosen
fish above median have tumors? That is, let
TB = # of tumor fish from the 10 fish sampled from below median BMI
TA = # of tumor fish from the 10 fish sampled from above median BMI
Find P(TB=2 and TA=3). (You may leave your answer as an unsimplified numerical
expression.)
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(2) (34 points) For people body mass indexes are computed as weight/(height^2). A
BMI of over 40 is considered obese. Let O = obese and C = die of cancer. Suppose a
person has a 15% chance of dying from cancer if obese and a 10% chance of dying from
cancer if not obese.
(a) Are the events O and C independent? Mutually exclusive? Explain.
(b) Suppose 20% of the population is obese and 80% of the population is not obese. If a
person is chosen randomly from this population, what is the probability that the person
will die of cancer?
First write the given and wanted information in symbolic form by filling in the following
blanks.
Given
O = Obese
P( ________ ) = 0.15
P( ________ ) = 0.2
Find
C=die of cancer
P( _________ ) = 0.10
P( _________ ) = 0.8
P(________)
Now find the probability that a random person from this population dies of cancer.
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(c) If a patient dies of cancer, what is the probability that the patient is obese?
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(3) (14 points)
Suppose a person has a 10% chance of dying from cancer if not obese. If we follow 100
independent non-obese people, what is the approximate probability that 15 or more of
these people die from cancer?
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(4) (32 points)
Suppose we track 100 people who eat a Mediterranean diet and 100 people who do not
eat a Mediterranean diet. We want to test the null hypothesis that people eating both
diets have the same chances of dying from cancer against the alternative that the groups
do not have the same chance of dying from cancer. We find 10 of 100 people dying of
cancer from the Mediterranean diet group and 15 of 100 people dying of cancer in the
non-Mediterranean diet group.
(a) Do the test at the level. Do you reject the null hypothesis? Show all work
and state explicitly what tabled value you compared to.
(b) Find the p-value for the test in part (a). Based on this p-value do you reject the null
hypothesis?
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(5) (14 points) We found 10 of 100 sample people dying of cancer for the
Mediterranean diet group. Give a 90% confidence interval for the population proportion
of people dying from cancer when eating a Mediterranean diet. (You may leave your
answer as an unsimplified numerical expression.)
(6) (8 points)
An ecologist takes a stratified sample of blackbirds in a wetland. The wetland is divided
into 2 strata. The first stratum has 10 acres. The second stratum has 20 acres. The
average numbers of blackbirds per acre are found in the sample to be
Stratum
1
2
Acres
10
20
Per acre
Blackbirds
2.2
3.4
The goal is to find an estimate for estimate for the average number of blackbirds per acre
in the entire wetland, all 30 acres. Chris says the estimate should be (2.2 + 3.4)/2 = 2.8.
Terry says the estimate should be (1/3)*2.2 + (2/3)*3.4 = 3.0. Do you agree with Chris
or Terry? Briefly explain your reason for your choice.
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(7) (26 points) An article in the American Journal of Clinical Nutrition investigated
whether eating oat bran would reduce LDL cholesterol levels. Part of the data used in
this study is given below. The data below are for 4 subjects. Each subject ate either corn
flakes or oat bran for breakfast for a 2-week period. Then the subject switched to the
other diet. LDL levels were measured after eating each diet. Thus for each of the 4
subjects there were two LDL values, one when eating corn flakes and a second LDL
measurement when eating oat bran. (Which diet was eaten first was decided randomly
for each subject.)
Subject
1
2
3
4
LDL (mmol/l)
Corn
Oat
Flakes
Bran
4.61
3.84
6.42
5.57
3.82
2.96
4.54
4.80
Test the null hypothesis that the diets are equivalent versus the alternative that oat bran
reduces LDL level. Test at the level. Do you reject the null hypothesis? Show
all work and state explicitly what tabled value you compared to.
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(8) (14 points) Serum cholesterol levels among 20 to 74 year old females are known to
have a standard deviation of roughly 45 mg/100 ml. How many subjects would we have
to check in order to be 95% confident that our estimate of the population average serum
cholesterol level will be off by at most 5 mg/100 ml? (You may leave your answer as an
unsimplified numerical expression.)
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