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C22.0103
FINAL EXAM
Name:________________________
Write your answers to the first five questions on the attached sheets, in the spaces
provided. Circle the choice which best answers questions 6-15. Do not write
anything else on this page (besides your name and the circles). When you are
finished, hand in the entire exam (both question sheets and answer sheets).
Please do not remove any pages from the exam paper. There are 15 questions,
each worth 5 points. Everyone receives 25 points for free. Good Luck!
1) WRITTEN
11) (A) (B) (C) (D) (E)
2) WRITTEN
12) (A) (B) (C) (D) (E)
3) WRITTEN
13) (A) (B) (C) (D) (E)
4) WRITTEN
14) (A) (B) (C) (D) (E)
5) WRITTEN
15) (A) (B) (C) (D) (E)
6) (A) (B) (C) (D) (E)
7) (A) (B) (C) (D) (E)
8) (A) (B) (C) (D) (E)
9) (A) (B) (C) (D) (E)
10) (A) (B) (C) (D) (E)
Please Leave This Page Blank
Answer For Question 1:
Answer for Question 2:
Answer for Question 3:
Answer for Question 4:
Answer for Question 5:
C22.0103
FINAL EXAM
Questions 1) to 3) are based on data from a study reported in the journal
"Neurology" in 1998. The study considers the Intelligence Quotient (IQ) for 10
pairs of twins, a total of 20 observations. The explanatory variables are total brain
volume (BrainVol, in cubic centimeters), head size (HeadSiz, the head
circumference in centimeters), and body weight (BodyWt, in kilograms). The
Minitab multiple regression output is given below.
Regression Analysis: IQ versus BrainVol, HeadSiz, BodyWt
The regression equation is
IQ = 29 - 0.0187 BrainVol + 1.69 HeadSiz - 0.014 BodyWt
Predictor
Constant
BrainVol
HeadSiz
BodyWt
Coef
28.6
-0.01874
1.686
-0.0144
S = 14.08
SE Coef
102.7
0.03017
2.079
0.1669
R-Sq = 4.3%
T
0.28
-0.62
0.81
-0.09
P
0.784
0.543
0.429
0.933
R-Sq(adj) = 0.0%
Analysis of Variance
Source
Regression
Residual Error
Total
DF
3
16
19
SS
144.0
3172.0
3316.0
MS
48.0
198.3
F
0.24
P
0.866
1)
A) Based on this output, discuss the strength of the evidence that links IQ to
these variables.
B) Give an interpretation of the coefficient of BodyWt given by Minitab.
2)
A) Construct a 99% confidence interval for the true coefficient of BodyWt in
the model.
B) Use the output to predict the IQ of someone with a brain volume of 1000
cubic centimeters, a head size of 60 centimeters, and a body weight of 65
kilograms.
3) The birth order of the twins was also recorded. So there are two groups. Group
1 contains the 10 twins who were born first, and Group 2 contains the 10 twins
who were born second. Below is the Minitab 2-sample t-test output, as well as the
boxplots for the two groups.
Two-Sample T-Test and CI: IQ, BrthOrd
Two-sample T for IQ
BrthOrd
1
2
N
10
10
Mean
100.4
101.6
StDev
14.5
12.5
SE Mean
4.6
4.0
Difference = mu (1) - mu (2)
Estimate for difference: -1.20
95% CI for difference: (-13.99, 11.59)
T-Test of difference = 0 (vs not =): T-Value = -0.20
= 17
P-Value = 0.845
DF
Boxplots of IQ by Birth Order of Twins
130
120
IQ
110
100
90
80
1
2
BrthOrd
A) Based on the boxplots and 2-sample t-test output, is there credible
evidence that the mean IQ score depends on birth order?
B) If we consider each pair of twins, and take the difference between IQs
(Group 1 minus Group 2) , these 10 differences have a sample mean of
1.2 and a sample standard deviation of 8.36. Use this information to test
the null hypothesis that the expected value of IQ does not depend on birth
order, at the 1% level of significance. Use a two-tailed test. Interpret the
results of the test. Explain why the test you are performing here is
different (in principle) from the one performed in the two-sample t-test.
4) A 1993 study reported in Forbes Magazine considered the top 59 small
companies for that year. For each company, the age and salary of the chief
executive officer were recorded. (Age is in years, Salary is in thousands of dollars
per year, including bonuses). A scatterplot of salary vs. age is given below, as
well as the Minitab regression output.
Salary of CEOs Versus Age
SAL
1000
500
0
30
35
40
45
50
55
60
65
70
75
AGE
Regression Analysis: SAL versus AGE
The regression equation is
SAL = 243 + 3.13 AGE
Predictor
Constant
AGE
Coef
242.7
3.133
SE Coef
168.8
3.226
S = 220.6
R-Sq = 1.6%
T
1.44
0.97
P
0.156
0.336
R-Sq(adj) = 0.0%
Analysis of Variance
Source
Regression
Residual Error
Total
DF
1
57
58
SS
45896
2774936
2820832
MS
45896
48683
F
0.94
P
0.336
A) Test whether there is a positive linear relationship between salary and age,
at the 1% level of significance.
B) Do you think that natural variability alone could account for such a large
value of ̂1 as actually found here? Explain.
C) What proportion of the variability in salary is explained by age?
D) Explain how the value of S=220.6 can be obtained from other numbers
given in the Minitab output
E) Explain how the F-statistic of 0.94 can be obtained from other numbers
given in the Minitab output.
5) For a random sample of 10 observations from a standard normal distribution,
what is the probability that the absolute value of the sample mean will exceed
2.821 SE, where SE is the estimated standard error of the mean?
6) Given a sample of 5 observations from a normal population, suppose we wish
to test H0 :   10 versus HA :   10 . If the sample standard deviation is 2,
and the right-tailed p-value based on the t-statistic is less than .05, then the
sample mean x must be:
A)
B)
C)
D)
E)
Greater than 12.300
Greater than 12.483
Greater than 11.803
Greater than 11.907
None of the Above
7) If three cards are drawn from a shuffled full deck of 52 cards (without
replacement), the probability that all three cards are Kings is:
A) .0577 B) .0192 C) .000181 D) .000136 E) None of the Above.
8) In problem 7) above, the events A={First card drawn is a King} and
B={Second card drawn is a King} are:
A) Mutually Exclusive B) Independent C) Complements of Each Other
D) Dependent E) None of the Above
9) Suppose for a certain data set we obtain a 99% confidence interval for the
population mean  of (4 , 6). If all the assumptions required for the validity
of the confidence interval are met, then
A) There is a 99% probability that the sample mean is between 4 and 6.
B) The sample mean must be equal to 5.
C) If we say that the population mean is between 4 and 6 then we will be
lying 1% of the time.
D) There is a 99% probability that the mean of the population is between 4
and 6.
E) None of the Above.
10) A spinner for a certain board game gives three possible outcomes, denoted 1,
2 and 3. Each outcome is equally likely. If X is the value obtained in spinning
the spinner one time, then the theoretical variance of X is:
A) 2/3 B) 14/3 C) 2 D) 1 E) None of the Above.
11) If we spin the spinner 100 times, the probability that the total is at most 225 is:
A) .9989 B) 0 C) .0011 D) .9878 E) 0.
12) If X is a binomial random variable with n=50 and expectation 5, then the
variance of X is:
A) 12.5 B) 5 C) 2.12 D) 3.54 E) None of the Above.
13) True or False: In linear regression, a point with small leverage must have
small influence.
A) True B) False
14) In a multiple linear regression with two explanatory variables and 15
observations, if the F-statistic is 6, then the coefficient of multiple
determination is:
A) 1/2 B) 1/3 C) 1/4 D) 1/5 E) It cannot be determined.
15) True or False: If A and B are independent events with nonzero probabilities,
then the probability that A or B will occur is equal to P(A)+P(B).
A) True B) False.