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ECON 2142 Graded Homework 2
Answer the questions in the spaces provided. If you run out of room for an answer,
continue on a new sheet of paper stapled to the end of this packet.
Name:
1. (6 points) A trucking company has found that its trucks average 0.2 breakdowns during round trips
from New York to Los Angeles.
(a) (2 points) What is the probability that a single truck will make the complete trip without experiencing a breakdown?
(b) (2 points) If 3 trucks are assigned to a NY/LA round trip, what is the probability that at least 2
of them will make the complete trip without experiencing a breakdown?
(c) (2 points) In general, how are the binomial and Poisson distributions related?
ECON 2142 Graded Homework 2
2. (5 points) The campaign manager for a political candidate claims that 55% of registered voters favor
the candidate over her strongest opponent. Assume that this claim is true and that we’re interested in
computing the probability that in a simple random sample of 300 voters, at least 60% would favor the
candidate over her strongest opponent.
(a) (1 point) What are the values of n and ⇡ to be used in the sampling distribution of the proportion?
A.
B.
C.
D.
⇡
⇡
⇡
⇡
= 0.40
= 0.45
= 0.55
= 0.60
and
and
and
and
n = 300
n = 300
n = 300
n = 300
(b) (1 point) What is the standard error of the sampling distribution of the proportion? Recall that
you need to use ⇡ if it’s available in the calculation of p .
A.
B.
C.
D.
p
p
p
p
⇡ 0.03
⇡ 0.04
⇡ 0.49
⇡ 0.50
(c) (1 point) What is the probability that at least 60% would favor the candidate over her strongest
opponent?
A.
B.
C.
D.
P (p
P (p
P (p
P (p
0.60) ⇡ 0.0062
0.60) ⇡ 0.0475
0.60) ⇡ 0.1056
0.60) ⇡ 0.4594
(d) (1 point) Suppose that only 49% of the registered voters in the sample favor the candidate over
her strongest opponent. If the campaign manager’s claim of 55% is correct, what is the probability
that the sample proportion would be no more than 0.49 for the same sample size of 300?
A.
B.
C.
D.
P (p 0.49) ⇡ 0.0001
P (p 0.49) ⇡ 0.0030
P (p 0.49) ⇡ 0.0228
P (p 0.49) ⇡ 0.0668
(e) (1 point) Based on your answer to part (d), speculate on whether the campaign manager’s claim
might be mistaken.
A.
B.
C.
D.
The
The
The
The
campaign manager is most likely correct in his claim.
campaign manager is most likely mistaken in his claim.
sample data is most likely correct.
sample date is most likely mistaken.
Page 2
ECON 2142 Graded Homework 2
3. (6 points) Historically, Shop-Mart has gotten an average of 2000 hours of use from its G&E fluorescent
lightbulbs. Because its fixtures are attached to the ceiling, the bulbs are rather cumbersome to replace,
and Shop-Mart is looking into the possibility of switching to Phipps bulbs, which cost the same. A sample
of Phipps bulbs lasted an average of 2080 hours, and the p-value in a right-tail test (H0 : µ 2000 versus
H1 : µ > 2000) is 0.012. Shop-Mart has shared these results with both G&E and Phipps.
(a) (1.5 points) Give Shop-Mart a recommendation. In doing so, use “0.012” in a sentence that would
be understood by a Shop-Mart executive who has had no statistical training but is comfortable with
probabilities as used by weather forecasters.
(b) (1.5 points) In interpreting these results, what level of significance might G&E like to use in reaching
a conclusion? In other words, for what level of ↵ would a G&E executive like to hold the test to
reach his desired outcome in the hypothesis test?
(c) (1.5 points) In interpreting these results, what level of significance might Phipps like to use in
reaching a conclusion? In other words, for what level of ↵ would a Phipps executive like to hold
the test to reach his desired outcome in the hypothesis test?
(d) (1.5 points) If the test had been two-tail (i.e., H0 : µ = 2000 versus H1 : µ 6= 2000) instead of
right-tail, would the p-value still be 0.012? If not, what would the p-value be? Explain.
Page 3
ECON 2142 Graded Homework 2
4. (5 points) Suspecting that television repair shops tend to charge women more than they do men, Emily
disconnected the speaker wire on her portable television and took it to a sample of 12 shops. She was
given repair estimates that averaged $85, with a standard deviation of $28. Her friend John, taking the
same set to another sample of 9 shops, was provided with an average estimate of $65, with a standard
deviation of $21. Assuming normal populations with equal standard deviations, use the 0.05 level in
evaluating Emily’s suspicion. The following multiple choice questions are geared towards evaluating this
hypothesis test.
(a) (1 point) What is the two-sample hypothesis test we must use for this question?
A.
B.
C.
D.
z-test for Independent Samples
Unequal variances t-test
Pooled-variances t-test
Comparing Proportions from Independent Samples
(b) (1 point) Suppose µ1 denotes the population mean for women and µ2 denotes the population mean
for men. What are the null and alternative hypotheses?
A.
B.
C.
D.
H0
H0
H0
H0
: µ1 = µ2 versus H1
: µ1 µ2 versus H1
: µ1 µ2 versus H1
: µ1 6= µ2 versus H1
: µ1
: µ1
: µ1
: µ1
6= µ2
< µ2
> µ2
= µ2
(c) (1 point) What is the value of s2p ?
A.
B.
C.
D.
10.69
25.29
114.33
639.58
(d) (1 point) If your answer to part (a) is a t test, what are tcalculated and tcritical , respectively? Alternatively, if your answer to part (a) is a z test, what are zcalculated and zcritical , respectively?
A.
B.
C.
D.
1.729;
1.793;
2.093;
1.793;
1.793
1.729
1.793
2.093
(e) (1 point) What is the conclusion at the 0.05 significance level? What can we say about the p value
for this test?
A.
B.
C.
D.
Reject H0 ; the p
Reject H0 ; the p
Fail to reject H0 ;
Fail to reject H0 ;
value
value
the p
the p
is > 0.05
is < 0.05
value is > 0.05
value is < 0.05
Page 4
ECON 2142 Graded Homework 2
5. (5 points) As a follow-up to the previous question, using the 0.05 level of significance in comparing the
sample standard deviations, were we justified in assuming that the population standard deviations were
equal? Would your conclusion change if the standard deviation of the estimates received by Emily had
been $35? (Hint: you need to run two hypothesis tests of whether the population standard deviations
are equal using the original value of $28 and the new value of $35)
Page 5
ECON 2142 Graded Homework 2
6. (5 points) A pharmaceutical manufacturer has come up with a new drug intended to provide greater
headache relief than the old formula. Of 250 patients treated with the previous medication, 130 reported
“fast relief from headache pain.” Of 200 individuals treated with the new formula, 128 said they got
“fast relief.” At the 0.05 level, can we conclude that the new formula is better than the old? Using
the appropriate statistical table, what is the approximate p-value for this test? The following multiple
choice questions are geared towards evaluating this hypothesis test.
(a) (1 point) What is the two-sample hypothesis test we must use for this question?
A.
B.
C.
D.
z-test for Independent Samples
Unequal variances t-test
Pooled-variances t-test
Comparing Proportions from Independent Samples
(b) (1 point) Suppose ⇡1 denotes the proportion of relieved individuals using the old medication and
⇡2 denotes the proportion of relieved individuals using the new medication. What are H0 and H1
for this hypothesis test?
A.
B.
C.
D.
H0
H0
H0
H0
: ⇡1 ⇡2 versus
: ⇡1 > ⇡2 versus
: ⇡1 = ⇡2 versus
: ⇡1 < ⇡2 versus
H1
H1
H1
H1
: ⇡1 < ⇡2
: ⇡1 ⇡2
: ⇡1 6= ⇡2
: ⇡1 ⇡2
(c) (1 point) What is the value of p̄?
A.
B.
C.
D.
p̄ = 0.57
p̄ = 1.16
p̄ = 225
p̄ = 450
(d) (1 point) What is the relevant test-statistic, i.e. zcalculated or tcalculated ?
A.
B.
C.
D.
-2.56
-1.65
0.05
0.57
(e) (1 point) What is the conclusion for this hypothesis test?
A.
B.
C.
D.
Fail to reject H0 and conclude that the old formula is better than the new formula
Reject H0 and conclude that the old formula is better than the new formula
Reject H0 and conclude that the new formula is better than the old formula
Fail to reject H0 and conclude that the new formula is better than the old formula
Page 6
ECON 2142 Graded Homework 2
7. (6 points) The idling speed of 14 gasoline-powered generators is measured with and without an oil
additive that is designed to lower friction. With the additive installed, the mean change in speed was
123 revolutions per minute (rpm), with a standard deviation of 3.5 rpm. At the 0.05 level of significance,
is the additive e↵ective in increasing engine rpm? Carry out the hypothesis test and include all the
relevant steps (write out H0 and H1 , determine the appropriate test-statistic with or without using
your calculator, determine the critical test-statistic or p-value using your calculator, and come to a
conclusion). Be sure to clearly include all relevant data points such as ↵, tcalculated , and p value.
Page 7
ECON 2142 Graded Homework 2
8. (5 points) Given the following data from three independent samples, use the 0.025 level in determining
whether the population means could be the same. The following multiple choice questions are geared
towards evaluating this hypothesis.
Sample
1
2
3
6
7 14
9 20 18
18 11 23
12 15 17
13 23 27
10 16
(a) (1 point) What is the formulation of H0 and H1 ?
A.
B.
C.
D.
H0
H0
H0
H0
: µ1
: µ1
: µ1
: µ1
= µ2
= µ2
= µ2
= µ2
= µ3
= µ3
= µ3
= µ3
= µ4 = µ5 = µ6 versus H1 : at least one of the µ’s does not equal another
= µ4 = µ5 = µ6 versus H1 : µ1 6= µ2 6= µ3 6= µ4 6= µ5 6= µ6
versus H1 : at least one of the µ’s does not equal another
versus H1 : µ1 6= µ2 6= µ3
(b) (1 point) How many treatments are there and is this a balanced experiment, respectively?
A.
B.
C.
D.
3;
3;
6;
6;
No
Yes
No
Yes
(c) (1 point) In the ANOVA test, how many degrees of freedom are associated with the treatments
and error, respectively?
A.
B.
C.
D.
2;
3;
2;
3;
14
14
17
17
(d) (1 point) What is the value of Fcalculated ?
A.
B.
C.
D.
0.025
0.048
3.80
4.86
(e) (1 point) At the 0.025 level, what is the conclusion?
A.
B.
C.
D.
We reject H0 and conclude that the means of the four di↵erent samples may be equal
We fail to reject H0 and conclude that the means of the four di↵erent samples may be equal
We reject H0 and conclude that the means of the four di↵erent samples may not be equal
We fail to reject H0 and conclude that the means of the four di↵erent samples may not be equal
Page 8
ECON 2142 Graded Homework 2
9. (6 points) Researchers have obtained and tested samples of four di↵erent brands of nylon rope that are
advertised as having a breaking strength of 100 pounds. Given the breaking strengths (in pounds) shown
below, use the 0.025 level in comparing the brands. Fill in the ANOVA table as part of your solution
(note that not every cell is required to be populated; you should know which ones are necessary and
which ones aren’t).
Brand A
103.1
108.9
106.7
114.3
113.3
110.5
Brand B
111.6
117.8
109.8
110.1
118.3
116.7
Brand C
109.0
111.8
113.0
109.7
108.6
114.7
Brand C
118.0
115.8
114.2
117.3
113.8
110.6
ANOVA Table:
Source of Variation
Between Groups
Within Groups
Total
SS
df
MS
Conclusion:
Page 9
Fcalculated
p-value
Fcritical
↵
ECON 2142 Graded Homework 2
10. (5 points) A national public television network has found that 35% of its contributions are for less than
$20, with 45% for $20–$50, and 20% for more than $50. In a random sample of 200 of the contributions
to a local station, 42% were for less than $20, 43% were for $20–$50, and 15% were over $50. At the
0.05 level, does the local station’s distribution of contributions di↵er significantly from that experienced
nationally? The following multiple choice questions are geared towards evaluating this hypothesis.
(a) (1 point) In this hypothesis test, the determination of 2critical relies on m = the number of parameters that had to be estimated in computing the expected frequencies. What is the value of
m?
A.
B.
C.
D.
4
2
1
0
(b) (1 point) What is the sum of the di↵erences between the observed and expected frequencies? What
is the sum of the squares of the di↵erences between observed and expected frequencies?
A.
B.
C.
D.
0; 0
0; 5.48
0; 312
5.48; 312
(c) (1 point) What is the conclusion of the hypothesis test?
A. We reject H0 and conclude that the distribution of contribution amounts for this network is
equal to the national level distribution.
B. We reject H0 and conclude that the distribution of contribution amounts for this network is
not equal to the national level distribution.
C. We fail to reject H0 and conclude that the distribution of contribution amounts for this network
is equal to the national level distribution.
D. We fail to reject H0 and conclude that the distribution of contribution amounts for this network
is not equal to the national level distribution.
(d) (1 point) Is the
A.
B.
C.
D.
statistic ever a negative value?
It is possible under some circumstances for the 2 statistic to be negative
Under no circumstances can the 2 statistic be negative
The 2 statistic is always negative
Cannot determine the determine the nature of the 2 statistic with limited information
(e) (1 point) Is the
A.
B.
C.
D.
2
2
distribution symmetric and if so, is it positively or negatively skewed?
symmetric; positive skew
not symmetric; negative skew
not symmetric; positive skew
symmetric; no skew
Page 10
ECON 2142 Graded Homework 2
11. (6 points) The personnel director of a large firm has summarized a random sample of last year’s absentee
reports in the accompanying contingency table. At the 0.01 level of significance, test the independence
of age versus length of absence. The following long answer questions are geared towards evaluating this
hypothesis. Note: you can disregard the fact that some of the cells have frequency < 5. In class, we
discussed this as being a requirement, but for the sake of this question, you can disregard this fact.
Age Group
25 years
26-40 years
40 years
Number of Days Absent
1
2-4 5-7 > 7
30
8
3
4
45
15 12
5
7
39
18 22
11
10
61
63 42
19
21
145
(a) (1.5 points) What are the null and alternative hypotheses?
(b) (1.5 points) Populate the following table using the expected frequencies if the age and length of
absence are independent.
1
Age Group
Number of Days Absent
2-4
5-7
>7
25 years
26-40 years
40 years
63
(c) (1.5 points) What is the value of
42
19
2
calculated ?
(d) (1.5 points) What is the conclusion at the 0.01 level of significance?
Page 11
21
45
39
61
145
ECON 2142 Graded Homework 2
12. (5 points) Safety researchers in a government agency believe that too much variability in the speeds of
vehicles on urban sections of interstate highways can contribute to accidents by causing a greater level
of interaction between vehicles traveling in the same direction. They believe that a standard deviation
in excess of 5 mph is undesirable. Observing a random sample of vehicles on an urban portion of the
interstate highway in their locale, they find the speeds to be as listed below:
78.9
70.1
58.1
54.7
64.3
51.4
58.9
55.2
59.4
52.9
48.1
57.5
58.4
52.2
58.8
48.1
71.9
57.8
63.1
55.3
53.6
59.1
64.4
55.4
57.7
64.4
59.2
59.6
68.6
60.0
62.3
64.8
57.0
49.9
58.2
55.9
68.1
70.0
63.2
59.9
55.0
60.6
59.7
64.4
62.9
52.4
65.6
61.0
(a) (2.5 points) At the 0.025 level, and assuming a normal distribution of vehicle speeds, could they be
mistaken in their conclusion that too much variability exists in the speeds of vehicles passing this
location?
(b) (2.5 points) Determine the 95% confidence interval for the population variance of the speeds of
vehicles passing this location on the urban interstate.
Page 12
61.7
62.1
61.7
ECON 2142 Graded Homework 2
13. (6 points) As of mid-June 2009, five U.S. Government Bond mutual funds were reported as having
generated the 1-year and 3-year (annualized) rates of return shown below.
U.S.
Government
Bond Fund
American Funds, AMUSX
Fidelity, FGOVX
Wells Fargo, STVSX
Morgan Stanley, USGBX
MFS Govt. Securities, MFGSX
x=3-Year
Annualized
Rate of Return
6.3%
7.4
6.5
2.4
7.2
y=1-Year Rate
of Return
7.1%
8.4
7.3
1.5
9.0
(a) (2 points) Determine the least-squares regression line and interpret its slope.
(b) (2 points) For a U.S. Government Bond mutual fund with a 3-year annualized rate of return of
5.0%, estimate the 1-year rate of return.
(c) (2 points) For a U.S. Government Bond mutual fund with a 3-year annualized rate of return of
7.0%, estimate the 1-year rate of return.
Page 13
ECON 2142 Graded Homework 2
14. (4 points) The following questions are two short answer questions related to the previous question and
to the topic of simple linear regression overall.
(a) (2 points) Using the previous problem, construct and interpret the 90% confidence and prediction
intervals associated with a 3-year annualized rate of return of 7.0%.
(b) (2 points) Describe the standard error of the estimate (SEE) and demonstrate how it relates to
the sum of squared errors (SSE).
Page 14
ECON 2142 Graded Homework 2
15. (5 points) A tire company has carried out tests in which rolling resistance (pounds) and inflation pressure
(pounds per square inch, or psi) have been measured for psi values ranging from 20 to 45. The regression
analysis is summarized in the following calculator output:
(a) (1 point) To the greatest number of decimal places in the output, what is the least-squares regression
line?
(b) (1 point) What proportion of the variation in rolling resistance is explained by the regression line?
(c) (1 point) At what level of significance does the slope of the line di↵er from zero? What type of test
did the calculator use in reaching this conclusion?
(d) (1 point) At what level of significance does the coefficient of correlation di↵er from zero? Compare
this with the level found in part (c) and explain either why they are di↵erent or why they are the
same.
(e) (1 point) Construct the 95% confidence interval for the slope of the population regression line.
Page 15
ECON 2142 Graded Homework 2
16. (5 points) For n = 5 data points, the following quantities have been calculated:
P
P
x = 457.8
x2 = 44, 195.74
(a) (1 point) What is the value of b1 ?
(a)
(b)
(c)
(d)
P
P 2y = 1030.1
y = 215, 344.8
P
P xy =2 96, 706.05
(y ŷ) = 617.5735
0.90
1.05
1.20
1.35
(b) (1 point) What is the value of b0 ?
(a)
(b)
(c)
(d)
90
100
110
120
(c) (1 point) What is the value of the SEE?
(a)
(b)
(c)
(d)
14.35
15.35
16.35
17.35
(d) (1 point) What is the coefficient of correlation, r?
(a)
(b)
(c)
(d)
0.50
0.70
0.90
1.10
(e) (1 point) What is the interpretation of r2 generally?
(a)
(b)
(c)
(d)
The
The
The
The
percent
percent
percent
percent
of
of
of
of
variation which is explained by the regression
correlation which is explained by the regression
variation which is NOT explained by the regression
correlation which is NOT explained by the regression
Page 16
ECON 2142 Graded Homework 2
17. (15 points) Annual per-capita consumption of all fresh fruits versus that of apples and grapes from 1998
through 2003 was as shown in the table below:
Year
1998
1999
2000
2001
2002
2003
y =All
Fresh Fruits
128.9 lb/person
129.8
128.0
125.7
126.9
126.7
x1 =Apples
19.0 lb/person
18.7
17.6
15.8
16.2
16.7
x2 = Grapes
7.1 lb/person
8.1
7.4
7.7
8.7
7.5
We get the following regression output from the calculator:
(a) (2 points) Interpret the partial regression coefficients.
(b) (2 points) What is the estimated per-capita consumption of all fresh fruits during a year when 17
pounds of apples and 6 pounds of grapes are consumed per person?
Page 17
ECON 2142 Graded Homework 2
(c) (2 points) Determine the 95% prediction interval for per-capita consumption of fresh fruits during
a year like the one described in part (b).
(d) (2 points) Determine the 95% confidence interval for mean per-capit ...

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