MGEB12: Quantitative Methods in Economics-II
Problem Set-2
Chpater-9: 36, 38, 40, 46, 48abc, 50, 54, 58
Supplemental:
1. Which of the following statements are true? Where alpha is the probability of a Type I error and beta is the
probability of a Type II error:
A. Alpha+Power=1
B. Power+Beta=1
C. Alpha+Beta=1
D. All of the above are true
E. None of the above is true
2. A survey was conducted at a movie theater to determine the preference for different kinds of popcorn. The
results of the survey showed that Brand A was preferred by 65 percent of the people with a margin of error of
plus or minus 3 percent. What is meant by the statement “plus or minus 3 percent”?
A. Three percent of the population that was surveyed will change their minds.
B. Three percent of the time the results of such a survey are not accurate.
C. Three percent of the population was surveyed.
D. The true proportion of the population who preferred Brand A popcorn could be determined if 3 percent more
of the population was surveyed.
E. It would be unlikely to get the observed sample proportion of 65 percent unless the actual percentage of
people in the studied population who prefer Brand A is between 62 percent and 68 percent.
3. For Ho:  = 15 and H1:   15, testing at the 10% level of significance, the maximum probability of rejecting
the null hypothesis when it is true is:
A. 0.1 and the maximum probability of committing a Type I error is 0.1.
B. 5% and the maximum probability of committing a Type II error is 95%.
C. 0.1 and the probability of committing a Type I error is 0.05.
D. 0.05 and the probability of committing a Type II error is 0.1.
E. 0.05 and the probability of committing a Type I error is 0.1.
4. The alternative hypothesis is that a process is producing more than the maximum allowable proportion of
defective items. A Type I error is to conclude that the process is
A. not producing too many defects when it might not
B. not producing too many defects when it is not.
C. producing too many defects when it actually is.
D. not producing too many defects when it actually is.
E. producing too many defects when it actually isn't.
5. (Not for Test-1, For Test-2) A sample of size 150 from population 1 has 40 successes. A sample of size 250
from population 2 has 30 successes. The value of the test statistic for testing the null hypothesis that the
proportion of successes in population one exceeds the proportion of successes in population two by 0.05 is:
A. 1.645
B. 2.327
C. 1.960
D. 1.977
E. 2.510
6. If the lower and upper confidence limits of the population proportion p, using a sample of size 1500, are 0.184
and 0.238, respectively, then the lower and upper confidence limits of the total number of successes in the
population, given that the population size is 750,000, are respectively:
A. 276 and 357
B. 137,724 and 178,143
C. 138,000 and 178,500
D. 138,276 and 179,857
E. 148,361 and 189,671
Questions 7-10: A survey claims that more than 8 out of 10 doctors recommend aspirin for their patients with
headaches. To test this claim, a random sample of 100 doctors’ results in 86 who indicate that they recommend
aspirin.
7. The null hypothesis and the alternative hypothesis for this test is:
A. H0: P = 0.80 H1: P # 0.80
B. H0: P = 0.80 H1: P < 0.80
C. H0: P = 0.80 H1: P > 0.80
D. H0: P < 0.80 H1: P > 0.80
E. H0: P = 0.80 H1: P ≥ 0.80
8. The p-value of the test statistic in this problem is approximately equal to:
A. 0.0521
B. 0.0802
C. 0.1231
D. 0.0721
E. 0.0668
9. For a 5% significance level, the power of the test for the alternative of 0.85 is approximately:
A. 0.67
B. 0.53
C. 0.44
D. 0.33
E. 0.47
10. If confidence interval was used to estimate the population proportion. What size sample would be necessary
if we wanted the margin of error to be 0.04 using 95% confidence level?
A. 141
B. 289
C. 302
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D. 331
E. 384
Questions 11-12: A normal population has a population standard deviation of 12. The hypotheses about the
population mean, H0: µ = 40 versus H1: µ > 40 are to be tested. A random sample of 100 observations will be
selected and the sample mean is denoted by x .
11. Suppose the p-value is 0.015, the value of x must be closest to
A. 40.9
B. 41.5
C. 42.6
D. 43.2
E. 44.2
12. If the true population mean is 43 and the hypotheses are tested at the 5% significance level, the probability
of a Type II error is closest to
A. 0.305
B. 0.273
C. 0.229
D. 0.195
E. 0.187
13. The 90% confidence interval for the proportion of busy signals encountered by customers is [0.05, 0.13].
What is the margin of error for this confidence interval?
A. This calculation requires a standard error.
B. 0.01
C. 0.08
D. 0.04
E. 0.02
14. A sample of 39 statistics students contained 14 females. What size sample would be necessary if we wanted
to estimate the true proportion of female statistics students to within ±0.08 with 95% confidence?
A. 2606
B. 564
C. 1300
D. 139
E. 269
15. A student claims the ability to correctly identify whether a person is a business major or an economics major
by the way the person dresses. Suppose she is presented with one person and considers this to be a hypothesis
test with the null hypothesis being that the person is a business major and the alternative that the person is an
economics major. With reference to this test, what would be a Type II error?
A. Identifying the person as an economics major when the person is an economics major.
B. Identifying the person as an economics major when the person is a business major.
C. Identifying the person as a business major when the person is a business major.
D. Identifying the person as a business major when the person is an economics major.
E. None of the above
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16. In a binomial experiment, which of the following parameters will produce a probability distribution with the
smallest standard deviation, given n is the same for each situation?
A. P = 0.5
B. P = 0.4
C. P = 0.3
D. P = 0.9
E. (1-P) =0.8
Questions 17-19: According to a poll, (51%) of surveyed employees agree with the statement that the people
who run most companies are "honest and ethical."
17. A 96% confidence interval estimate of the percentage of employees who agree that those who run most
companies are “honest and ethical” was calculated to be [0.48, 0.54]. What is the sample size of the poll?
A. 851
B. 1003
C. 1167
D. 1444
E. 1647
18. Which of the following is the correct interpretation of the confidence interval estimate in the previous
question, [0.48, 0.54]?
A. We are 96% confident that the interval 48% to 54% includes the true proportion of US employees who agree
that those who run most companies are “honest and ethical”.
B. We are confident that 96% of US employees agree that those who run most companies are “honest and
ethical” between 48% and 54% of the time.
C. Of all US employees, we are confident that 48% to 54% of them agree that those who run most companies
are “honest and ethical” 96% of the time.
D. We are 96% confident that the interval 48% to 54% includes the sample proportion of US employees who
agree that those who run most companies are “honest and ethical”.
E. All of the above
19. Using the estimated 96% interval, [0.48, 0.54], what conclusion would we reach if we test H0: P = 0.56
versus H1: P # 0.56 at α = 0.04?
A. Accept the null and reject the alternative.
B. Fail to reject the null and conclude the alternative.
C. Reject the null and conclude the alternative.
D. Reject the null but do not accept the alternative
E. We cannot tell what our decision will be from the information given
Questions 20-21: The mayor of a city claims that of the 80,000 residents of the city, at least 28,000 are students.
A counselor believes that there are fewer students than that. He polls 1,000 residents and finds that 340 of them
are students.
20. What is the appropriate set of hypotheses to determine if the Mayors claim is valid?
A. H0: P = 0.35 H1: P # 0.35
B. H0: P # 0.34 H1: P > 0.34
C. H0: μ = 28,000 H1: μ > 28,000
D. H0: P ≥ 0.35 H1: P < 0.35
E. H0: μ # 28,000 H1: μ < 28,000
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21. What is the standardized test statistic for this hypothesis test?
A. (0.35-0.34)/100
B. (0.34-0.35)/0.015
C. (0.35-0.34)/0.2275
D. (0.34-0.35)/0.06
E. (0.34-0.35)/0.0.36
Questions 22-24: In the wake of the current gasoline price increases, there is reasons to believe that the price of
gasoline is higher in Toronto than in Calgary where the average price per gallon for regular unleaded gasoline is
currently $1.75 per gallon with a standard deviation of $.06 per gallon. To test the claim, a group of students
from the UOFT statistics class sample 36 gasoline stations in Toronto area and find that regular unleaded
gasoline sells at a mean price of $1.77 per gallon. (Assume that the standard deviation remains constant)
22. A test statistic that the price of gasoline in Toronto is higher than that in Calgary will be:
A. H0:  = 1.75 and H1:  > 1.75
B. H0:  = 1.77 and H1:  > 1.77
C. H0:  = 1.75 and H1:  < 1.75
D. H0:  = 1.77 and H1:  < 1.77
E. H0:  = 1.75 and H1:  > 1.77
23. The P-value of the test is:
A. 0.0721
B. 0.0456
C. 0.0371
D. 0.0256
E. 0.0228
24. If in testing the hypothesis we wish the probability of a type I error to be α = 0.05 and the probability of a
type II error to be β = 0.01 when the alternative is 1.78, the sample size would have to be at least:
A. 64
B. 71
C. 82
D. 93
E. 96
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MC: Solution Keys
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B
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Long Questions
Question-1 There is a consensus emerging in the business community that the ups and downs of the stock
market cannot be predicted. In your sample of 36 financial analysts, 20 have correctly predicted the stock
market correctly. You are interested in testing whether the probability of predicting the market correctly is more
than 50%.
a) Construct your null hypothesis and the alternative hypothesis?
b) Calculate your test statistic?
c) Find the p-value? Test your hypothesis at 5% significance level. State your conclusion.
d) Assuming that the number of analysts that predicted the stock market correctly remains constant at
20, what is the sample size for which your p-value is 5%.
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Solutions
Question – 1 There is a consensus emerging in the business community that the ups and downs of the stock
market cannot be predicted. In your sample of 36 financial analysts, 20 have correctly predicted the stock
market correctly. You are interested in testing whether the probability of predicting the market correctly is more
than 50%.
a) Construct your null hypothesis and the alternative hypothesis?
b) Calculate your test statistic?
c) Find the p-value? Test your hypothesis at 5% significance level. State your conclusion.
d) Assuming that the number of analysts that predicted the stock market correctly remains constant at
20, what is the sample size for which your p-value is 5%.
a.
H o : P = 0.5
H1 : P  0.5
b.
Check the condition: nP>5 n(1-P).5, both hold (calculate):
 20
p=
 0.556
36
0.556 − 0.5
Z stat =
= 0.672
(0.5)(1 − 0.5)
36
c.
P-value = 0.2514
P-value > significance level => will not reject the null – will not reject that the chance of predicting the market
correctly is only 50%=> cannot conclude that the stock market can be predicted correctly more than 50%.
d.
z stat =
20
− 0. 5
n
= 1.645
.5  (1 − 0.5)
n
20
− n  0.5 = 1.645  0.5 = 0.8225
n
d.
400
n 2 + n  0.25 − 20 = 0.8225 2
n
400
+ n  0.25 = 20.6765
n
n
n  31
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