hypothesis testing
xoon
stats2.pdf
Dr. V.R. Bencivenga and Dr. Stephen Donald
Economics 329
HYPOTHESIS TESTING (ONE POPULATION)
Problems and CQ’s – SET TWO
Instructions:
Here are four problems. A sequence of multiple choice questions is asked for each of them. Your
professor will tell you whether these will be done for credit, and if so, whether they will be done
as CQ’s in class, or as SCANTRON Q’S to be done outside of class.
2
PROBLEM 1
In India, both average clearing time and the variance of clearing times are large (“clearing time”
is time for funds to be transferred when a payment is authorized). You are a consultant who
has been hired by India to improve efficiency of the banking system by reducing both the
average and the variance. You install new hardware and software, and reorganize work flows.
Before your reforms, mean clearing time between any pair of large or medium-sized cities was
3.9 days, and the population standard deviation of check-clearing times was 1.5 days. (These
values were known.)
Now you want to demonstrate the impact of your reforms on average clearing time. You take a
random sample of 41 clearing times since your reforms were implemented. In this sample, the
mean clearing time is 3.5 days, and the sample standard deviation of clearing times is 1.1 days.
A normal plot indicates that clearing times are approximately normally distributed.
CQ1. Which are the appropriate null and alternative hypotheses?
a. 9.3:H.vs9.3:H A0 c. 9.3:H.vs9.3:H A0
b. 9.3:H.vs9.3:H A0 d. 5.3:H.vs9.3:H A0
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CQ2. What test statistic should we use?
a. 1nt~
n s
X
b.
X ~ N 0,1
s n
c.
X ~ N 0,1
n
CQ3. What is the form of the rejection region?
a. Reject H0 if cvX
b. Reject H0 if cvX
c. Reject H0 if 1cvX or if 2cvX
Calculate the rejection region that should be used to conduct this hypothesis test at a 5%
significance level. (I.e., what rejection region controls the maximum probability of Type I error
at 5%?)
CQ4. What is the conclusion of the test?
a. Accept H0 b. Reject H0 in favor of HA
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CQ5. You also want to demonstrate the impact of your reforms on the variance of clearing times. Which are the appropriate null and alternative hypotheses?
a. 22
A 22
0 5.1:H.vs5.1:H
b. 22
A 22
0 5.1:H.vs5.1:H
c. 22
A 22
0 5.1:H.vs5.1:H
d. 22
A 22
0 1.1:H.vs5.1:H
CQ6. What test statistic should we use?
a. X = number of clearing times less than 1.5 2 in the sample
b. p = sample proportion of clearing times less than 1.5 2 in the sample
c. 1nt~
n s
X
d. 21n2
2
~ s)1n(
5
CQ7. What is the form of the rejection region?
a. Reject H0 if cvs 2
b. Reject H0 if cvs 2
c. Reject H0 if 1 2 cvs or if 2
2 cvs
What rejection region should be used to conduct this hypothesis test at the 5% significance
level? (I.e., what rejection region controls the maximum probability of Type I error at 5%?)
CQ8. What is the conclusion of the test?
a. accept H0
b. Reject H0 in favor of HA
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PROBLEM 2
It is known that life expectancy (population mean age at death) of males in urban areas of a less
developed country is 55 years. A demographer suspects that life expectancy in rural areas of this
country is less than 55 years. She identifies a random sample of 196 recent deaths of males in
rural areas. In this sample, the mean age at death is 51.5 years, and the standard deviation of
ages at death is 21 years. (This problem does not reflect the actuarial definition of life
expectancy, which is the expected number of years until death, computed at birth, using the age-
specific mortality rates prevailing at the time of birth.)
CQ9. Which are the appropriate null and alternative hypotheses?
a. 55:H.vs55:H A0
b. 55:H.vs55:H A0
c. 55:H.vs55:H A0
CQ10. What test statistic should we use?
a. 1nt~
n s
X
b.
X ~ N 0,1
s n
c.
X ~ N 0,1
n
7
CQ11. What is the form of the rejection region?
a. Reject H0 if cvX .
b. Reject H0 if cvX .
c. Reject H0 if 1cvX or if 2cvX .
CQ12. What is the conclusion of the test?
a. Accept H0
b. Reject H0 in favor of HA
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PROBLEM 3
Suppose the military has designed a new missile-guidance system. With the old system, 80% of
missiles landed within thirty meters of the target at a range of five hundred miles. The military
claims that the new system is better (that more than 80% of missiles will land within thirty
meters at five hundred miles). Ten randomly-chosen, identical missiles will be launched using
the new guidance system in order to test the null hypothesis that the accuracy rate of the new
system is no better than that of the old system against the alternative hypothesis that the
accuracy rate of the new system is better.
State the null and alternative hypotheses in terms of the parameter of a probability model for
the sample observations.
CQ13. Which are the appropriate null and alternative hypotheses?
a. 0 AH : p 0.8 H : p 0.8
b. 0 AH : p 0.8 H : p 0.8
c. 0 AH : p 0.8 H : p 0.8
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CQ14. What is the form of the rejection region? The test statistic is the binomial random
variable, X.
a. Reject H0 if cvX .
b. Reject H0 if cvX .
c. Reject H0 if 1cvX or if 2cvX .
CQ15. What is the minimum sample size that permits the maximum probability of Type I error to be controlled at 10%?
a. 9 c. 11
b. 10 d. 12
CQ16. The military decides to increase the sample size to fifteen. What rejection region should be used to control the maximum probability of Type I error at 10%?
a. 12 c. 14
b. 13 d. 15
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PROBLEM 4
In order to shrink its labor force, a large company (such as IBM) is considering offering an early
retirement package to its employees who are over fifty-five and who have been on the payroll at
least ten years. However, this plan will not be cost-effective if more than 25% of such employees
choose to accept this package. A random sample of 100 eligible employees indicates that 30 of
them would accept the early retirement package, if it were offered. Using a 1% level of
significance, test the null hypothesis that the fraction of all eligible employers who would accept
the package is no greater than 0.25, against the alternative that it is greater.
CQ17. Which are the appropriate null and alternative hypotheses?
a. 0 AH : p 0.25 H : p 0.25
b. 0 AH : p 0.25 H : p 0.25
c. 0 AH : p 0.25 H : p 0.25
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CQ18. What test statistic should we use?
a. p p
~ N 0,1 p(1 p)
n
b.
p p ~ N 0,1
p(1 p)
n
c. 1nt~
n
)p1(p
pp
d. X ~ binomial (n = 30, p)
CQ19. What is the form of the rejection region?
a. Reject H0 if cvp .
b. Reject H0 if cvp .
c. Reject H0 if 1cvp or if 2cvp .
CQ20. What is the conclusion of the test?
a. Accept H0
b. Reject H0 in favor of HA
