CHAPTER 11: INTRODUCTION TO HYPOTHESIS TESTING
TRUE/FALSE
1. A null hypothesis is a statement about the value of a population parameter.
2. An alternative or research hypothesis is an assertion that holds if the null hypothesis is false.
3. A Type I error is represented by
; it is the probability of rejecting a true null hypothesis.
4. A Type I error is represented by
.
5. Reducing the probability of a Type I error also reduces the probability of a Type II error.
6. Increasing the probability of a Type I error will increase the probability of a Type II error.
7. It is possible to commit a Type I error and a Type II error at the same time.
8. A Type II error is represented by
; it is the probability of rejecting a true null hypothesis.
9. In a criminal trial, a Type I error is made when an innocent person is convicted.
10. The probability of making a Type I error and the level of significance are the same.
11. In a criminal trial, a Type II error is made when an innocent person is acquitted.
12. A Type II error is represented by
; it is the probability of failing to reject a false null hypothesis.
13. In testing a hypothesis, statements for the null and alternative hypotheses as well as the selection of the
level of significance should precede the collection and examination of the data.
14. The statement of the null hypothesis always includes an equals sign (=).
15. There is an inverse relationship between the probabilities of Type I and Type II errors; as one
increases, the other decreases, and vice versa.
MULTIPLE CHOICE
1. A professor of linguistics refutes the claim that the average student spends 3 hours studying for the
midterm exam. She thinks they spend more time than that. Which hypotheses are used to test the
claim?
a.
H0:
= 3 vs. H1:
3
c.
H0:
3 vs. H1:
= 3
b.
H0:
= 3 vs. H1:
3
d.
H0:
= 3 vs. H1:
3
2. The probability of a Type I error is denoted by:
a.
c.
b.
1 −
d.
1 −
3. A Type I error is committed if we make:
a.
a correct decision when the null hypothesis is false.
b.
a correct decision when the null hypothesis is true.
c.
an incorrect decision when the null hypothesis is false.
d.
an incorrect decision when the null hypothesis is true.
4. A Type II error is committed if we make:
a.
a correct decision when the null hypothesis is false.
b.
a correct decision when the null hypothesis is true.
c.
an incorrect decision when the null hypothesis is false.
d.
an incorrect decision when the null hypothesis is true.
5. The hypothesis of most interest to the researcher is:
a.
the alternative hypothesis.
c.
both hypotheses are of equal interest.
b.
the null hypothesis.
d.
Neither hypothesis is of interest.
6. A spouse suspects that the average amount of money spent on Christmas gifts for immediate family
members is above $1,200. The correct set of hypotheses is:
a.
H0:
= 1200 vs. H1:
1200
c.
H0:
= 1200 vs. H1:
1200
b.
H0:
1200 vs. H1:
= 1200
d.
H0:
1200 vs. H1:
= 1200
7. Which of the following conclusions is not an appropriate conclusion from a hypothesis test?
a.
Reject H0. Sufficient evidence to support H1.
b.
Fail to reject H0. Insufficient evidence to support H1.
c.
Accept H0. Sufficient evidence to support H0.
d.
All of these choices are true.
8. A Type I error occurs when we:
a.
reject a false null hypothesis.
c.
don’t reject a false null hypothesis.
b.
reject a true null hypothesis.
d.
don’t reject a true null hypothesis.
9. A Type II error is defined as:
a.
rejecting a true null hypothesis.
c.
not rejecting a true null hypothesis.
b.
rejecting a false null hypothesis.
d.
not rejecting a false null hypothesis.
10. The probability of a Type II error is denoted by:
a.
c.
1 −
b.
d.
1 −
11. In a criminal trial, a Type I error is made when:
a.
a guilty defendant is acquitted.
c.
a guilty defendant is convicted.
b.
an innocent person is convicted.
d.
an innocent person is acquitted.
12. In a criminal trial, a Type II error is made when:
a.
a guilty defendant is acquitted.
c.
a guilty defendant is convicted.
b.
an innocent person is convicted.
d.
an innocent person is acquitted.
13. We cannot commit a Type I error when the:
a.
null hypothesis is true.
c.
null hypothesis is false.
b.
level of significance is 0.10.
d.
test is a two-tail test.
14. The level of significance can be:
a.
any number between −1.0 and 1.0.
b.
any number greater than zero.
c.
any number greater than 1.96 or less than −1.96.
d.
None of these choices.
15. Which of the following is an appropriate null hypothesis?
a.
The mean of a population is equal to 60.
b.
The mean of a sample is equal to 60.
c.
The mean of a population is not equal to 60.
d.
All of these choices are true.
16. Which of the following statements is not true?
a.
The probability of making a Type II error increases as the probability of making a Type I
error decreases.
b.
The probability of making a Type II error and the level of significance are the same.
c.
The power of the test decreases as the level of significance decreases.
d.
All of these choices are true.
17. Which of the following would be an appropriate alternative hypothesis?
a.
The mean of a population is equal to 70.
b.
The mean of a sample is equal to 70.
c.
The mean of a population is greater than 70.
d.
The mean of a sample is greater than 70.
18. If a test of hypothesis has a Type I error probability of .05, this means that:
a.
if the null hypothesis is true, we don’t reject if 5% of the time.
b.
if the null hypothesis is true, we reject it 5% of the time.
c.
if the null hypothesis is false, we don’t reject it 5% of the time.
d.
if the null hypothesis is false, we reject it 5% of the time.
19. Suppose we wish to test H0:
= vs. H1:
45. What will result if we conclude that the mean is
greater than 45 when the actual mean is 50?
a.
We have made a Type I error.
b.
We have made a Type II error.
c.
We have made both a Type I error and a Type II error.
d.
We have made the correct decision.
20. Which of the following probabilities is equal to the significance level
?
a.
Probability of making a Type I error.
b.
Probability of making a Type II error.
c.
Probability of rejecting H0 when you are supposed to.
d.
Probability of not rejecting H0 when you shouldn’t.
21. If we reject the null hypothesis when it is false, then we have committed:
a.
a Type II error.
c.
both a Type I error and a Type II error.
b.
a Type I error.
d.
neither a Type I error nor a Type II error.
22. Researchers claim that 40 tissues is the average number of tissues a person uses during the course of a
cold. The company who makes Puffs brand tissues thinks that fewer of their tissues are needed. What
are their null and alternative hypotheses?
a.
H0:
= 40 vs. H1:
40
c.
H0: = 40 vs. H1: 40
b.
H0:
= 40 vs. H1:
40
d.
H0:
40 vs. H1:
= 40
23. The owner of a local Jazz Club has recently surveyed a random sample of n = 200 customers of the
club. She would now like to determine whether or not the mean age of her customers is over 30. If so,
she plans to alter the entertainment to appeal to an older crowd. If not, no entertainment changes will
be made. The appropriate hypotheses to test are:
a.
H0:
= 30 vs. H1:
30.
c.
H0: = 30 vs. H1: 30.
b.
H0:
= 30 vs. H1:
30.
d.
H0: = 30 vs. H1: 30.
COMPLETION
1. If a researcher rejects a true null hypothesis, she has made a(n) ____________________ error.
2. If a researcher fails to reject a false null hypothesis he has made a(n) ____________________ error.
3. If a researcher rejects a false null hypothesis, she has made a(n) ____________________ decision.
4. If a researcher fails to reject a true null hypothesis, he has made a(n) ____________________
decision.
5. The probability of a Type I error is denoted by ____________________.
6. The probability of a Type II error is denoted by ____________________.
7. You cannot commit a(n) ____________________ error when the null hypothesis is true.
8. You cannot commit a(n) ____________________ error when the null hypothesis is false.
9. The hypothesis testing procedure begins with the assumption that the null hypothesis is
____________________.
10. After you set up the hypotheses and collect your data, you calculate the statistic that serves as the
criterion for making your decision. This number is called the ____________________ statistic.
SHORT ANSWER
1. Formulate the null and alternative hypothesis in each case:
a.
Ten years ago the average American drank 2.5 cups of coffee per day. You suspect that
amount has changed since then.
b.
A researcher at Ohio State University speculates that the average SAT score for entering
freshmen is higher than the national average of 1,650.
c.
The manager of the University of Iowa bookstore suspects the average amount of money a
student spends on books is less than the reported $400.
a.
H0:
= 2.5, H1:
H0:
= 1650, H1:
H0:
= 400, H1:
2. For each of the following pairs of null and alternative hypotheses, determine whether or not they are
statistically correct to conduct a hypothesis test.
a.
H0:
= 25, H1:
25
b.
H0:
= 30, H1:
30
c.
H0: = 35, H1: 35
d.
H0: = 40, H1: 40
e.
H0:
60, H1:
= 60
a.
appropriate
b.
appropriate
c.
inappropriate
d.
inappropriate
e.
inappropriate
3. Suppose an auto manufacturer states that their car goes from 0 to 60 miles per hour in 10 seconds on
average, and you suspect that time is longer.
a.
Set up the null and alternative hypotheses to test this claim.
b.
Explain how you know which is the null hypothesis and which is the alternative
hypothesis.
unless there is sufficient evidence against it. It represents the original claim. The
providing sufficient evidence against H0.
4. Suppose a pickup and delivery company states that their packages arrive within two days or less on
average. You want to find out whether the actual average delivery time is longer than this. You
conduct a hypothesis test.
a.
Set up the null and alternative hypotheses.
b.
Suppose you conclude wrongly that the company’s statement about average delivery time
is within two days. What type of error is being committed and what is the impact of that
error?
c.
Suppose you conclude wrongly that the delivery company’s average time to delivery is in
fact longer than two days. What type of error did you commit and what is the impact of
this error?
d.
Which error is worse from the company’s standpoint, a Type I or a Type II error? Why?
e.
Which error is worse from a consumer standpoint, a Type I or a Type II error? Why?
not get their packages delivered within 2 days on average, and could eventually become
dissatisfied.
caused a false alarm. The impact is that the company loses its credibility needlessly. This
could cause some real problems for the person who made the false alarm.
5. Explain the difference between accepting H0 and failing to reject H0.
6. Explain why a Type I error and a Type II error have an inverse relationship.
7. Think about a situation where you have a test for a virus. First, you are tested positive or negative.
Second, you either really do have the virus or you don’t.
a.
If you actually have the virus but the test did not catch it, which error has been made and
what is the impact of that error?
b.
If you actually don’t have the virus but the test says you did, which error is being made
and what is the impact of this error?
c.
Which error is the worst one to commit in this situation and why?
a.
A Type II error has been committed, which is a very costly error. You are being told you
are OK when you really have the virus, and you are going untreated.
8. The p-value of a test is the probability of observing a test statistic at least as extreme as the one
computed given that the null hypothesis is true.
9. A p-value is usually set at 0.05.
10. The p–value of a test is the smallest
at which the null hypothesis can be rejected.
11. The p–value is the probability that the null hypothesis is true.
12. In order to determine the p-value, it is necessary to know the level of significance.
13. A one-tail p-value is two times the size of a two-tail test.
14. In a one-tail test, the p-value is found to be equal to 0.054. If the test had been two-tail, then the
p-value would have been 0.027.
15. For a given level of significance, if the sample size is increased, the probability of committing a Type
II error will decrease.
16. The critical values will bound the rejection and non-rejection regions for the null hypothesis.
17. If we do not reject the null hypothesis, we conclude that there is enough statistical evidence to infer
that the null hypothesis is true.
18. If a null hypothesis is rejected at the 0.05 level of significance, it must be rejected at the 0.025 level.
19. A sample is used to obtain a 95% confidence interval for the mean of a population. The confidence
interval goes from 78.21 to 87.64. If the same sample had been used to test the null hypothesis that the
mean of the population differs from 90, the null hypothesis could be rejected at a level of significance
of 0.05.
20. If we reject a null hypothesis at the 0.05 level of significance, then we must also reject it at the 0.10
level.
21. If your p-value is greater than 0.900 you should reject H0 at the 0.10 level.
22. A p–value is a probability, and must be between 0 and 1.
23. A one-tail test for the population mean
produces a test-statistic z = −0.75. The p-value associated
with the test is 0.7734.
24. Using the confidence interval when conducting a two-tail test for the population mean
, we do not
reject the null hypothesis if the hypothesized value for
falls between the lower and upper confidence
limits.
25. A two-tail test for the population mean
produces a test-statistic z = 1.89. The p-value associated with
the test is 0.0588.
26. For a given level of significance, if the sample size is increased, the probability of committing a Type I
error will decrease.
27. A sample is used to obtain a 95% confidence interval for the mean of a population. The confidence
interval goes from 10.89 to 13.21. If the same sample had been used to test H0:
= 12 vs. H1:
12,
H0 could not be rejected at the 0.05 level.
28. If we reject the null hypothesis, we conclude that there is enough statistical evidence to infer that the
alternative hypothesis is true.
29. The larger the p-value, the more likely one is to reject the null hypothesis.
30. In order to determine the p-value, which of the following is not needed?
a.
The level of significance.
b.
Whether the test is one-tail or two-tail.
c.
The value of the test statistic.
d.
All of these choices are true.
31. Which of the following p-values will lead us to reject the null hypothesis if the level of significance
equals 0.05?
a.
0.150
b.
0.100
c.
0.051
d.
0.025
32. In testing the hypotheses H0:
= 50 vs. H1:
50, the following information is known: n = 64, =
53.5, and
= 10. The standardized test statistic z equals:
a.
1.96
b.
−2.80
c.
2.80
d.
−1.96
33. If a hypothesis is not rejected at the 0.10 level of significance, it:
a.
must be rejected at the 0.05 level.
b.
may be rejected at the 0.05 level.
c.
will not be rejected at the 0.05 level.
d.
must be rejected at the 0.025 level.
34. In testing the hypotheses H0:
= 75 vs. H1:
< 75, if the value of the test statistic z equals −2.42, then
the p-value is:
a.
0.5078
b.
2.4200
c.
0.9922
d.
0.0078
35. For a two-tail test, the null hypothesis will be rejected at the 0.05 level of significance if the value of
the standardized test statistic z is:
a.
smaller than 1.96 or greater than −1.96
b.
greater than −1.96 or smaller than 1.96
c.
smaller than −1.96 or greater than 1.96
d.
greater than 1.645 or less than −1.645
36. In testing the hypotheses H0:
= 800 vs. H1:
800, if the value of the test statistic equals 1.75, then
the p-value is:
a.
0.0401
b.
0.0802
c.
0.4599
d.
0.9599
37. If a hypothesis is rejected at the 0.025 level of significance, it:
a.
must be rejected at any level.
b.
must be rejected at the 0.01 level.
c.
must not be rejected at the 0.01 level.
d.
may or may not be rejected at the 0.01 level.
38. Suppose that we reject a null hypothesis at the 0.05 level of significance. Then for which of the
following
-values do we also reject the null hypothesis?
a.
0.06
b.
0.04
c.
0.03
d.
0.02
39. The critical values z
or z
/ 2 are the boundary values for:
a.
the rejection region(s).
b.
the level of significance.
c.
Type I error.
d.
Type II error.
40. In a two-tail test for the population mean, if the null hypothesis is rejected when the alternative
hypothesis is true:
a.
a Type I error is committed.
b.
a Type II error is committed.
c.
a correct decision is made.
d.
a one-tail test should be used instead of a two-tail test.
41. Using a confidence interval when conducting a two-tail test for
, we do not reject H0 if the
hypothesized value for
:
a.
is to the left of the lower confidence limit (LCL).
b.
is to the right of the upper confidence limit (UCL).
c.
falls between the LCL and UCL.
d.
falls in the rejection region.
42. In a two-tail test for the population mean, the null hypothesis will be rejected at
level of significance
if the value of the standardized test statistic z is such that:
a.
z > z
b.
z < −z
c.
−z
< z < z
d.
| z | > z
/ 2
43. In testing the hypothesis H0:
= 100 vs. H1:
> 100, the p-value is found to be 0.074, and the sample
mean is 105. Which of the following statements is true?
a.
The probability of observing a sample mean at least as large as 105 from a population
whose mean is 100 is 0.074.
b.
The probability of observing a sample mean smaller than 105 from a population whose
mean is 100 is 0.074.
c.
The probability that the population mean is larger than 100 is 0.074.
d.
None of these choices.
44. If we reject the null hypothesis, we conclude that:
a.
there is enough statistical evidence to infer that the alternative hypothesis is true.
b.
there is not enough statistical evidence to infer that the alternative hypothesis is true.
c.
there is enough statistical evidence to infer that the null hypothesis is true.
d.
there is not enough statistical evidence to infer that the null hypothesis is true.
45. Suppose that in a certain hypothesis test the null hypothesis is rejected at the .10 level; it is also
rejected at the .05 level; however it cannot be rejected at the .01 level. The most accurate statement
that can be made about the p-value for this test is that:
a.
p-value = 0.01.
b.
p-value = 0.10.
c.
0.01 < p-value < 0.05.
d.
0.05 < p-value < 0.10.
46. Statisticians can translate p-values into several descriptive terms. Suppose you typically reject H0 at
level 0.05. Which of the following statements is correct?
a.
If the p-value < 0.001, there is overwhelming evidence to infer that the alternative
hypothesis is true.
b.
If 0.01 < p-value < 0.05, there is evidence to infer that the alternative hypothesis is true.
c.
If p-value > 0.10, there is no evidence to infer that the alternative hypothesis is true.
d.
All of these choices are true.
47. If we do not reject the null hypothesis, we conclude that:
a.
there is enough statistical evidence to infer that the alternative hypothesis is true.
b.
there is not enough statistical evidence to infer that the alternative hypothesis is true.
c.
there is enough statistical evidence to infer that the null hypothesis is true.
d.
there is not enough statistical evidence to infer that the null hypothesis is true.
48. In a one-tail test, the p-value is found to be equal to 0.068. If the test had been two-tail, the p-value
would have been:
a.
0.932
b.
0.466
c.
0.034
d.
0.136
49. If the value of the sample mean is close enough to the hypothesized value
0 of the population mean
, then:
a.
the value of
0 is definitely correct.
b.
the value of
0 is definitely wrong.
c.
we reject the null hypothesis.
d.
we cannot reject the null hypothesis.
50. The p–value of a test is the:
a.
smallest
at which the null hypothesis can be rejected.
b.
largest
at which the null hypothesis can be rejected.
c.
smallest
at which the null hypothesis cannot be rejected.
d.
largest
at which the null hypothesis cannot be rejected.
51. We have created a 95% confidence interval for
with the result (8, 13). What conclusion will we
make if we test H0:
= 15 vs. H1:
15 at
= 0.05?
a.
Reject H0 in favor of H1
b.
Accept H0 in favor of H1
c.
Fail to reject H0 in favor of H1
d.
We cannot tell what our decision will be from the information given
52. The p–value criterion for hypothesis testing is to reject the null hypothesis if: