Chapter 13—Hypothesis testing: Describing a single population
MULTIPLE CHOICE
1. In order to determine the p-value, which of the following items of information is not needed?
A.
The level of significance.
B.
Whether the test is one– or two–tailed.
C.
The value of the test statistic.
D.
All of the above are needed.
2. In testing the hypotheses:
=
:
0
H
35
H1 :
< 35,
the following information is known: n = 49,
x
= 37 and
= 16. The standardised test statistic equals:
A.
0.33.
B.
–0.33.
C.
–2.33.
D.
2.33.
3. 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.
4. In testing the hypotheses
=
:
0
H
75.
:
1
H
< 75.
if the value of the Z test statistic equals 1.78, then the p-value is:
A.
0.0.375.
B.
0.4625.
C.
0.9625.
D.
0.5375.
5. For a two-tail Z test, the null hypothesis will be rejected at the 0.05 level of significance if the value of
the standardised test statistic is:
A.
smaller than –1.645.
B.
greater than 1.96.
C.
smaller than –1.645 or greater than 1.645.
D.
smaller than –1.96 or greater than 1.96.
6. In testing the hypotheses:
=
:
0
H
500
:
1
H
500,
if the value of the Z test statistic equals 2.03, then the p-value is:
A.
0.0424.
B.
0.4788.
C.
0.9576.
D.
0.0212.
7. 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 be rejected or not rejected at the 0.01 level.
8. The power of a test is the probability of making:
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.
9. A Type II error is committed if we make:
A.
a correct decision when the null hypothesis is false.
B.
correct decision when the null hypothesis is true.
C.
incorrect decision when the null hypothesis is false.
D.
incorrect decision when the null hypothesis is true.
10. A Type I error is committed if we make:
A.
a correct decision when the null hypothesis is false.
B.
correct decision when the null hypothesis is true.
C.
incorrect decision when the null hypothesis is false.
D.
incorrect decision when the null hypothesis is true.
11. Which of the following p-value will not lead us to reject the null hypothesis if the level of significance
equals 0.10?
A.
0.15.
B.
0.01.
C.
0.05.
D.
0.025.
12. Suppose you intend to test the claim that a typical high school student in Sydney spends more than $50
a week on mobile phone calls. Which hypotheses are used to test the claim?
A.
:
0
H
50.
H1 :
< 50.
B.
H0 :
= 50.
H1 :
50.
C.
:
0
H
50.
H1 :
= 50.
D.
H0 :
= 50.
H1 :
> 50.
13. Suppose that we reject a null hypothesis at the 0.05 level of significance. 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.
14. The critical values z
or z
2/
are the boundary values for the:
A.
rejection region(s).
B.
level of significance.
C.
power of the test.
D.
Type II error.
15. Using the confidence interval when conducting a two-tail test for the population mean
we do not
reject the null hypothesis if the hypothesised 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.
16. 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.
17. In a one-tail test for the population mean, if the null hypothesis is not 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 two-tail test should be used instead of a one-tail test.
18. If the research question is not an equality statement, then in hypothesis testing it is specified as:
A.
the null hypothesis.
B.
either the null or the alternative hypothesis.
C.
the alternative hypothesis.
D.
the test statistic.
19. In a two-tail test for the population mean, the null hypothesis will be rejected at the
level of
significance if the value of the standardised test statistic z is such that:
A.
z > z
.
B.
z < –z
.
C.
–z
< z < z
.
D.
|z| > z
2/
.
20. In testing the hypotheses
H0 :
= 75.
H1 :
< 75.
the p-value is found to be 0.042, and the sample mean is 80. Which of the following statements is true?
A.
The probability of observing a sample mean at most as large as 75 from a population
whose mean is 100 is 0.042.
B.
The smallest value of
that would lead to the rejection of the null hypothesis is 0.042.
C.
The probability that the population mean is smaaer than 75 is 0.042.
D.
None of the above statements is correct.
21. Statisticians can translate p-values into several descriptive terms. Which of the following statements is
correct?
A.
If p-value < 0.01, there is overwhelming evidence to infer that the alternative hypothesis is
true.
B.
If 0.01 < p-value < 0.05, there is strong evidence to infer that the alternative hypothesis is
true.
C.
If 0.05 < p-value < 0.10, there is weak evidence to infer that the alternative hypothesis is
true.
D.
All of the above statements are correct.
22. A spouse stated that the average amount of money spent on Christmas gifts for immediate family
members is above $1200. The correct set of hypotheses is:
A.
200:
0=
H
.
1200:
1
H
.
B.
1200:
0
H
.
1200:
1=
H
.
C.
1200:
0=
H
.
1200:
1
H
.
D.
1200:
0
H
.
1200:
1=
H
.
23. The confidence interval approach can be employed to conduct tests of hypotheses. Which of the
following statements is false?
A.
The confidence interval approach is equivalent to the rejection region approach.
B.
The confidence interval approach has the disadvantage of complexity.
C.
One-sided confidence intervals can be used when conducting a one-tail test.
D.
The confidence interval approach does not yield a p-value.
24. Whenever the null hypothesis is not rejected:
A.
the null hypothesis is true.
B.
the alternative hypothesis is false.
C.
the null hypothesis is maintained.
D.
the null hypothesis is accepted.
25. The probability of a Type I error is denoted by:
A.
.
B.
1 –
.
C.
.
D.
1 –
.
26. A Type I error occurs when we:
A.
reject a false null hypothesis.
B.
reject a true null hypothesis.
C.
don’t reject a false null hypothesis.
D.
don’t reject a true null hypothesis.
27. A Type II error is defined as:
A.
rejecting a true null hypothesis.
B.
rejecting a false null hypothesis.
C.
not rejecting a true null hypothesis.
D.
not rejecting a false null hypothesis.
28. The probability of a Type II error is denoted by:
A.
.
B.
.
C.
1 –
.
D.
1 –
.
29. The power of a test is the probability that it will lead us to:
A.
reject the null hypothesis when it is true.
B.
reject the null hypothesis when it is false.
C.
fail to reject the null hypothesis when it is true.
D.
fail to reject the null hypothesis when it is false.
30. The power of a test is denoted by:
A.
.
B.
.
C.
1 –
.
D.
1 –
.
31. 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.
the test is statistically insignificant at whatever level of significance the test was conducted
at.
32. In a given hypothesis test, the null hypothesis can be rejected at the 0.10 level of significance, but
cannot be rejected at the 0.05 and 0.01 levels. The most accurate statement about the p-value for this
test is:
A.
p-value > 0.01.
B.
0.05 < p-value < 0.10.
C.
0.01 < p-value < 0.10.
D.
0.01 < p-value < 0.05.
33. In a criminal trial, a Type I error is made when:
A.
a guilty defendant is acquitted.
B.
an innocent person is convicted.
C.
a guilty defendant is convicted.
D.
an innocent person is acquitted.
34. 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.
the test is statistically insignificant at whatever level of significance the test was conducted
at.
35. Which of the following statements is (are) 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.
None of the above statements are true.
36. In a criminal trial, a Type II error is made when:
A.
a guilty defendant is acquitted.
B.
an innocent person is convicted.
C.
a guilty defendant is convicted.
D.
an innocent person is acquitted.
37. In a one-tail test, the p-value is found to be equal to 0.032. If the test had been two-tailed, the p-value
would have been:
A.
0.064.
B.
0.080.
C.
0.016.
D.
0.066.
38. If the value of the sample mean
x
is close enough to the hypothesised value of the population mean
, then:
A.
the hypothesised value is definitely true.
B.
the hypothesised value is definitely false.
C.
we reject the null hypothesis.
D.
we don’t reject the null hypothesis.
39. We cannot commit a Type I error when the:
A.
null hypothesis is true.
B.
level of significance is 0.10.
C.
null hypothesis is false.
D.
test is a two-tail test.
40. For a given level of significance, if the sample size increases, the probability of a Type II error will:
A.
remain the same.
B.
increase.
C.
decrease.
D.
be equal to 1.0 regardless of the value of
.
41. The p–value of a test is the:
A.
smallest value of
at which the null hypothesis can be rejected.
B.
largest value of
at which the null hypothesis can be rejected.
C.
smallest value of
at which the null hypothesis cannot be rejected.
D.
largest value of
at which the null hypothesis cannot be rejected.
42. The rejection region for testing the hypotheses
=
:
0
H
100.
:
1
H
100.
at the 0.05 level of significance is:
A.
|z| < 0.95.
B.
|z| > 1.96.
C.
z > 1.65.
D.
z < 2.33.
43. The rejection region for testing the hypotheses
=
:
0
H
80.
:
1
H
< 80.
at the 0.10 level of significance is:
A.
z > 1.96.
B.
z < 0.90.
C.
z > –1.65.
D.
z < –1.28.
44. The level of significance can be:
A.
any value between –1.04 and 1.04.
B.
any positive value.
C.
any value smaller than 1.645.
D.
None of the above answers is correct.
45. The p–value criterion for hypothesis testing is to reject the null hypothesis if:
A.
p-value =
.
B.
p-value <
.
C.
p-value >
.
D.
–
< p-value <
.
TRUE/FALSE
1. 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.
2. A one-tail p-value is two times the size of that for a two-tail test.
3. The power of a test is the probability that a true null hypothesis will be rejected.
4. The p-value is usually 0.05.
5. A null hypothesis is a statement about the value of a population parameter; it is put up for testing in the
face of numerical evidence.
6. An alternative or research hypothesis is an assertion that holds if the null hypothesis is false.
7. A two-tail test is a test in which a null hypothesis can be rejected by an extreme result occurring in
either direction.
8. A Type I error is represented by
, and is the probability of not rejecting a false null hypothesis.
9. A Type II error is represented by
, and is the probability of rejecting a true null hypothesis.
10. In any given test, it is impossible to commit the Type I and Type II errors at the same time.
11. The critical values will bound the rejection and non-rejection regions for the null hypothesis.
12. The power of a test refers to the probability of rejecting a false null hypothesis.
13. Reducing the probability of a Type I error also reduces the probability of a Type II error.
14. In any test, the probability of a Type I error and the probability of a Type II error add up to 1.
15. A Type I error is represented by
, and is the probability of rejecting a true null hypothesis.
16. If we reject the null hypothesis, we conclude that there is enough statistical evidence to infer that the
alternative hypothesis is true.
17. In a criminal trial, a Type I error is made when an innocent person is convicted.
18. A Type II error is represented by
and is the probability of failing to reject a false null hypothesis.
19. If we do not reject the null hypothesis, we conclude that there is enough statistical evidence to infer
that the null hypothesis is true.
20. The probability of making a Type I error and the level of significance are the same.
21. The p–value of a test is the smallest value of
at which the null hypothesis can be rejected.
22. If a null hypothesis is rejected at the 0.05 level of significance, it must be rejected at the 0.10 level.
23. In a criminal trial, a Type II error is made when an innocent person is acquitted.
24. In a one-tail test, the p-value is found to be equal to 0.018. If the test had been two-tailed, the p-value
would have been 0.036.
25. In order to determine the p-value, it is necessary to know the level of significance.
26. A professor of statistics refutes the claim that the average student spends 6 hours studying for the final.
To test the claim, the hypotheses H0:
= 6, H1:
6 should be used.
27. If we reject a null hypothesis at the 0.05 level of significance, then we must also reject it at the 0.10
level.
28. Using the confidence interval when conducting a two-tail test for the population mean
, we do not
reject the null hypothesis if the hypothesised value for
is smaller than the upper confidence limit.
29. In a two-tail test for the population mean, if the null hypothesis is rejected when the alternative
hypothesis is true, a Type I error is committed.
30. In a one-tail test for the population mean, if the null hypothesis is not rejected when the alternative
hypothesis is true, a Type II error is committed.
31. There is an inverse relationship between the probabilities of Type I and Type II errors.
32. There is a direct relationship between the power of a test and the probability of a Type II error.
33. A test for the population mean
produces a test-statistic z = –0.75. The p-value associated with the
test is 0.2266 if the test is a left-tail test, it is 0.7734 if the test is a right-tail test, and it is 0.4533 if the
test is a two-tail test.
34. A two-tail test for the population mean
produces a test-statistic z = –1.43. The p-value associated
with the test is 0.0764.
35. If a sample size is increased at a given
level, the probability of committing a Type II error
increases.
SHORT ANSWER
1. A random sample of 100 observations from a normal population whose standard deviation is 50
produced a mean of 75. Does this statistic provide sufficient evidence at the 5% level of significance to
infer that the population mean is not 80?
2. A drug company is interested in the effectiveness of a new sleeping pill. A random sample of 50
people try the new sleeping pill and the number of additional hours of sleep (compared with the nights
without any sleeping pill), X, are recorded. Assume that the population standard deviation of X is 3
hours.
a. State the null and alternative hypotheses for the claim that the new drug increases the number of
hours of sleep at least by 2 hours on average.
b. Using a standardised test statistic, test the hypothesis at the 5% level of significance if the sample
mean of additional hours of sleep is 2.2 hours.
3. A social scientist claims that the average adult watches less than 26 hours of television per week. He
collects data on 25 individuals’ television viewing habits, and finds that the mean number of hours that
the 25 people spent watching television was 22.4 hours. If the population standard deviation is known
to be 8 hours, can we conclude at the 1% significance level that he is right?