Chapter 9
1 of 23
The time that it takes an untrained rat to run a standard maze has a Normal distribution with mean 65 seconds
and standard deviation 15 seconds. The researchers want to use a test of hypotheses to determine whether
training significantly improves the rats' completion times. An appropriate alternative hypothesis would have the
form
A. Ha: μ > 65.
B. Ha:
< 65.
C. Ha: μ < 65.
2 of 23
A researcher collects infant mortality data from a random sample of villages in a certain country. It is claimed
that the average death rate in this country is the same as that of a neighboring country, which is known to be
17 deaths per 1000 live births. To test this claim using a test of hypotheses, what should the null and
alternative hypotheses be?
A. H0: μ = 17, Ha: μ ≠ 17
B. H0: μ ≠ 17, Ha: μ = 17
C. H0: μ = 17, Ha: μ > 17
3 of 23
A social psychologist reports that "in our sample, ethnocentrism was significantly higher (P < 0.05) among
church attendees than among nonattendees." This means that
A. ethnocentrism was at least 5% higher among church attendees than among nonattendees.
B. the observed differences between church attendees and nonattendees account for all but 5% of those
sampled. These results are quite meaningful and should be investigated further.
C. if there were actually no difference in ethnocentrism between church attendees and nonattendees, then
the chance that we would have observed a difference at least as extreme as the one we did is less than 5%.
4 of 23
Which of the following P-values obtained from a test of hypotheses constitutes the least amount of evidence
against the null hypothesis?
A. 0.107
B. 0.207
C. 0.017
5 of 23
Suppose we conduct a test of hypotheses and find that the test results are significant at level
of the following statements then must be true?
A. The results are significant at level
= 0.025. Which
= 0.05.
B. The results are not significant at level
= 0.01.
C. The test results are important.
6 of 23
A noted psychic is tested for ESP. The psychic is presented with 400 cards, all face down, and asked to
determine if each card is marked with one of four symbols: a star, a cross, a circle, or a square. The psychic is
correct in 120 of the 400 cases. Let p represent the probability that the psychic correctly identifies the symbol
on the card in a random trial. Suppose we wish to see if there is evidence to suggest that the psychic is doing
significantly better than he would be if he were just guessing. To do so, we test H0: p = 0.25 against Ha: p >
0.25. The P-value of the test is
A. 0.0104.
B. 0.0146.
C. 0.9896.
7 of 23
Jamaal, a player on a college basketball team, made only 50% of his free throws last season. During the offseason, he worked on developing a softer shot in the hope of improving his free-throw accuracy. This season,
Jamaal made 54 of 95 free throws. Can we conclude that Jamaal's free-throw percentage p this season is
significantly different from last year's percentage? The approximate P-value for an appropriate test is
A. 0.8164.
B. 0.0918.
C. 0.1836.
8 of 23
Which of the following would have no effect on the P-value of a z test for a population proportion p?
A. increasing the sample size
B. decreasing the significance level of the test,
C. getting a different value of the sample proportion
from the sample data
9 of 23
Suppose that we are conducting a test for a population mean μ based on the standard Normal distribution. We
originally do a one-sided test but then decide that a two-sided test might be more appropriate (typically, we
prefer to use a two-sided test unless there is some reason to believe that an effect in a particular direction
exists). How will the results of the test change if we use a two-sided test instead of a one-sided test, assuming
that the same data are used?
A. The test statistic will have a larger value.
B. The P-value of the test will be smaller.
C. We will require stronger evidence against H0 to reject H0.
10 of 23
The distribution of times that a company's technicians take to respond to trouble calls is Normal with mean μ
and standard deviation σ = 0.25 hours. The company advertises that its technicians take an average of no more
than 2 hours to respond to trouble calls from customers. We wish to conduct a test to assess the amount of
evidence against the company's claim. In a random sample of 25 trouble calls, the average amount of time that
technicians took to respond was 2.1 hours. From these data, the P-value of the appropriate test is
A. 0.9772.
B. 0.0228.
C. 0.0456.
11 of 23
On a national compliance test for diabetics, scores have mean μ = 72 and standard deviation σ = 6. A doctor
has developed a new program for informing diabetics how to manage the disease and wants to know whether
diabetics using this program have a significantly different average score on the test than the national average.
For a sample of 60 diabetics who used the program, the sample mean test score was 73.5. A boxplot of the 60
scores reveals considerable skewness. Which of the following statements is true?
A. The P-value of the test is 0.0524.
B. The P-value of the test is 0.0262.
C. We cannot use the test based on the standard Normal distribution in this case because the distribution of
scores is not Normal.
12 of 23
We wish to test H0: μ = 10 against Ha: μ > 10, where μ is the unknown mean of a Normal population for which
the standard deviation σ is also unknown. We draw an SRS of size n = 13 from the population and compute the
value of the test statistic, t. From the table of critical values of the t distribution, find the critical value t* that
we would compare against the value of t to make a decision about the significance of the test results at
significance level = 0.05.
A. 1.771
B. 1.782
C. 2.179
13 of 23
We wish to see whether the dial temperature for a certain model of oven is properly calibrated. Four ovens of a
certain model are selected at random. The dial on each oven is set to 300°F. After one hour, the actual
temperature of each oven is measured. The observed temperatures are 305°, 310°, 300°, and 305°. Assuming
that actual temperatures for this model when the dial is set to 300° are Normally distributed with mean μ, we
test to see whether the oven is properly calibrated by testing H0: μ = 300 against a two-sided alternative. From
the data, the P-value for this test is
A. between 0.01 and 0.025.
B. between 0.025 and 0.05.
C. between 0.05 and 0.10.
14 of 23
A local teachers' union claims that the average number of school days missed due to illness by the city's school
teachers is fewer than 5 per year. A random sample of 28 city school teachers missed an average of 4.5 days
last year, with a sample standard deviation of 0.9 days. Assume that "days missed" follow a Normal distribution
with mean μ. A test conducted to see whether there is evidence to support the union's claim would have a Pvalue of
A. between 0.0025 and 0.005.
B. between 0.001 and 0.0025.
C. between 0.005 and 0.01.
15 of 23
You are thinking of using the t test to test a hypothesis about the mean of a population. A random sample of
size n from the population has a slightly skewed distribution with no apparent outliers. For which of the
following sample sizes n could you not justify using the t test?
A. 5
B. 20
C. neither (A) nor (B). (That is, we can justify using the t test in either case.)
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In which of the following situations would it not be appropriate to use the t test for a population mean? (Assume
that standard deviations are unknown and Normality conditions are satisfied in all cases; the question relates to
whether the procedure itself would be appropriate.)
A. We compare the average blood pressure of a random sample of patients before using a new drug to
control high blood pressure and after the patients have been using the new drug for a specified period of time.
B. We compare the average blood pressure of a random sample of adult male patients to a known "baseline"
value for that particular population.
C. We compare the average blood pressures of independently gathered random samples of men and women.
17 of 23
An advertisement for Chain X, a certain regional supermarket chain, claimed that the chain has had consistently
lower prices than its regional competitors. As part of a survey conducted by an independent price-checking
company, the average weekly grocery bill (based on the prices of approximately 95 commonly purchased items)
was recorded for Chain X and one of its leading competitors during four randomly selected weeks. The bills
(rounded to the nearest dollar) were as follows:
We wish to conduct a test of H0: μd = 0 vs. Ha: μd < 0, where μd = the average difference between the Chain X
bill and the competitor's bill during a particular week. (Assume that differences are Normally distributed.) The Pvalue for this test is
A. less than 0.01.
B. between 0.025 and 0.05.
C. between 0.05 and 0.10.
18 of 23
We want to use the t test for a population mean difference μd to test the claim H0: μd = 1. For five paired
observations, the differences are 4, -1, 4, 0, and 3. The approximate value of the test statistic in this case is
A. 0.95.
B. 1.91.
C. 2.13.
19 of 23
Suppose that we would like to test H0: μ = 50 against Ha: μ ≠ 50, where μ is the mean of a Normal population,
using a test based on the standard Normal distribution. The 95% confidence interval for μ is found to be (51.3,
54.7). Which of the following must then be true?
A. The P-value of the test is greater than 0.05.
B. The P-value of the test is at most 0.05.
C. The P-value of the test could be either greater than or at most 0.05. It can't be determined without
knowing the sample size.
20 of 23
Does taking garlic tablets twice a day provide significant health benefits? A researcher conducted a study of 50
adult subjects who took garlic tablets twice a day for a period of six months. At the end of the study, 100
variables related to the health of the subjects were measured for each subject, and the means were compared
to known means for these variables in the population of all adults. Four of these 100 variables were significantly
better (in the sense of statistical significance) at the 5% level for the group taking the garlic tablets compared
to the population as a whole. One variable was significantly better at the 1% level for the group taking the
garlic tablets compared to the population as a whole. From these results, it would be correct to conclude that
A. there is good statistical evidence that taking garlic tablets twice a day provides some health benefits.
B. there is good statistical evidence that taking garlic tablets twice a day provides benefits in the case of the
variable that was significant at the 1% level. However, we should be somewhat cautious about making claims
for the variables that were significant at the 5% level.
C. Neither (A) nor (B) is true.
21 of 23
The Department of Health plans to test the lead level in a public park. The park will be closed if the lead level
exceeds the allowed limit. Otherwise, the park will be kept open. The department conducts the test using soil
samples gathered at randomly selected locations. Which of the following decisions would constitute making a
Type I error in this situation?
A. keeping the park open when the average lead level exceeds the allowed limit
B. closing the park when the average lead level is acceptable
C. closing the park when the average lead level exceeds the allowed limit
22 of 23
In a test of hypotheses, the probability that a false null hypothesis should be rejected is also known as the
A. probability of committing a Type II error.
B. power of the test.
C. significance level of the test.
23 of 23
Which of the following will cause the power of a test to increase?
A. increasing the sample size
B. decreasing the significance level
of the test
C. increasing the probability of committing a Type II error, β
1. C The alternative hypothesis corresponds to what the researchers are trying to show (the research hypothesis). If training
improves times, then the rats should run the maze in less time, which means that the mean completion time would
decrease.
2. A When directly testing a claim, the claim should be used as the null hypothesis. No direction was specified in this case,
so a two-sided alternative hypothesis should be used.
3. C Recall that the P-value is the probability that the test statistic would take a value at least as extreme as that actually
observed if the null hypothesis were true.
4. B Recall that the P-value summarizes the exact amount of evidence against H0. The smaller the P-value, the stronger the
evidence against H0, so the largest of these three P-values would constitute the least amount of evidence against H0.
5. A For the results to be significant at level α = 0.025, the P-value must be at most 0.025. The P-value is, therefore, also at
most 0.05, and thus the results are significant at level α = 0.05.
6. A
Recall that to test the null hypothesis H0: p = p0, we use a test based on the test statistic
. Here,
=
120/400 = 0.30, n = 400, and p0 = 0.25, so
The P-value is the probability that the standard Normal random variable takes a value greater than 2.31, that is, the area
under the standard Normal curve to the right of z = 2.31. Using the table of Normal curve areas, we find this value to be 1 0.9896 = 0.0104.
7. C In this case, we want to test H0: p = 0.5 against the two-sided alternativeHa: p ≠ 0.5. The test statistic for this test is
. In this case, = 54/95
0.568, p0 = 0.5, and n = 95, so
Because the test is
two-sided and the test value is positive, the P-value is twice the area under the standard Normal curve to the right of z =
1.33; that is, 2(1 - 0.9082) = 0.1836.
8. B Changing is a decision made by the individual conducting the test and would have no effect on the observed data,
which determines the P-value (through the test statistic value).
9. C If the same data are used, the P-value of the test will be larger (specifically, it will double). Therefore, to reject H0, the
value of the test statistic would have to be larger (in either the positive or the negative direction), indicating stronger
evidence against H0.
10. B We test the claim by testing H0: μ = 2 against Ha: μ > 2, where μ = the mean response time for all trouble calls. Recall
that the test statistic for testing H0: μ = μ0 in this situation is
. In this case,
= 2.1, μ0 = 2, n = 25, and σ = 0.25,
so the test statistic value is
Because the alternative is right-tailed, the P-value is the area
under the standard Normal curve to the right of z = 2. This value is 1 - 0.9772 = 0.0228.
11. A We test H0: μ = 72 against Ha: μ ≠ 72, where μ = the mean score for all diabetics using the program. Recall that the test
statistic for testing H0: μ = μ0 in this situation is
. Here,
= 73.5, μ0 = 72, n = 60, and σ = 6. The test statistic value is
. Because this test
is two-sided and the test value z = 1.94 is positive, the P-value is twice the area under the standard Normal curve to the right
of 1.94. This value is 2(1 - 0.9738) = 0.0524.
12. B The appropriate critical value to use is the upper 0.05 critical value of the t(n - 1) = t(12) distribution, which is 1.782.
13. C
Recall that the test statistic for testing H0: μ = μ0 in this situation is
From the n = 4 observed values, we find that
therefore,
= 305 and s
.
4.08. Also, μ0 = 300. The value of the test statistic is,
The test statistic has the t(n - 1) = t(3) distribution. According to the table of t
critical values, 2.45 lies between the values 2.353 and 3.182, which are the upper 0.05 and 0.025 critical values of the t(3)
distribution. Because the alternative hypothesis is two-sided, the P-value must lie between 2(0.025) = 0.05 and 2(0.05) =
0.10.
14. A The hypotheses in this case are H0: μ = 5 and Ha: μ < 5 (the union's belief). The test statistic for testing H0: μ = μ0 in this
situation is
. Here, = 4.5, μ0 = 5, s = 0.9, and n = 28, so the test statistic value is
. The
test is left-tailed, so the P-value is P(t ≤ -2.94), where t has the t(n - 1) = t(27) distribution. By the symmetry of the t curve
about 0, this is the same as P(t ≥ 2.94). In the table of t critical values, 2.94 falls between 2.771, the upper 0.005 critical
value of the t(27) distribution, and 3.057, the upper 0.0025 critical value. The P-value is, therefore, between 0.0025 and
0.005.
15. A Recall that if n < 15, then the distribution must be Normal or near-Normal in order to justify using a t-procedure.
16. C In this case, we are comparing population means for two independent samples, and there are two distinct means to
consider (the male mean and the female mean). The t test for a single mean would not be appropriate in this case.
17. B The test statistic for this test is
, where = the sample mean difference, sd = the sample standard
deviation of differences, and n = the number of differences. In this case, the n = 4 differences are -1, -15, -23, and -27. From
this, we get = -16.5 and sd
11.4746, so the value of the test statistic is
and the P-value is
P(t ≤ -2.88), where t has the t(n - 1) = t(3) distribution. By the symmetry of the t curve about 0, this is the same as P(t ≥
2.88). The value 2.88 falls between 2.353 and 3.182, the upper 0.05 and 0.025 critical values of the t(3) distribution,
respectively, so the P-value is between 0.025 and 0.05.
18. A
The test statistic for this test is
because in this case, the hypothesized value of μd under H0 is equal to 1.
From the given difference data, we get n = 5, = 2, and sd
2.345, so the test statistic value is
19. B Recall that a test of H0: μ = μ0 against a two-sided alternative rejects H0 at level exactly when the value μ0 falls outside
a level 1 - confidence interval for μ. The interval here suggests that we should reject H0: μ = 50 in favor of Ha: μ ≠ 50
because the interval doesn't include the value 50. Because we reject H0 at level = 0.05, the P-value of the test must be at
most 0.05.
20. C Even if garlic had no effect on any of the 100 variables measured, we would expect to see about five variables
appearing to be statistically significant at the 5% level and about one variable appearing to be significant at the 1% level by
chance alone. This is consistent with what is observed, indicating that the results may simply be due to chance as opposed
to being an effect of using the garlic tablets.
21. B A Type I error is committed when a true null hypothesis is incorrectly rejected. In this case, the null hypothesis is that
the lead level is acceptable and the alternative is that it exceeds the allowed limit, so the answer you chose would, in fact,
correspond to a Type I error.
22. B The power of the test (for a particular alternative) is the probability of rejecting H0 when H0 is false.
23. A The power of a test is a measure of the sensitivity of the test to a particular effect (difference between the value of the
parameter hypothesized under H0 and the true value of the parameter). Increasing the sample size makes the test more
sensitive to the effect and increases the power.