Ch. 9 Review: Testing a Claim
1. Given α = 0.05, which of the following is true?
1.
2.
3.
4.
5.
P(Type II error) = 0.95
The power of the test is 0.95.
P(rejecting H0 when H0 is true) = 0.05
P(failing to reject H0 when H0 is false) = 0.05
The probability of a Type II error is independent of the value of α.
2. The mayor of a large city will run for governor if he believes that more than 30% of the
voters in the state already support him. He will have a survey firm ask a random
sample of n voters whether or not they support him. He will use a large sample test for
proportions to test the null hypothesis that the proportion of all voters who support
him is 30% or less against the alternative that the percentage is higher than 30%.
Suppose that 35% of all voters in the state actually support him. In which of the
following situations would the power for the test be the highest?
a.
b.
c.
d.
e.
The mayor uses a significance level of 0.01 and n=250 voters.
The mayor uses a significance level of 0.01 and n= 500 voters.
The mayor uses a significance level of 0.01 and n= 1,000 voters.
The mayor uses a significance level of 0.05 and n= 500 voters.
The mayor uses a significance level of 0.05 and n= 1,000 voters.
3. When performing a test of significance for a null hypothesis, Ho, against an alternative
Ha, the p-value is
a.
b.
c.
d.
The probability that Ho is true.
The probability that Ha is true.
The probability that Ho is false.
The probability of observing a value of a test statistic at least as extreme as that
observed in the sample if Ho is true.
e. The probability of observing a value of a test statistic at least as extreme as that
observed in the sample if Ha is true.
For problems 4-6, state H0 and HA in words and in symbols and define the parameter.
You do not need to answer the question posed in each problem.
4. A company has developed a new deluxe AAA battery that is supposed to last longer
than its regular AAA battery. Based on years of experience, the company knows that
its regular AAA batteries last for 30 hours of continuous use on average with a
standard deviation of 2 hours. The company selects an SRS of 50 new batteries and
uses them continuously until they are completely drained. The batteries in this sample
last an average of 30.7 hours. What conclusion would you make about the new
batteries?
5. Isaiah, who plays for the basketball team at South, says he is an 80% free-throw
shooter. Kevin, who has seen most of Isaiah’s games, is skeptical that Isaiah shoots
that well. Kevin starts keeping track and finds out that Isaiah makes 32 of his next 50
free-throws. Is this evidence that Isaiah is not as good as he claims to be?
6. One study chose 18 subjects at random from a company with over 200 workers who
assembled electronic devices. Half of the workers were assigned at random to each of
two groups. Both groups did similar assembly work, but one group was allowed to
pace themselves while the other group used an assembly line that moved at a fixed
pace. After two weeks, all the workers took a test of job satisfaction. Then they
switched work setups and took the test again after two more weeks. (This experiment
uses a matched pairs design.) The authors of the study want to investigate whether
job satisfaction of assembly-line workers differs when their work is machine-paced
rather than self-paced. They plan to find the mean difference in job satisfaction scores
for each worker in the sample in order to answer this question.
7. A car dealership believes the percentage of its customers who are satisfied with their service
is higher than the industry standard of 67%. Null and alternative hypotheses for testing this
claim are given below. What error do these hypotheses contain?
H0: p̂ = .67
Ha: p̂ > .67
8. In a significance test of the hypotheses H0: μ = 15 vs. Ha: μ > 15, what is the P-value if
the test statistic z is 1.2? (MAKE A SKETCH OF THE SAMPLING DISTRIBUTION AND
SHADE THE AREA UNDER THE CURVE THAT IS THE P-VALUE.)
9. What is the P-value if the test statistic z is the same as in part #8 but the alternative
hypothesis is Ha: μ ≠ 15? (MAKE A SKETCH OF THE SAMPLING DISTRIBUTION AND
SHADE THE AREA UNDER THE CURVE THAT IS THE P-VALUE.)
10. Suppose the people of Lake Wobegon have always believed that the mean height of
their eighth grade girls was 60 inches. A sample of the heights of 23 randomly selected
eighth grade girls gave a mean of 62.35. A one-sample t-test with a one-sided
alternative was done and the resulting P-value was 0.0498. Explain what 0.0498 is the
probability of in the context of the problem.
11. A random sample of 15 cigarettes of a certain brand was tested for nicotine content.
The average content of these 15 cigarettes was found to be 20.3 mg. The standard
deviation for the nicotine of all cigarettes of this brand is 3.0 mg. It is assumed that the
sample is from a normal population. A 95% confidence interval for the mean nicotine
content for all cigarettes of this brand is calculated and found to be 18.782 to 21.818
mg.
The manufacturer claims that the mean nicotine content in its cigarettes is 18.3 mg. Is
there evidence at the 5% level that μ ≠ 18.3?
12. In recent years, the mean yield of corn in the United States has been about 120 bushels
per acre. A survey of 40 farmers this year gives a sample mean yield of 123.8 bushels
per acre and a standard deviation of 10 bushels per acre. The farmers surveyed are a
SRS from the population of all commercial corn growers. We want to know whether the
sample result is good evidence that the national mean this year is not 120 bushels per
acre. Use a = 0.01. Follow the 4-step process.
b. What values of t would lead you to reject Ho?
13. When a law firm represents a group of people in a class action lawsuit and wins that lawsuit,
the firm receives a percentage of the group’s monetary settlement. That settlement amount
is based on the total number of people in the group – the larger the group and the larger the
settlement, the more money the firm will receive.
A law firm is trying to decide whether to represent car owners in a class action lawsuit
against the manufacturer of a certain make and model for a particular defect. If 5 percent or
less of the cars of this make and model have the defect, the firm will not recover its
expenses. Therefore, the firm will handle the lawsuit only if it is convinced that more than 5
percent of cars of this make and model have the defect. The firm plans to take a random
sample of 1,000 people who bought this car and ask them if they experienced this defect in
their cars.
a) Define the parameter of interest and state the null and alternative hypotheses that the law
firm should tests.
b) In the context of this situation, describe Type I and Type II errors and describe the
consequences of each of these for the law firm.
c) Give two ways to reduce the probability of a Type II error.
d) Give two ways to increase the power of the test.
14. The effect of exercise on the amount of lactic acid in the blood was examined in the article
“A Descriptive Analysis of Elite-Level Racquetball.” Eight males were selected at random
from those attending a week-long training camp. Blood lactate levels were measured
before and after playing three games of racquetball, as shown in the accompanying table.
Is there significant evidence that blood lactate levels increase as a result of exercise?
Follow the Inference Toolbox.
Player
1
2
3
4
5
6
7
8
Before
13
20
17
13
13
16
15
16
After
18
37
40
35
30
20
33
19