Deck 9: Hypothesis Tests for One Population Mean

Question 1

Provide an appropriate response.
Traditionally in hypothesis testing the null hypothesis represents the "status quo" which will be overturned only if there is evidence against it. Which of the statements below might represent a
Null hypothesis?

A)The teaching method raises SAT scores.
B)The treatment has no effect.
C)The defendant is guilty.
D)None of the above

Answer 1

B

Question 2

Provide an appropriate response.
Suppose that you wish to perform a hypothesis test for a population mean using the
critical value method. The test is right-tailed. Suppose that the population standard
deviation is unknown. The correct procedure to use is the t-test. If you mistakenly use the
standard normal table to obtain the critical value, will the value that you obtain be larger
or smaller than the correct value? Does the mistaken use of the normal table make it more
or less likely that the null hypothesis will be rejected?

Answer 2

The critical value obtained using the standard normal table will be smaller than the
correct critical value. This will make it more likely that the null hypothesis will be
rejected.

Question 3

Provide an appropriate response.
A right-tailed hypothesis test for a population mean is to be performed. If the null
hypothesis is rejected at the 5% level of significance, does this necessarily mean that it
would be rejected at the 1% level of significance? at the 10% level of significance? Explain
your reasoning. In your explanation, refer to the critical values corresponding to the
different significance levels.

Answer 3

A null hypothesis which is rejected at the 5% level of significance will certainly be
rejected at the 10% level of significance but not necessarily at the 1% level of
significance. If the null hypothesis is rejected at the 5% level of significance, the test
statistic is greater than the critical value of 1.645. This means that the test statistic is
certainly greater than 1.28 which is the critical value corresponding to a 10% level of
significance. The test statistic is not necessarily greater than 2.33 which is the critical
value corresponding to a 1% level of significance.

Question 4

Provide an appropriate response.
A hypothesis test for a population mean is to be performed. The hypotheses are

Question 5

Solve the problem.
A high school biology student wishes to test the hypothesis that hummingbird feeders can affect the mean mass of ruby-throated hummingbirds in the area surrounding the feeder. She captures
And weighs several of the hummingbirds near a science museum where several feeders are located.
She obtains the following masses in grams:

Question 6

Perform a hypothesis test for the population mean. Assume that preliminary data analyses indicate that it is reasonable toapply the z-test. Use the critical-value approach.
A researcher claims that the amounts of acetaminophen in a certain brand of cold tablets
have a mean different from the 600 mg claimed by the manufacturer. Test this claim at the
2% level of significance. The mean acetaminophen content for a random sample of n = 48
tablets is 605 mg. Assume that the population standard deviation is 4.8 mg.

Question 7

Decide whether it appears reasonable to use a t-test to perform a hypothesis test for the mean in question. Explain youranswer and include any relevant graphs, such as a normal probability plot, histogram, or stem-and-leaf diagram. If thereare outliers, explain how to proceed.

Question 8

Provide an appropriate response.
A man is on trial accused of murder in the first degree. The prosecutor presents evidence
that he hopes will convince the jury to reject the hypothesis that the man is innocent. This
situation can be modeled as a hypothesis test with the following hypotheses: If convicted, the defendant will receive the death penalty. Do you think that a Type I error
or a Type II error would be more serious? Why?

Question 9

Perform a one-sample z-test for a population mean using the P-value approach. Be sure to state the hypotheses and thesignificance level, to compute the value of the test statistic, to obtain the P-value, and to state your conclusion. Also,assess the strength of the evidence against the null hypothesis.
In one city, the average amount of time that tenth-graders spend watching television each
week is 21.6 hours. The principal of Birchwood High School believes that at his school,
tenth-graders watch less television. For a sample of 28 tenth-graders from Birchwood
High School, the mean amount of time spent watching television per week was 19.4 hours.
Do the data provide sufficient evidence to conclude that for tenth-graders at Birchwood
High School, the mean amount of time spent watching television per week is less than the
city average of 21.6 hours? Perform the appropriate hypothesis test using a significance
level of 5%. Assume that = 7.2 hours.

Question 10

Solve the problem.
A high school biology student wishes to test the hypothesis that hummingbird feeders can affect the mean mass of ruby-throated hummingbirds in the area surrounding the feeder. She captures
And weighs several of the hummingbirds near a science museum where several feeders are located.
She obtains the following masses in grams:

Question 11

Provide an appropriate response.
Give an example of a hypothesis test for which it is important to have a small β probability.
Explain why it is important to have a small value for

Question 12

Provide an appropriate response.
For a fixed sample size, how will increasing the significance level of a hypothesis test affect
the probability of a Type I error? How will it affect the probability of a Type II error?

Question 13

Construct a graph portraying the decision criterion for the specified hypothesis test.
A hypothesis test for a population mean is conducted. The hypotheses are: T he significance level is 0.04 and the critical value is 1.75. Sketch a normal curve displaying
t he decision criterion. This curve will represent the normal curve for the test statistic under
t he assumption that the null hypothesis is true. On your graph indicate the area in the tail,
the critical value, the rejection region, and the nonrejection region.

Question 14

Two graphical displays are given for a set of data. A hypothesis test is to be performed for the mean of the populationfrom which the data were obtained. Would it be reasonable to use a one-mean z-test? Explain your reasoning. Assumethat the population standard deviation is known.
A normal probability plot and a stem-and-leaf diagram of the data are given below.

Question 15

Solve the problem. Use the critical-value approach.
A hypothesis test for a population mean is to be performed. If the sample size is small (less
than 15), under what conditions is it reasonable to use the z-test? If the sample size is
moderate (between 15 and 30), under what conditions is it reasonable to use the z-test?

Question 16

Construct a graph portraying the decision criterion for the specified hypothesis test.
In tests of a computer component, it is found that the mean time between failures is
520 hours. A modification is made which is supposed to increase the time between failures.
Tests on a random sample of 10 modified components resulted in the following times (in
hours)between failures. At the 5% significance level, test the claim that for the modified components, the mean
time between failures is greater than 520 hours.

Question 17

Solve the problem. Use the critical-value approach.
A sheet-metal press stamps out bolt washers with a nominal inner diameter of 0.25 inches. Measurement of the inner diameters of a random sample of 14 washers produced the following
Results (in inches): The normal scores of the data are summarized below:

Question 18

Solve the problem. Use the critical-value approach.
A machine that fills soda bottles is supposed to fill them to a mean volume of 16.2 fluid ounces. A random sample of 20 filled bottles produced the following volumes in fluid ounces: Using technology, perform the following hypothesis test: at the 5% significance level, determine
Whether the fill volume is less than the supposed value. Comment on the appropriateness of the
Test.

Question 19

Provide an appropriate response.
Suppose that you want to perform a hypothesis test for a population mean. Give an
example of a situation in which both the z-test and the Wilcoxon signed-rank test are
permissible but the Wilcoxon signed-rank test is preferable. Specify the distribution of the
variable under consideration, the sample size, and whether or not the population standard
deviation is known.

Question 20

Provide an appropriate response.
A hypothesis test for a population mean is to be performed. The hypotheses are Which of the sketches below could represent the power curve for the test?

Question 21

Two graphical displays are given for a set of data. A hypothesis test is to be performed for the mean of the populationfrom which the data were obtained. Would it be reasonable to use a one-mean z-test? Explain your reasoning. Assumethat the population standard deviation is known.
A normal probability plot and a histogram plot of the data are given below.

Question 22

Provide an appropriate response.
In 1995, the mean math SAT score for students at one school was 488. A teacher introduces
a new teaching method to prepare students for the SAT. One year later, he performs a
hypothesis test to determine whether the mean math SAT score has increased. The
hypotheses are If the null hypothesis is rejected at the 10% level of significance, do you think the teacher
would feel confident that his teaching method works? What about if the null hypothesis is
rejected at the 1% level of significance? Which of these two results would constitute
stronger evidence that his teaching method works? Explain your thinking.

Question 23

Construct a graph portraying the decision criterion for the specified hypothesis test.
A one-sample z-test for a population mean is to be performed. Why might it be more
useful for those interpreting the results to know the P-value rather than simply whether or
not the null hypothesis was rejected?

Question 24

Preliminary data analyses indicate that it is reasonable to use a t-test to carry out the specified hypothesis test. Performthe t-test using the critical-value approach.
In one state, the mean time served in prison by convicted burglars is 18.7 months. A
researcher would like to perform a hypothesis test to determine whether the mean amount
of time served by convicted burglars in her hometown is different from 18.7 months. She
takes a random sample of 11 such cases from court files in her home town and finds that = 21 months and s = 7 months. Use a 5% significance level to perform the test.

Question 25

Construct a graph portraying the decision criterion for the specified hypothesis test.
Suppose that you wish to perform a hypothesis test for a population mean. Suppose that
the population standard deviation is unknown, the population is normally distributed, and
the sample size is small. Would you perform a z-test or a t-test? Why? Would the test be
exact or approximate?

Question 26

Two graphical displays are given for a set of data. A hypothesis test is to be performed for the mean of the populationfrom which the data were obtained. Would it be reasonable to use a one-mean z-test? Explain your reasoning. Assumethat the population standard deviation is known.
A normal probability plot and a stem-and-leaf diagram of the data are given below.

Question 27

Preliminary data analyses indicate that it is reasonable to use a t-test to carry out the specified hypothesis test. Performthe t-test using the P-value approach.
A large software company gives job applicants a test of programming ability and the mean
for that test has been 160 in the past. Twenty-five job applicants are randomly selected
from a large university and they produce a mean score of 183 with a standard deviation of
12. Use a 0.05 level of significance to test whether the mean score for students from this
university is greater than 160.

Question 28

Provide an appropriate response.
Explain the difference between a Type I error and a Type II error.

Question 29

Construct a graph portraying the decision criterion for the specified hypothesis test.
A hypothesis test for a population mean is conducted. The hypotheses are: The significance level is 0.08 and the critical value is -1.41. Sketch a normal curve
displaying the decision criterion. This curve will represent the normal curve for the test
statistic under the assumption that the null hypothesis is true. On your graph indicate the
area in the tail, the critical value, the rejection region, and the nonrejection region.

Question 30

Construct a graph portraying the decision criterion for the specified hypothesis test.
A hypothesis test for a population mean is conducted. The hypotheses are: The significance level is 0.06 and the critical values are -1.88 and 1.88. Sketch a normal
curve displaying the decision criterion. This curve will represent the normal curve for the
test statistic under the assumption that the null hypothesis is true. On your graph indicate
the area in each tail, the critical values, the rejection region, and the nonrejection region.

Question 31

Provide an appropriate response.
A hypothesis test for a population mean is to be performed. Suppose that the sample size is
50 but that the data contain outliers. Is it reasonable to use the z-test? How should you
proceed?

Question 32

Provide an appropriate response.
What assumption is required for the Wilcoxon signed-rank test? Why do you think this
assumption is required?

Question 33

Construct a graph portraying the decision criterion for the specified hypothesis test.
In tests of a computer component, it is found that the mean time between failures is
983 hours. A modification is made which is supposed to increase reliability by increasing
the time between failures. Tests on a sample of 36 modified components produce a mean
time between failures of 983 hours. Using a 1% level of significance, perform a hypothesis
test to determine whether the mean time between failures for the modified components is
greater than 937 hours. Assume that the population standard deviation is 52 hours.

Question 34

Preliminary data analyses indicate that it is reasonable to use a t-test to carry out the specified hypothesis test. Performthe t-test using the P-value approach.
compute the value of the test statistic, and find the P-value for the sample. State your
conclusion. Use a significance level of 0.05.

Question 35

Provide an ap propriate response.
When performing a one-sample z-test for a population mean, what criterion do you use
for rejecting the null hypothesis using the critical value approach? using the P-value
approach? Assume that a right-tailed test is being performed.

Question 36

Provide an appropriate response.
Suppose that you want to perform a hypothesis test for a population mean. Give an
example of a situation in which both the t-test and the Wilcoxon signed-rank test are
permissible but the t-test is preferable. Specify the distribution of the variable under
consideration, the sample size, and whether or not the population standard deviation is
known.

Question 37

Perform a hypothesis test for the population mean. Assume that preliminary data analyses indicate that it is reasonable toapply the z-test. Use the critical-value approach.
A manufacturer makes steel bars that are supposed to have a mean length of 50 cm. A
retailer suspects that the bars are running too long. A sample of 37 bars is taken and their
mean length is determined to be 51 cm. Using a 1% level of significance, perform a
hypothesis test to determine whether the population mean is greater than 50 cm. Assume
that the population standard deviation is 3.6 cm.

Question 38

Perform a one-sample z-test for a population mean using the P-value approach. Be sure to state the hypotheses and thesignificance level, to compute the value of the test statistic, to obtain the P-value, and to state your conclusion. Also,assess the strength of the evidence against the null hypothesis.
Last year, the mean running time for a certain type of flashlight battery was 8.5 hours. This
year, the manufacturer has introduced a change in the production method which he hopes
will increase the mean running time. A random sample of 40 of the new light bulbs was
obtained and the mean running time was found to be 8.7 hours. Do the data provide
sufficient evidence to conclude that the mean running time, µ, of the new light bulbs is
larger than last year's mean of 8.5 hours? Perform the appropriate hypothesis test using a
significance level of 5%. Assume that = 0.5 hours.

Question 39

Preliminary data analyses indicate that it is reasonable to use a t-test to carry out the specified hypothesis test. Performthe t-test using the P-value approach.

Question 40

A hypothesis testing situation is given. The population standard deviation, sample size, and significance level are given.Complete the table to give the probability of a Type II error and the power for each of the given values of µ. Use the tableto draw the power curve.
In 1990, the average math SAT score for students at one school was 472. Five years later, a
teacher wants to perform a hypothesis test to determine whether the average SAT score of
students at the school has changed from the 1990 mean of 472. Preliminary data analyses
indicate that it is reasonable to apply a z-test. The hypotheses are

Question 41

Provide an appropriate response.
Outline the six steps involved in performing a one-sample z-test for a population mean.

Question 42

A hypothesis testing situation is given. The population standard deviation, sample size, and significance level are given.Complete the table to give the probability of a Type II error and the power for each of the given values of µ. Use the table to draw the power curve.
A health insurer has determined that the "reasonable and customary" fee for a certain
medical procedure is $1200. They suspect that the average fee charged by one particular
clinic for this procedure is higher than $1200. The insurer wants to perform a hypothesis
test to determine whether their suspicion is correct. Preliminary data analyses indicate that
it is reasonable to apply a z-test. The hypotheses are

Question 43

Preliminary data analyses indicate that it is reasonable to use a t-test to carry out the specified hypothesis test. Performthe t-test using the P-value approach.
A test of sobriety involves measuring a subject's motor skills. The mean score for men who
are sober is known to be 35.0. A researcher would like to perform a hypothesis test to
determine whether the mean score for sober women differs from 35.0. Twenty randomly
selected sober women take the test and produce a mean score of 41.0 with a standard
deviation of 3.7. Perform the hypothesis test at the 0.01 level of significance.

Question 44

Provide an appropriate response.
A hypothesis test is performed at the 5% significance level to determine whether the mean
body temperature for a certain population differs from 37.1° C. The hypotheses are Explain the difference between statistical significance and practical significance.

Question 45

Perform a hypothesis test for the population mean. Assume that preliminary data analyses indicate that it is reasonable toapply the z-test. Use the critical-value approach.
A car insurance company has determined that the mean annual car insurance cost for a
family in the town of Watlington is $1716. A researcher wants to perform a hypothesis test
to determine whether the mean insurance cost for a family in the town of Putford is higher
than this. The mean insurance cost for a random sample of 32 families in Putford was
$1761. At the 10% significance level, do the data provide sufficient evidence to conclude
that the mean insurance cost for a family in Putford is higher than $1716, the mean cost for
a family in Watlington.? Assume that the population standard deviation is $35.50.

Question 46

Two graphical displays are given for a set of data. A hypothesis test is to be performed for the mean of the populationfrom which the data were obtained. Would it be reasonable to use a one-mean z-test? Explain your reasoning. Assumethat the population standard deviation is known.
A normal probability plot and a stem-and-leaf diagram of the data are given below.

Question 47

Provide an appropriate response.
Jenny is conducting a hypothesis test concerning a population mean. The hypotheses are as
follows.

Question 48

Provide an appropriate response.
A pharmaceutical company has a new drug which relieves headaches. However, there is
some indication that the drug may have the side effect of increasing blood pressure.
Suppose the drug company conducts a hypothesis test to determine whether the
medication raises blood pressure. The hypotheses are: Do you think that for doctors and patients it is more important to have a small α
probability or a small β probability? Why? Do you think that the pharmaceutical company
would prefer to have a small α probability or a small β probability? Why?

Question 49

Preliminary data analyses indicate that it is reasonable to use a t-test to carry out the specified hypothesis test. Performthe t-test using the critical-value approach.

Question 50

Provide an appropriate response.
List the advantages and disadvantages of nonparametric tests.

Question 51

Preliminary data analyses indicate that it is reasonable to use a t-test to carry out the specified hypothesis test. Performthe t-test using the P-value approach.

Question 52

Provide an appropriate response.
Describe parametric and nonparametric tests. Explain why nonparametric tests are
important.

Question 53

Provide an appropriate response.
A hypothesis test for a population mean is to be performed. The hypotheses are

Question 54

Use the Wilcoxon signed-rank test to perform the required hypothesis test. Be sure to state the hypotheses and thesignificance level, to obtain the critical value(s), to compute the value of the test statistic, and to state your conclusion.
The forced vital capacity (FVC)is often used by physicians to assess a person's ability to
move air in and out of their lungs. It is the maximum amount of air that can be exhaled
after a deep breath. For adult males, the average FVC is 5.0 liters. A researcher wants to
perform a hypothesis test to determine whether the average forced vital capacity for
women differs from this value. A random sample of 12 women yielded the following
values for FVC, in liters. At the 1% significance level, do the data provide sufficient evidence to conclude that the
mean forced vital capacity for women differs from the mean value for men of 5.0 liters?

Question 55

Preliminary data analyses indicate that it is reasonable to use a t-test to carry out the specified hypothesis test. Performthe t-test using the P-value approach.

Question 56

Preliminary data analyses indicate that it is reasonable to use a t-test to carry out the specified hypothesis test. Performthe t-test using the P-value approach.
In tests of a computer component, it is found that the mean time between failures is
520 hours. A modification is made which is supposed to increase the time between failures.
Tests on a random sample of 10 modified components resulted in the following times (in
hours)between failures. At the 0.05 significance level, test the claim that for the modified components, the mean
time between failures is greater than 520 hours.

Question 57

Provide an appropriate response.
A one-sample z-test for a population mean is to be performed. Compare the steps
involved in using the critical-value approach and the steps involved in using the P-value
approach. Which steps do the two methods have in common? How do the two methods
differ?

Question 58

Provide an appropriate response.
Suppose that you wish to use a Wilcoxon signed-rank test to perform a hypothesis test for
a population mean. If the absolute differences contain ties, how should you assign the
ranks? For example if two absolute differences are tied for fourth place, what ranks should
they receive? If three absolute differences are tied for eighth place, what ranks should they
receive?