Download Testbank 7

AP Statistics Testbank 7
Multiple-Choice Questions
1) In formulating hypotheses for a statistical test of significance, the null hypothesis is often
a) a statement of "no effect" or "no difference."
b) the probability of observing the data you actually obtained.
c) a statement that the data are all 0.
d) a statement that the mean of the data is 0.
e) usually stated as a strict inequality.
2) In their advertisements, the marketers of a new diet program would like to claim that their methods
result in a mean weight loss of more than 10 pounds in two weeks. In order to determine if this is a
valid claim, they hire an independent testing agency which then selects 25 people to be placed on this
diet. The agency should be testing the null hypothesis H 0 : µ = 10 and the alternative hypothesis
a) H a : µ < 10 .
b) H a : µ > 10 .
c) H a : µ > 25.
d) H a : µ ≠ 10 .
e) H a : µ ≠ 10 ± σ / n .
3) Which of the following are true statements?
(I)
If there is evidence sufficient to reject a null hypothesis at the 10%, then there is sufficient
evidence to reject this null hypothesis at the 5% level.
(II)
Whether to use a one- or a two-sided alternative is typically decided after the data are
gathered.
(III) If the hypothesis test is conducted at the 1% level, there is a 1% chance of rejecting the null
hypothesis.
a) I only
b) II only
c) III only
d) I, II, and III
e) None are true.
4) SAS high-school counselor Mr. Williams once told me that our students study mathematics on average 2
hours per day. However, because of the “math addiction” diagnosed at this school, I firmly believe that
the average is higher. An appropriate set of hypotheses for this situation would be
a) H 0 : µ < 2, H a : µ > 2;
b)
c)
d)
e)
H0
H0
H0
H0
:µ
:µ
:µ
:µ
≥ 2, H a : µ < 2;
= 2, H a : µ ≠ 2;
= 2, H a : µ > 2;
= 2.8, H a : µ > 2.8.
5) SAS high-school counselor Mr. Williams once told me that our students study mathematics on average 2
hours per day. However, because of the “math addiction” diagnosed at this school, I firmly believe that
the average is higher. As a result, I decided to have Kaz interview 18 to determine their average
G
mathematics study time; Kaz’s results yield x = 2.8 and s x = 1.155 . Assume that the underlying
population of mathematics study times are approximately normally distributed. In which interval will the
significance, or P -value be found?
a) P < .0025
b) .0025 < P < .005
c) .005 < P < .01
d) .01 < P < .05
e) .05 < P
6) We continue with the situation in problem 5, above. Suppose that Billy now takes Kaz’s results and uses
the TInterval function on the TI-83 to compute a 90% confidence interval, with the resulting interval
being 2.8 ± .47 . Consider the following statements:
(I)
(II)
(III)
a)
b)
c)
d)
e)
Billy’s results are irrelevant to our hypothesis testing.
Billy’s results imply that we can reject the hypothesis H 0 : µ = 2 in favor of the alternative
H a : µ > 2 at the 5% level.
Billy’s results imply that we can reject the hypothesis H 0 : µ = 2 in favor of the alternative
H a : µ > 2 at the 1% level.
I only
II only
III only
II and III
I, II, and III
7) A patient claims that he consumes only 2000 calories per day, but a dietician suspects that the actual
figure is higher. The dietician plans to check his food intake for 30 days and will reject the patient’s
claim if the 30-day mean is more than 2100 calories. Assuming that the patient true standard deviation is
σ =350 calories per day, what is the probability that the dietician will mistakenly reject a patient’s true
claim?
a) .03
b) .06
c) .12
d) .28
e) .44
8) Suppose that a test of hypothesis is performed and that the data revealed a P -value of .0087. Consider
the following statements:
(I)
(II)
(III)
a)
b)
c)
d)
e)
The null hypothesis can be rejected at the 1% level.
The null hypothesis can be rejected at the 5% level.
We need to know if the alternative is one- or two-sided.
I only
II only
III only
I and II
I, II, and III
9) A researcher wishes to determine if the majority of American adults over the age of 65 plan to vote
Republican in the next presidential election. Let p represent the proportion of the population of all
American adults over the age of 65 who plan to vote Republican in the next presidential election. In
terms of p, the researcher should test which of the following null and alternative hypotheses.
a) H 0
b) H 0
c) H 0
d) H 0
e) H 0
: p = 0.5 vs. H a : p > 0.5.
: p = 0.5 vs. H a : p ≠ 0.5.
: p = 0.5 vs. H a : p < 0.5.
: p ≥ 0.5 vs. H a : p < 0.5.
: p = 0.5 vs. H a : p = 0.5 ± 0.03, since 0.03 is the accepted margin of error for most polls.
10) A company has put into production a new printer which they claim will reliably print at least one million
characters for any failures. To test this, the following data were gathered representing the number of
printed characters (in millions of characters) before each of 15 printers failed.
1.05
.98
1.05
.92
1.18
1.01
1.11
1.07
1.12
1.20
1.03
1.18
1.22
.93
.85
We assume that the distribution of printed characters before first failure is roughly normally distributed.
Now consider the following statements:
(I)
(II)
(III)
a)
b)
c)
d)
e)
The null hypothesis can be rejected at the level α = .05 .
The null hypothesis can be rejected at the level α = .01
The hypothesis test is based on the t-statistic.
I only.
II only.
III only.
I and III.
II and III.
11) Mr. S has an appetite for a certain brand of potato chips, which are typically packaged in 14 oz. bags.
However, Mr. S has come to suspect that the net weight of the chips in each bag is significantly less than
the advertised 14 ounces, leading him to want to test the hypotheses
H 0 : µ = 14, H a : µ < 14.
He sends Erica out to perform some measurements, who returns with the sample mean of x = 13.82 oz.
and the sample deviation of s x = 0.24 oz. for 16 bags of these chips. Based on Erica’s measurements,
and assuming that the weights are approximately normally distributed, we would
a) reject H 0 at significance level 0.10 but not at 0.05.
b) reject H 0 at significance level 0.05 but not at 0.025.
c) reject H 0 at significance level 0.025 but not at 0.01.
d) reject H 0 at significance level 0.01.
e) not reject H 0 at any level of significance.
12) Suppose that you would like to test a hypothesis about the mean of a population using a significance
level of 0.05. Suppose further that you would like to use the t-statistic even though you suspect that the
population is slightly skewed (and therefore not normal). Which of the following is correct?
a) You should not the t-statistic since the population does not have a normal distribution.
b) You may use the t-statistic provided that your sample size is large―say, at least 50.
c) You may use the t-statistic, but you should probably only claim the significance level is 0.10.
d) You may not use the t-statistic in this situation only for confidence intervals but not for tests of
hypotheses.
e) None of the above is correct.
13) Suppose that Mr. S sends two students, Tommy and Myung-Soo, out to test a hypothesis about a
population proportion. Tommy returns with a measurement with P-value of 0.03 and Myung-Soo returns
with a measurement with P-value of 0.022.
(I)
(II)
(III)
a)
b)
c)
d)
e)
Tommy’s results are more significant than Myung-Soo’s.
Myung-Soo’s results are more significant than Tommy’s.
The null hypothesis can be rejected at the 5% level of significance.
I only
II only
III only
I and III
II and III
14) A service station advertises that its mechanics can change a muffler in only 15 minutes. A consumers’
group doubts this claim and runs a hypothesis test using an SRS or 60 cars needing new mufflers. In this
sample the mean changing time is 16.25 minutes with a standard deviation of 3.5 minutes. Is this strong
evidence against the 15-minute claim?
a) Yes, because the P-value is only .0028.
b) No, because the P-value is .0028.
c) Yes, because the P-value is .28.
d) No, because 15 is within 16.25 ± 3.5.
e) Yes, because 16.25 is larger than the claimed 15 minutes.
15) In leaving for school on an overcast April morning you make a judgement on the null hypothesis:
H 0 : The weather will remain dry.
What would the results be of Type I and Type II errors?
a) Type I error: get drenched
Type II error: needlessly carry around an umbrella
b) Type I error: needlessly carry around an umbrella
Type II error: get drenched
c) Type I error: carry an umbrella, and it rains
Type II error: carry no umbrella, but the weather remains dry
d) Type I error: get drenched
Type II error: carry no umbrella, but the weather remains dry
e) Type I error: get drenched
Type II error: carry an umbrella, and it rains
16) An assembly-line machine is supposed to turn out ball bearings with a diameter of 1.25 centimeters.
Each morning the first 30 ball bearings produced are pulled and measured. If their mean diameter is
under 1.23 centimeters or over 1.27 centimeters, the machinery is stopped and an engineer is called to
make adjustments before production is resumed. The quality control procedure may be viewed as a
hypothesis test with
H 0 : µ = 1.25, H a : µ ≠ 1.25.
The engineer is asked to make adjustments when the null hypothesis is rejected. In test terminology,
what would be the result of a Type II error?
a) A warranted halt in production to adjust the machinery
b) An unnecessary stoppage of the production process
c) Continued production of wrong size ball bearings
d) Continued production of proper size ball bearings
e) Continued production of ball bearings that randomly are the right or wrong size
17) A piece of medical equipment is not functioning properly, however, in running operational checks, a lab
technician does not find evidence of the malfunction. The lab technician has committed
a) a Type I error
b) a Type II error
c) both a Type I and a Type II error
d) neither a Type I nor a Type II error
e) A random sampling error
18) Mr. S’s grading policies have come under attack by the Central Administration as well as by the Board
of Directors of SAS. To analyze the situation, a null hypothesis, together with an alternative hypothesis
have been formulated:
H 0 : Mr. S' s grading policies are fair
H a : Mr S plays favorites in awarding grades
The Board of Directors finds no irregularities, and therefore takes no actions against him, but the rumors
among the students is that it is advantageous for Mr. S’s students to regularly give him chocolatecovered expresso coffee beans. It is conceivable that
a)
b)
c)
d)
e)
a Type II error has been made
a Type II error has been made
a Type I and a Type II error have been made
neither type of error was made
the null and alternative hypotheses were incorrectly formulated
Free-Response Questions
19) 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
test.
b) In the context of this situation, describe Type I and Type II errors and describe the consequences of
each type of error for the law firm.
20) A pharmaceutical company has developed a new drug to reduce cholesterol. A regulatory agency will
recommend the new drug for use if there is convincing evidence that the mean reduction in cholesterol
level after one month of use is more then 20 milligrams/deciliter (mg/dl). This is because a mean
reduction of this magnitude would be greater than the mean reduction for the current most widely used
drug.
The pharmaceutical company collected data by giving the new drug to a random sample of 50 people
from the population of people with high cholesterol. The reduction in cholesterol level after one month
of use was recorded for each individual in the sample, resulting in a sample mean reduction of 24 mg/dl
and a standard deviation of 15 mg/dl.
a) The regulatory agency decides to use a confidence interval estimate for the population mean
reduction in cholesterol level for the new drug. Provide the 95% confidence interval for the mean
reduction in cholesterol level.
b) Because the 95% confidence interval includes 20, the regulatory agency is not convinced that the
new drug is better than the current best seller. The pharmaceutical company tested the following
hypotheses:
H 0 : µ = 20 versus H a : µ > 20,
where µ represents the population mean reduction in cholesterol level for the new drug, and where
the null hypothesis would be rejected at a level α = .05 . What conclusion was reached? Be sure to
explain which test you used and why you used it.
c) The test procedure resulted in a z-value of 1.89 and a P-value of 0.03. Because the P-value is less
than 0.05, the company believes that there is convincing evidence that the mean reduction in
cholesterol level for the new drug is more than 20. Explain why the confidence interval and the
hypothesis test led to different conclusions.
21) Mr. Corbett and Mr. Surowski both believe that the average mathematically-inclined student at SAS can
score significantly higher than 90 on the American Mathematics Examination 12 (AMC 12). This was
put to the test two weeks ago when 70 SAS students sat for the AMC 12 and scored an overall average
of 91.3 with a standard deviation of 6.54.
a) State null and alternative hypotheses suitable for testing Mr. Corbett and Mr. Surowski’s claim.
b) Compute the P-value associated with x = 91.3 . Do you feel that the resulting average score on the
AMC 12 is significant? Why or why not?
c) If we are to reject your null hypothesis in part a) at a level of significance α = .05 , what is the
outcome?
d) At what confidence level will the confidence interval about x yield the same conclusion as in c)?
Compute this confidence interval.
e) A non-mathematically inclined person could conceivably argue that 91.3 is not much greater than 90
and hence Mr. Corbett and Mr. Surowski’s claim cannot possibly be true. How would you respond
to this?
22) The Shanghai Municipal Government would like to claim that significantly fewer than 5% of its
residents drive private automobiles to work.
a) State null and alternative hypothesis suitable to test this claim.
Next, assume that the Municipal Government commissioned a study to investigate the transportation
habits of 813 residents, with the following results:
Regularly drive a private automobile
Regularly ride in taxis
Regularly take public transportation
Regularly use other forms of transportation
4.6%
15.9%
57.2%
22.0%
100%
b) Can a normal distribution reasonably be applied to this situation? Give supporting calculations.
c) What is the P-value associated with the above data? Do you feel that this is significant?
d) Do the above results justify the Shanghai Municipal Government’s claim? Use the level of
significance of α = .05 .
23) Every once in a while I hear someone saying that left-handed people especially capable at mathematics. I’m not sure I
believe this, but perhaps this is something worth studying. In order to gather some evidence for this, suppose that we
administered a standardized mathematics test to 20 left-handed subjects, where we know that the overall mean on this
test is µ = 12 .3 .
a) State null and alternative hypotheses appropriate to this situation.
Next, assume that the mean score on this test is x = 13.2 with a standard deviation of s=3.32.
b) Compute the P-value of the mean and comment on its relative significance.
c) Would you reject your null hypothesis at any “reasonable” level of significance? Why or why not?
d) What hypotheses, if any, did you make in the application of your statistical methods?