Download Chapter 12: Inference for Proportions

Ch. 12 Review
AP/IB Statistics
1. The weights (in pounds) of four adult males are 150, 220, 205, and 190. The standard
error of these weights is
A) 200. B) 15.05. C) 30.10. D) 17.99. E) 906.25.
2. The one sample t statistic from a sample of n = 17 observations for the two-sided test of
H0:  = 6, Ha:   6 has the value t = 1.95. Based on this information, which of the
following would be true?
A) We would reject the null hypothesis at  = 0.10.
B) 0.025 < P-value < 0.05.
C) We would reject the null hypothesis at  = 0.05.
D) Both B) and C) are correct.
E) We would not reject the null hypothesis in a two-sided test, but would reject it in a
one-sided test at  = 0.01.
3. The one sample t statistic from a sample of n = 10 observations for the one-sided test of
H0:  = 6, Ha:   6 has the value t = 2.79. Based on this information, which of the
following would be true?
A) We would reject the null hypothesis at  = 0.01.
B) We would reject the null hypothesis at  = 0.05.
C) We would reject the null hypothesis at  = 0.10.
D) Both B) and C) are correct.
E) We would fail to reject the null hypothesis at  = 0.10.
4. The weights (in pounds) of three adult males are 170, 215, and 195. The standard error
of the mean of these three weights is
A) 22.546. B) 15.851. C) 13.017. D) 508. E) 193.33.
5. The heights (in inches) of males in the United States are believed to be normally
distributed with mean . The average height of a random sample of 100 American adult
males is found to be X = 69.72 inches, and the standard deviation of the 100 heights is
found to be s = 4.75 inches. The standard error of X is
A) 0.0475. B) 0.475. C) 4.75. D) 47.50. E) None of these.
6. Scores on the Math SAT (SAT-M) are believed to be normally distributed with mean .
The scores of a random sample of three students who recently took the exam are 550,
620, and 480. A 90% confidence interval for  based on these data is
A) 550.00 ± 173.88.
D) 550.00 ± 118.01.
B) 550.00 ± 142.00.
E) 550.00 ± 79.21.
C) 550.00 ± 128.58.
7. Scores on the Math SAT (SAT-M) are believed to be normally distributed with mean .
The scores of a random sample of three students who recently took the exam are 550,
620, and 480. A 95% confidence interval for  based on these data is
A) 550.00 ± 173.88.
D) 550.00 ± 118.01.
B) 550.00 ± 142.00.
E) 550.00 ± 79.21.
C) 550.00 ± 128.58.
Use the following to answer questions 8 through 11:
An SRS of 100 postal employees found that the average amount of time these employees had
worked for the U.S. Postal Service was X = 7 years, with a standard deviation of s = 2 years.
Assume the distribution of the time the population of all postal employees has worked for the
Postal Service is approximately normal with mean . Do the observed data represent evidence
that  has changed from its value of 7.5 years of 20 years ago? To determine this, we test the
hypotheses H0:  = 7.5, Ha:   7.5 using the one-sample t test.
8. The appropriate degrees of freedom for this test are
A) 9. B) 10. C) 19. D) 99. E) 100.
9. The P-value for the one-sample t test is
A) larger than 0.10.
B) between 0.05 and 0.10.
C) between 0.01 and 0.05.
D) below 0.01.
E) impossible to determine, since the standard deviation of the study conducted 20
years ago is not given.
10. A 95% confidence interval for the mean number of years  that a current Postal Service
employee has spent with the Postal Service is
A) 7 ± 2. B) 7 ± 1.984. C) 7 ± 0.4. D) 7 ± 0.3. E) 7 ± 0.2.
11. Suppose the mean and standard deviation we obtained were based on a sample of 25
postal workers, rather than 100. The P-value would be
A) smaller.
B) larger.
C) unchanged, since the difference between X and the hypothesized value  = 7.5 is
unchanged.
D) unchanged, since both groups of workers have the same type of job.
E) unchanged, since the variability measured by the standard deviation stays the same.
12. We wish to see if the dial indicating the oven temperature for a certain model oven is
properly calibrated. Four ovens of this 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 temperatures observed are 310°, 315°, 302°, and 290°. Assuming that the actual
temperatures for this model when the dial is set to 300° are normally distributed with
mean , we test whether the dial is properly calibrated by testing the hypotheses
H0:  = 300, Ha:   300. Based on the data, the value of the one-sample t statistic is
A) 0.7795. B) 4.9521. C) 2.8223. D) 1.5678. E) 1.2345.
Use the following to answer questions 13 and 14:
Bags of a certain brand of tortilla chips are claimed to have a net weight of 14 ounces. Net
weights actually vary slightly from bag to bag and are normally distributed with mean . A
representative of a consumer advocate group wishes to see if there is any evidence that the mean
net weight is less than advertised and so intends to test the hypotheses H0:  = 14, Ha:  < 14.
To do this, he selects 16 bags of this brand at random and determines the net weight of each. He
finds the sample mean to be X = 13.88 ounces and the sample standard deviation to be s = 0.24
ounces.
13. Based on the data above,
A) we would reject H0 at significance level 0.10 but not at level 0.05.
B) we would reject H0 at significance level 0.05 but not at level 0.025.
C) we would reject H0 at significance level 0.025 but not at level 0.01.
D) we would reject H0 at significance level 0.01 but not at level 0.001.
E) we would reject H0 at significance level 0.001.
14. Referring to the information above, suppose we were not sure if the distribution of net
weights was normal. In which of the following circumstances would it not be safe to use
a t procedure in this problem?
A) The mean and median of the data are nearly equal.
B) A stemplot of the data is roughly bell-shaped.
C) A stemplot of the data displays a large outlier.
D) The sample standard deviation is large.
E) A histogram of the data displays moderate skewness.
15. The water diet requires the dieter to drink two cups of water every half hour from when
he gets up until he goes to bed, but otherwise allows him to eat whatever he likes. Four
adult volunteers agree to test the diet. They are weighed prior to beginning the diet and
after six weeks on the diet. The weights (in pounds) are
Person
1
2
3
4
Weight before the diet
180
125
240
140
Weight after six weeks
175
129
220
125
For the population of all adults, assume that weight loss (in pounds) after six weeks on
the diet (weight before beginning the diet – weight after six weeks on the diet) is
normally distributed with mean . To determine if the diet leads to significant weight
loss, we test the hypotheses H0:  = 0, Ha:  > 0. Based on these data,
A) we would fail to reject H0 at significance level 0.10.
B) we would reject H0 at significance level 0.10 but not at level 0.05.
C) we would reject H0 at significance level 0.05 but not at level 0.01.
D) we would reject H0 at significance level 0.01 but not at level 0.001.
E) we would reject H0 at significance level 0.001.
16. Do students tend to improve their Math SAT (SAT-M) score the second time they take
the test? A random sample of four students who took the test twice received the
following scores.
Student
1
2
3
4
First score
450
520
720
600
Second score
440
600
720
630
Assume that the change in SAT-M score (second score – first score) for the population
of all students taking the test twice is normally distributed with mean . A 90%
confidence interval for  is
A) 25.0 ± 64.29. B) 25.0 ± 47.56. C) 25.0 ± 43.08. D) 25.0 ± 33.24.
E) 25.0 ± 28.79.
17. You are thinking of using a t-procedure to test hypotheses about the mean of a
population using a significance level of 0.05. You suspect the distribution of the
population is not normal and may be moderately skewed. Which of the following
statements is correct?
A) You should not use the t-procedure since the population does not have a normal
distribution.
B) You may use the t-procedure provided you discard the high and low observations to
eliminate any outliers.
C) You may use the t-procedure, but you should probably only claim the significance
level is 0.10.
D) You should not use the t-procedure. t-procedures are robust to nonnormality for
confidence intervals but not for tests of hypotheses.
E) You may use the t-procedure provided your sample size is large, say at least 50.
18. To estimate the mean salary  of full professors at American colleges and universities,
you obtain the salaries of a random sample of 400 full professors. The sample mean is
X = $73,220 and the sample standard deviation is s = $4400. A 99% confidence
interval for  is
A) 73,220 ± 11,440. B) 73,220 ± 569. C) 73,220 ± 5567. D) 73,220 ± 431.
E) 73,220 ± 28.6.
19. The heights (in inches) of adult males in the United States are believed to be normally
distributed with mean . The average height of a random sample of 25 American adult
males is found to be X = 69.72 inches, and the standard deviation of the 25 heights is
found to be s = 4.15 inches. A 90% confidence interval for  is
A) 69.72 ± 0.70. B) 69.72 ± 1.09. C) 69.72 ± 1.37. D) 69.72 ± 1.42.
E) 69.72 ± 4.15.
20. A medical researcher wishes to investigate the effectiveness of exercise versus diet in
losing weight. Two groups of 25 overweight adult subjects are used, with a subject in
each group matched to a similar subject in the other group on the basis of a number of
physiological variables. One of the groups is placed on a regular program of vigorous
exercise but with no restriction on diet, while the other group is placed on a strict diet
but with no requirement to exercise. The weight losses after 20 weeks are determined
for each subject, and the difference between each matched pair of subjects (weight loss
of subject in exercise group – weight loss of matched subject in diet group) is computed.
The mean of these differences in weight loss is found to be –1.5 pounds with standard
deviation s = 4 pounds. Is this evidence of a difference in mean weight loss for the two
methods? Let  be the mean of the population of all possible differences and assume
that the distribution of this population is approximately normal. We test the hypotheses
H0:  = 0, Ha:   0 using the matched-pairs t test. The P-value for this test is
A) larger than 0.10.
D) between 0.001 and 0.01.
B) between 0.05 and 0.10.
E) below 0.001.
C) between 0.01 and 0.05.
Use the following to answer questions 21 through 22:
The college newspaper of a large Midwestern university periodically conducts a survey of
students on campus to determine the attitude on campus concerning issues of interest. Pictures of
the students interviewed, along with quotes of their responses, are printed in the paper. Students
are interviewed by a reporter “roaming” the campus who selects students to interview
“haphazardly.” On a particular day the reporter interviews five students and asks them if they
feel there is adequate student parking on campus. Four of the students say no. The sample
proportion p̂ that respond “no” is thus 0.8.
21. Referring to the information above, the standard error of p̂ is about
A) 0.18. B) 0.64. C) 0.4. D) 0.8. E) 0.032.
22. Referring to the information above, which of the following assumptions for inference
about a proportion using a confidence interval are violated in this example?
A) n is so large that both npˆ and n(1 - p̂ ) are at least 10.
B) The population is at least 10 times as large as the sample.
C) We are interested in inference about a proportion.
D) The data are an SRS from the population of interest.
E) Conditions stated in A. and D. are both correct.
Use the following to answer questions 23 and 24:
A radio talk show host with a large audience is interested in the proportion p̂ of adults in his
listening area that think the drinking age should be lowered to 18. To find out, he poses the
following question to his listeners: “Do you think that the drinking age should be reduced to 18
in light of the fact that 18-year-olds are eligible for military service?” He asks listeners to phone
in and vote “yes” if they agree the drinking age should be lowered and “no” if not.
23. You are told that the sample proportion p̂ of those who phoned in and answered yes is
p̂ = 0.70 and the standard error of the sample proportion is 0.0459. The number of
people who phoned in is approximately
A) 50.
B) 100.
C) 150.
D) 200.
E) cannot be determined from the information given.
24. Of the 100 people who phoned in, 70 answered “yes.” Which of the following
assumptions for inference about a proportion using a confidence interval are violated?
A) The desired confidence level is not given.
B) The population is at least 10 times as large as the sample.
C) n is so large that both the count of successes npˆ and the count of failures n(1 – p̂ )
are 10 or more.
D) The population is assumed to be Normal.
E) There appear to be no violations.
25. One hundred rats whose mothers were exposed to high levels of tobacco smoke during
pregnancy were put through a simple maze. The maze required the rats to make a choice
between going left or going right at the outset. Seventy of the rats went right when
running the maze for the first time. Assume that the 100 rats can be considered an SRS
from the population all rats born to mothers exposed to high levels of tobacco smoke
during pregnancy. (Note that this assumption may or may not be reasonable, but
researchers often assume lab rats are representative of such larger populations since lab
rats are often bred to have very uniform characteristics.) The standard error for the
sample proportion p̂ of rats who went right the first time when running the maze is
A) 0.0023. B) 0.0484. C) 0.0584. D) 0.0458. E) 0.0548.
Use the following to answer questions 26 and 27:
A newspaper conducted a statewide survey concerning the 1998 race for state senator. The
newspaper took a random sample (assume it is an SRS) of 1200 registered voters and found that
620 would vote for the Republican candidate. Let p represent the proportion of registered voters
in the state that would vote for the Republican candidate.
26. Referring to the information above, a 90% confidence interval for p is
A) 0.517 ± 0.014. B) 0.517 ± 0.022. C) 0.517 ± 0.024. D) 0.517 ± 0.028.
E) 0.517 ± 0.249.
27. Referring to the information above, what sample size would you need in order to
estimate p with margin of error 0.01 with 95% confidence? Use the guess p = 0.5 as the
value for p.
A) 49. B) 1500. C) 4800. D) 4900. E) 9604.
Use the following to answer questions 28 through 30:
A noted psychic was tested for ESP. The psychic was presented with 300 cards face down and
asked to determine if the card featured one of five symbols: star, cross, circle, square, or three
wavy lines. The psychic was correct in 75 cases. Let p represent the probability that the psychic
correctly identifies the symbol on the card in a random trial.
28. Referring to the information above, and assuming that the 300 trials can be treated as an
SRS from the population of all guesses the psychic would make in his lifetime a 90%
confidence interval for p is
A) 0.25 ± 0.06016.
B) 0.25 ± 0.04112.
C) 0.25 ± 0.07559.
D) 0.25 ± 0.09485.
E) None of these.
29. Referring to the information above, suppose you wished to see if there is evidence that
the psychic is doing better than if he were just guessing. To do this, you test the
hypotheses H0: p = 0.20, Ha: p > 0.20. The P-value of your test is
A) greater than 0.10.
D) between 0.001 and 0.01.
B) between 0.05 and 0.10.
E) below 0.001.
C) between 0.01 and 0.05.
30. Referring to the information above, what sample size would you need in order to
estimate p with margin of error 0.01 with 95% confidence? Use the guess p = 0.20 as
the value for p.
A) 7203. B) 4330. C) 9604. D) 1176. E) 6147.
Use the following to answer questions 31 and 32:
After a certain college football team once again lost a game to the college’s arch rival, the alumni
association conducted a survey to see if alumni were in favor of firing the coach. An SRS of 100
alumni from the population of all living alumni was taken. Sixty-four of the alumni in the sample
were in favor of firing the coach. Let p represent the proportion of all living alumni who favor
firing the coach.
31. Referring to the information above, a 95% confidence interval for p is
A) 0.64 ± 0.009. B) 0.64 ± 0.079. C) 0.64 ± 0.094. D) 0.64 ± 0.124.
E) 0.64 ± 0.360.
32. Referring to the information above, suppose you wished to see if the majority of alumni
are in favor of firing the coach. To do this, you test the hypotheses H0: p = 0.50, Ha: p >
0.50. The P-value of your test is
A) greater than 0.10.
D) between 0.001 and 0.01.
B) between 0.05 and 0.10.
E) below 0.001.
C) between 0.01 and 0.05.
Use the following to answer questions 33 and 34:
An inspector inspects large truckloads of potatoes to determine the proportion p in the shipment
with major defects prior to using the potatoes to make potato chips. Unless there is clear
evidence that this proportion is less than 0.10, the inspector will reject the shipment. To reach a
decision, she will test the hypotheses H0: p = 0.10, Ha: p < 0.10 using the large sample test for a
population proportion. She selects an SRS of 100 potatoes from the shipment of over 2000
potatoes on a truck. Suppose that only 8 of the potatoes sampled are found to have major defects.
33. Referring to the information above, the P-value of the test is
A) 0.2525. B) 0.0456. C) 0.4544. D) 0.0011. E) less than 0.0002.
34. Referring to the information above, which of the following assumptions for inference
about a proportion using a hypothesis test is violated?
A) The data are an SRS from the population of interest.
B) The population is at least 10 times as large as the sample.
C) The population is assumed to be Normal.
D) n is so large that both np0 and n(1 – p0) are 10 or more, where p0 is the proportion
with major defects if the null hypothesis is true.
E) There appear to be no violations.
35. A newspaper conducted a statewide survey concerning the 1998 race for governor. The
newspaper took a random sample (assume it is an SRS) of 1200 registered voters and
found that 640 would vote for the Democratic candidate. Is this evidence that a clear
majority of the population would vote for the Democratic candidate? To answer this,
test the hypotheses H0: p = 0.50, Ha: p > 0.50. The P-value of your test is
A) 0.4920. B) 0.0330. C) 0.0209. D) 0.0105. E) less than 0.0002.
Use the following to answer questions 36 and 37:
A sociologist is studying the effect of having children within the first two years of marriage on
the divorce rate. Using hospital birth records, she selects a random sample of 200 couples that
had a child within the first two years of marriage. Following up on these couples, she finds that
80 couples are divorced within five years.
36. Referring to the information above, a 80% confidence interval for the proportion p of all
couples that had a child within the first two years of marriage and are divorced within
five years is
A) 0.40 ± 0.0004. B) 0.40 ± 0.035. C) 0.40 ± 0.0444. D) 0.40 ± 0.057.
E) 0.40 ± 0.068.
37. Referring to the information above, in order to determine if having children within the
first two years of marriage increases the divorce rate, we should
A) test the hypotheses H0: p = 0.50, Ha: p  0.50.
B) test the hypotheses H0: p = 0.40, Ha: p ≠ 0.40.
C) test the hypotheses H0: p = 0.40, Ha: p > 0.40.
D) test the hypotheses H0: p = 0.50, Ha: p > 0.50.
E) do none of the above.
38. An inspector inspects large truckloads of potatoes to determine the proportion p in the
shipment with major defects prior to using the potatoes to make potato chips. She
intends to compute a 95% confidence interval for p. To do so, she selects an SRS of 50
potatoes from a shipment of over 2000 potatoes on a truck. Suppose that only two of the
potatoes sampled are found to have major defects. Which of the following assumptions
for inference about a proportion using a confidence interval are violated?
A) n is so large that both np0 < 10 and n(1 – p0) < 10.
B) n is so large that both npˆ and n(1 – p̂ ) are 10 or more.
C) The population size is too small.
D) The population is at least 10 times as large as the sample.
E) There appear to be no violations.
39. Eighty rats whose mothers were exposed to high levels of tobacco smoke during
pregnancy were put through a simple maze. The maze required the rats to make a choice
between going left or going right at the outset. Sixty of the rats went right when running
the maze for the first time. Assume that the 80 rats can be considered an SRS from the
population all rats born to mothers exposed to high levels of tobacco smoke during
pregnancy. (Note that this assumption may or may not be reasonable, but researchers
often assume lab rats are representative of such larger populations since lab rats are
often bred to have very uniform characteristics.) The standard error for the sample
proportion p̂ of rats who went right the first time when running the maze is
A) 0.0023. B) 0.0484. C) 0.0548. D) 0.0458. E) 0.0448.
A noted psychic was tested for ESP. The psychic was presented with 200 cards face down and
asked to determine if the card featured one of five symbols: star, cross, circle, square, or three
wavy lines. The psychic was correct in 50 cases. Let p represent the probability that the psychic
correctly identifies the symbol on the card in a random trial.
40. Referring to this information, and assuming that the 200 trials can be treated as an SRS
from the population of all guesses the psychic would make in his lifetime a 95%
confidence interval for p is
A) 0.25 ± 0.06001.
B) 0.25 ± 0.070.
C) 0.25 ± 0.07595.
D) 0.25 ± 0.090.
E) None of these.
41. A sociologist is studying the effect of having children within the first two years of
marriage on the divorce rate. Using hospital birth records, she selects a random sample of 200
couples that had a child within the first two years of marriage. Following up on these couples,
she finds that 80 couples are divorced within five years. A 95% confidence interval for the
proportion p of all couples that had a child within the first two years of marriage and are
divorced within five years is
A) 0.40 ± 0.0024. B) 0.40 ± 0.035. C) 0.40 ± 0.554. D) 0.40 ± 0.0587.
E) 0.40 ± 0.0679.
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