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Ch. 5 Normal Probability Distributions

5.1 Introduction to Normal Distributions and the Standard Normal Distribution

1 Find Areas Under the Standard Normal Curve

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Provide an appropriate response.

1) Find the area of the indicated region under the standard normal curve.

A) 0.0968 B) 0.9032 C) 0.4032 D) 0.0823

2) Find the area of the indicated region under the standard normal curve.

A) 0.9032 B) 0.0968 C) 0.0823 D) 0.9177

3) Find the area of the indicated region under the standard normal curve.

A) 0.6562 B) 1.309 C) 0.3438 D) 0.309

4) Find the area of the indicated region under the standard normal curve.

A) 0.1504 B) 0.1292 C) 0.8489 D) 0.0212

5) Find the area under the standard normal curve to the left of z = 1.5.

A) 0.9332 B) 0.0668 C) 0.5199 D) 0.7612

6) Find the area under the standard normal curve to the left of z = 1.25.

A) 0.8944 B) 0.1056 C) 0.2318 D) 0.7682

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7) Find the area under the standard normal curve to the right of z = 1.

A) 0.1587 B) 0.8413 C) 0.1397 D) 0.5398

8) Find the area under the standard normal curve to the right of z = -1.25.

A) 0.8944 B) 0.5843 C) 0.6978 D) 0.7193

9) Find the area under the standard normal curve between z = 0 and z = 3.

A) 0.4987 B) 0.9987 C) 0.0010 D) 0.4641

10) Find the area under the standard normal curve between z = 1 and z = 2.

A) 0.1359 B) 0.8413 C) 0.5398 D) 0.2139

11) Find the area under the standard normal curve between z = -1.5 and z = 2.5.

A) 0.9270 B) 0.7182 C) 0.6312 D) 0.9831

12) Find the area under the standard normal curve between z = 1.5 and z = 2.5.

A) 0.0606 B) 0.9938 C) 0.9332 D) 0.9816

13) Find the area under the standard normal curve between z = -1.25 and z = 1.25.

A) 0.7888 B) 0.8817 C) 0.6412 D) 0.2112

14) Find the sum of the areas under the standard normal curve to the left of z = -1.25 and to the right of z = 1.25.

A) 0.2112 B) 0.7888 C) 0.1056 D) 0.3944

2 Find Probabilities Using the Standard Normal Curve

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Find the probability of z occurring in the indicated region.

15)

0 1.82 z

A) 0.9656 B) 0.0344 C) 0.9772 D) 0.4656

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16)

-0.59 0 z

A) 0.2776 B) 0.7224 C) 0.2224 D) 0.1894

17)

-1.33 0 z

A) 0.9082 B) 0.0918 C) 0.9332 D) 0.0668

18)

0 1.75 z

A) 0.0401 B) 0.0668 C) 0.9599 D) 0.0228

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19)

-2 0 3 z

A) 0.9772 B) 0.0228 C) 0.9544 D) 0.0456

20)

0 1.50 z

A) 0.4332 B) 0.5668 C) 0.0668 D) 0.9332

3 Find Probabilities Using the Standard Normal Distribution

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Provide an appropriate response.

21) Use the standard normal distribution to find P(0 < z < 2.25).

A) 0.4878 B) 0.5122 C) 0.8817 D) 0.7888

22) Use the standard normal distribution to find P(-2.25 < z < 0).

A) 0.4878 B) 0.5122 C) 0.6831 D) 0.0122

23) Use the standard normal distribution to find P(-2.25 < z < 1.25).

A) 0.8822 B) 0.0122 C) 0.4878 D) 0.8944

24) Use the standard normal distribution to find P(-2.50 < z < 1.50).

A) 0.9270 B) 0.8822 C) 0.6167 D) 0.5496

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25) Use the standard normal distribution to find P(z < -2.33 or z > 2.33).

A) 0.0198 B) 0.9802 C) 0.7888 D) 0.0606

26) For the standard normal curve, find the z-score that corresponds to the third quartile.

A) 0.67 B) -0.67 C) 0.77 D) -0.23

27) For the standard normal curve, find the z-score that corresponds to the first quartile.

A) -0.67 B) 0.67 C) 0.77 D) -0.23

28) For the standard normal curve, find the z-score that corresponds to the first decile.

A) -1.28 B) 1.28 C) -2.33 D) 0.16

5.2 Normal Distributions: Finding Probabilities

1 Find Probabilities for Normally Distributed Variables

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Provide an appropriate response. Use the Standard Normal Table to find the probability.

1) IQ test scores are normally distributed with a mean of 100 and a standard deviation of 15. An individuals IQ

score is found to be 110. Find the z-score corresponding to this value.

A) 0.67 B) -0.67 C) 1.33 D) -1.33

2) IQ test scores are normally distributed with a mean of 100 and a standard deviation of 15. An individuals IQ

score is found to be 90. Find the z-score corresponding to this value.

A) -0.67 B) 0.67 C) 1.33 D) -1.33

3) IQ test scores are normally distributed with a mean of 100 and a standard deviation of 15. An individuals IQ

score is found to be 120. Find the z-score corresponding to this value.

A) 1.33 B) -0.67 C) 0.67 D) -1.33

4) IQ test scores are normally distributed with a mean of 100 and a standard deviation of 15. Find the IQ score

that corresponds to a z-score of 1.96.

A) 129.4 B) 115.6 C) 122.4 D) 132.1

5) IQ test scores are normally distributed with a mean of 102 and a standard deviation of 19. An individuals IQ

score is found to be 124. Find the z-score corresponding to this value.

A) 1.16 B) -1.16 C) 0.86 D) -0.86

6) IQ test scores are normally distributed with a mean of 100 and a standard deviation of 12. An individuals IQ

score is found to be 127. Find the z-score corresponding to this value.

A) 2.25 B) -2.25 C) 0.44 D) -0.44

7) The lengths of pregnancies of humans are normally distributed with a mean of 268 days and a standard

deviation of 15 days. Find the probability of a pregnancy lasting more than 300 days.

A) 0.0166 B) 0.9834 C) 0.2375 D) 0.3189

8) The lengths of pregnancies of humans are normally distributed with a mean of 268 days and a standard

deviation of 15 days. Find the probability of a pregnancy lasting less than 250 days.

A) 0.1151 B) 0.1591 C) 0.0606 D) 0.0066

9) The distribution of cholesterol levels in teenage boys is approximately normal with = 170 and = 30 (Source:

U.S. National Center for Health Statistics). Levels above 200 warrant attention. Find the probability that a

teenage boy has a cholesterol level greater than 200.

A) 0.1587 B) 0.8413 C) 0.3419 D) 0.2138

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10) The distribution of cholesterol levels in teenage boys is approximately normal with = 170 and = 30 (Source:

U.S. National Center for Health Statistics). Levels above 200 warrant attention. Find the probability that a

teenage boy has a cholesterol level greater than 225.

A) 0.0336 B) 0.0606 C) 0.0718 D) 0.0012

11) An airline knows from experience that the distribution of the number of suitcases that get lost each week on a

certain route is approximately normal with = 15.5 and = 3.6. What is the probability that during a given

week the airline will lose less than 20 suitcases?

A) 0.8944 B) 0.3944 C) 0.1056 D) 0.4040

12) An airline knows from experience that the distribution of the number of suitcases that get lost each week on a

certain route is approximately normal with = 15.5 and = 3.6. What is the probability that during a given

week the airline will lose more than 20 suitcases?

A) 0.1056 B) 0.3944 C) 0.4040 D) 0.8944

13) An airline knows from experience that the distribution of the number of suitcases that get lost each week on a

certain route is approximately normal with = 15.5 and = 3.6. What is the probability that during a given

week the airline will lose between 10 and 20 suitcases?

A) 0.8314 B) 0.3944 C) 0.1056 D) 0.4040

14) Assume that the salaries of elementary school teachers in the United States are normally distributed with a

mean of $32,000 and a standard deviation of $3000. If a teacher is selected at random, find the probability that

he or she makes more than $36,000.

A) 0.0918 B) 0.9082 C) 0.1056 D) 0.4040

15) Assume that the salaries of elementary school teachers in the United States are normally distributed with a

mean of $32,000 and a standard deviation of $3000. If a teacher is selected at random, find the probability that

he or she makes less than $28,000.

A) 0.0918 B) 0.9981 C) 0.2113 D) 0.9827

16) Assume that the heights of women are normally distributed with a mean of 63.6 inches and a standard

deviation of 2.5 inches. The cheerleaders for a local professional basketball team must be between 65.5 and 68.0

inches. If a woman is randomly selected, what is the probability that her height is between 65.5 and 68.0

inches?

A) 0.1844 B) 0.9608 C) 0.7881 D) 0.3112

17) The lengths of pregnancies of humans are normally distributed with a mean of 268 days and a standard

deviation of 15 days. A baby is premature if it is born three weeks early. What percent of babies are born

prematurely?

A) 8.08% B) 6.81% C) 9.21% D) 10.31%

18) The distribution of cholesterol levels in teenage boys is approximately normal with = 170 and = 30 (Source:

U.S. National Center for Health Statistics). Levels above 200 warrant attention. What percent of teenage boys

have levels between 170 and 225?

A) 3.36% B) 6.06% C) 46.64% D) 56.13%

19) Assume that blood pressure readings are normally distributed with = 120 and = 8. A blood pressure

reading of 145 or more may require medical attention. What percent of people have a blood pressure reading

greater than 145?

A) 0.09% B) 99.91% C) 6.06% D) 11.09%

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20) Assume that the heights of American men are normally distributed with a mean of 69.0 inches and a standard

deviation of 2.8 inches. The U.S. Marine Corps requires that men have heights between 64 and 78 inches. Find

the percent of men meeting these height requirements.

A) 96.26% B) 3.67% C) 99.93% D) 31.12%

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

21) Assume that the heights of women are normally distributed with a mean of 63.6 inches and a standard

deviation of 2.5 inches. The U.S. Army requires that the heights of women be between 58 and 80 inches. If a

woman is randomly selected, what is the probability that her height is between 58 and 80 inches?

2 Interpret Normal Distributions

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Provide an appropriate response. Use the Standard Normal Table to find the probability.

22) The lengths of pregnancies are normally distributed with a mean of 268 days and a standard deviation of 15

days. Out of 50 pregnancies, how many would you expect to last less than 250 days?

23) The distribution of cholesterol levels in teenage boys is approximately normal with = 170 and = 30. Levels

above 200 warrant attention. If 95 teenage boys are examined, how many would you expect to have cholesterol

levels greater than 225?

24) An airline knows from experience that the distribution of the number of suitcases that get lost each week on a

certain route is approximately normal with = 15.5 and = 3.6. In one year, how many weeks would you

expect the airline to lose between 10 and 20 suitcases?

25) Assume that the heights of women are normally distributed with a mean of 63.6 inches and a standard

deviation of 2.5 inches. The U.S. Army requires that the heights of women be between 58 and 80 inches. If 200

women want to enlist in the U.S. Army, how many would you expect to meet the height requirements?

26) Assume that the heights of men are normally distributed with a mean of 69.0 inches and a standard deviation

of 2.8 inches. The U.S. Marine Corps requires that the heights of men be between 64 and 78 inches. If 500 men

want to enlist in the U.S. Marine Corps, how many would you not expect to meet the height requirements?

5.3 Normal Distributions: Finding Values

1 Find a z-score Given the Area Under the Normal Curve

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Provide an appropriate response.

1) Find the z-score that corresponds to the given area under the standard normal curve.

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2) Find the z-score that corresponds to the given area under the standard normal curve.

3) Find the z-score that corresponds to the given area under the standard normal curve.

4) Find the z-score that corresponds to the given area under the standard normal curve.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

5) Find the z-scores for which 90% of the distributions area lies between -z and z.

A) (-1.645, 1.645) B) (-2.33, 2.33) C) (-1.96, 1.96) D) (-0.99, 0.99)

6) Find the z-scores for which 98% of the distributions area lies between -z and z.

A) (-2.33, 2.33) B) (-1.645, 1.645) C) (-1.96, 1.96) D) (-0.99, 0.99)

7) Find the z-score for which 70% of the distributions area lies to its right.

A) -0.53 B) -0.98 C) -0.81 D) -0.47

8) Find the z-score that is greater than the mean and for which 70% of the distributions area lies to its left.

A) 0.53 B) 0.98 C) 0.81 D) 0.47

9) Use a standard normal table to find the z-score that corresponds to the cumulative area of 0.01.

A) -2.33 B) 2.33 C) 0.255 D) -0.255

10) Find the z-score that has 84.85% of the distributions area to its right.

A) -1.03 B) 1.03 C) -0.39 D) 0.39

11) Find the z-score for which 99% of the distributions area lies between -z and z.

A) (-2.575, 2.575) B) (-1.28, 1.28) C) (-1.645, 1.645) D) (-1.96, 1.96)

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2 Find a z-score Given a Percentile

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Provide an appropriate response.

12) For the standard normal curve, find the z-score that corresponds to the 90th percentile.

A) 1.28 B) 0.28 C) 1.52 D) 2.81

13) For the standard normal curve, find the z-score that corresponds to the 30th percentile.

A) -0.53 B) -0.98 C) -0.47 D) -0.12

14) For the standard normal curve, find the z-score that corresponds to the 7th decile.

A) 0.53 B) 0.98 C) 0.47 D) 0.12

15) Use a standard normal table to find the z-score that corresponds to the 30th percentile.

A) -0.525 B) 1.88 C) -2.75 D) 0.84

3 Use Normal Distributions to Answer Questions

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Provide an appropriate response.

16) IQ test scores are normally distributed with a mean of 100 and a standard deviation of 15. Find the x -score that

corresponds to a z-score of 2.33.

A) 134.95 B) 125.95 C) 139.55 D) 142.35

17) IQ test scores are normally distributed with a mean of 100 and a standard deviation of 15. Find the x -score that

corresponds to a z-score of -1.645.

A) 75.3 B) 79.1 C) 82.3 D) 91.0

18) The scores on a mathematics exam have a mean of 77 and a standard deviation of 8. Find the x-value that

corresponds to the z-score 2.575.

A) 97.6 B) 56.4 C) 85.0 D) 79.6

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

19) A mathematics professor gives two different tests to two sections of his college algebra courses. The first class

has a mean of 56 with a standard deviation of 9 while the second class has a mean of 75 with a standard

deviation of 15. A student from the first class scores a 62 on the test while a student from the second class

scores an 83 on the test. Compare the scores.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

20) Compare the scores: a score of 75 on a test with a mean of 65 and a standard deviation of 8 and a score of 75 on

a test with a mean of 70 and a standard deviation of 4.

A) The two scores are statistically the same.

B) A score of 75 with a mean of 65 and a standard deviation of 8 is better.

C) A score of 75 with a mean of 70 and a standard deviation of 4 is better.

D) You cannot determine which score is better from the given information.

21) Compare the scores: a score of 88 on a test with a mean of 79 and a score of 78 on a test with a mean of 70.

A) You cannot determine which score is better from the given information.

B) The two scores are statistically the same.

C) A score of 75 with a mean of 65 and a standard deviation of 8 is better.

D) A score of 75 with a mean of 70 and a standard deviation of 4 is better.

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22) Compare the scores: a score of 220 on a test with a mean of 200 and a standard deviation of 21 and a score of 90

on a test with a mean of 80 and a standard deviation of 8.

A) A score of 90 with a mean of 80 and a standard deviation of 8 is better.

B) A score of 220 with a mean of 200 and a standard deviation of 21 is better.

C) The two scores are statistically the same.

D) You cannot determine which score is better from the given information.

23) Two high school students took equivalent language tests, one in German and one in French. The student taking

the German test, for which the mean was 66 and the standard deviation was 8, scored an 82, while the student

taking the French test, for which the mean was 27 and the standard deviation was 5, scored a 35. Compare the

scores.

A) A score of 82 with a mean of 66 and a standard deviation of 8 is better.

B) A score of 35 with a mean of 27 and a standard deviation of 5 is better.

C) The two scores are statistically the same.

D) You cannot determine which score is better from the given information.

24) SAT scores have a mean of 1026 and a standard deviation of 209. ACT scores have a mean of 20.8 and a

standard deviation of 4.8. A student takes both tests while a junior and scores 1130 on the SAT and 25 on the

ACT. Compare the scores.

A) A score of 25 on the ACT test was better.

B) A score of 1130 on the SAT test was better.

C) The two scores are statistically the same.

D) You cannot determine which score is better from the given information.

25) SAT scores have a mean of 1026 and a standard deviation of 209. ACT scores have a mean of 20.8 and a

standard deviation of 4.8. A student takes both tests while a junior and scores 860 on the SAT and 16 on the

ACT. Compare the scores.

A) A score of 860 on the SAT test was better.

B) A score of 16 on the ACT test was better.

C) The two scores are statistically the same.

D) You cannot determine which score is better from the given information.

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

26) Assume that blood pressure readings are normally distributed with = 111 and = 7. A researcher wishes to

select people for a study but wants to exclude the top and bottom 10 percent. What would be the upper and

lower readings to qualify people to participate in the study?

27) Assume that the salaries of elementary school teachers in the United States are normally distributed with a

mean of $32,000 and a standard deviation of $4000. What is the cutoff salary for teachers in the top 10%?

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

28) Assume that the salaries of elementary school teachers in the United States are normally distributed with a

mean of $28,000 and a standard deviation of $3000. What is the cutoff salary for teachers in the bottom 10%?

A) $24,160 B) $31,840 C) $23,065 D) $32,935

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

29) The times for completing one circuit of a bicycle course are normally distributed with a mean of 64.5 minutes

and a standard deviation of 7.8 minutes. An association wants to sponsor a race but will cut the bottom 25% of

riders. In a trial run, what should be the cutoff time?

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30) Assume that the heights of men are normally distributed with a mean of 70.6 inches and a standard deviation

of 2.2 inches. If the top 5 percent and bottom 5 percent are excluded for an experiment, what are the cutoff

heights to be eligible for this experiment? Round your answers to one decimal place.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

31) Assume that the heights of women are normally distributed with a mean of 63.5 inches and a standard

deviation of 2.5 inches. Find Q3, the third quartile that separates the bottom 75% from the top 25%.

A) 65.2 B) 61.8 C) 66.4 D) 66.7

32) The body temperatures of adults are normally distributed with a mean of 98.6 F and a standard deviation of

0.19 F. What temperature represents the 95th percentile?

A) 98.91 F B) 98.29 F C) 98.84 F D) 98.97 F

33) In a certain normal distribution, find the standard deviation when = 50 and 10.56% of the area lies to the

right of 55.

A) 4 B) 2 C) 3 D) 5

34) In a certain normal distribution, find the mean when = 5 and 5.48% of the area lies to the left of 78.

A) 86 B) 70 C) 94 D) 62

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

35) In a certain normal distribution, 6.3% of the area lies to the left of 36 and 6.3% of the area lies to the right of 42.

Find the mean and the standard deviation .

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

36) A tire company finds the lifespan for one brand of its tires is normally distributed with a mean of 48,400 miles

and a standard deviation of 5000 miles. If the manufacturer is willing to replace no more than 10% of the tires,

what should be the approximate number of miles for a warranty?

A) 42,000 B) 54,800 C) 40,175 D) 56,625

5.4 Sampling Distributions and the Central Limit Theorem

1 Interpret Sampling Distributions

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Provide an appropriate response.

1) The distribution of room and board expenses per year at a four-year college is normally distributed with a

mean of $5850 and standard deviation of $1125. Random samples of size 20 are drawn from this population

and the mean of each sample is determined. Which of the following mean expenses would be considered

unusual?

A) $5180 B) $6350 C) $6180 D) none of these

2 Find Probabilities

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Provide an appropriate response.

2) The lengths of pregnancies are normally distributed with a mean of 273 days and a standard deviation of 20

days. If 64 women are randomly selected, find the probability that they have a mean pregnancy between 273

days and 275 days.

A) 0.2881 B) 0.7881 C) 0.2119 D) 0.5517

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3) The lengths of pregnancies are normally distributed with a mean of 268 days and a standard deviation of 15

days. If 64 women are randomly selected, find the probability that they have a mean pregnancy between 266

days and 268 days.

A) 0.3577 B) 0.7881 C) 0.2881 D) 0.5517

4) Assume that the heights of women are normally distributed with a mean of 63.6 inches and a standard

deviation of 2.5 inches. If 100 women are randomly selected, find the probability that they have a mean height

greater than 63.0 inches.

A) 0.9918 B) 0.0082 C) 0.2881 D) 0.8989

5) Assume that the heights of women are normally distributed with a mean of 63.6 inches and a standard

deviation of 2.5 inches. If 75 women are randomly selected, find the probability that they have a mean height

between 63 and 65 inches.

A) 0.9811 B) 0.3071 C) 0.0188 D) 0.2119

6) Assume that the heights of men are normally distributed with a mean of 68.4 inches and a standard deviation

of 2.8 inches. If 64 men are randomly selected, find the probability that they have a mean height greater than

69.4 inches.

A) 0.0021 B) 0.8188 C) 0.9005 D) 9.9671

7) Assume that the heights of men are normally distributed with a mean of 69.0 inches and a standard deviation

of 2.8 inches. If 64 men are randomly selected, find the probability that they have a mean height between 68

and 70 inches.

A) 0.9958 B) 0.9979 C) 0.0021 D) 0.9015

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

8) The body temperatures of adults are normally distributed with a mean of 98.6 F and a standard deviation of

0.60 F. If 25 adults are randomly selected, find the probability that their mean body temperature is less than

99 F.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

9) The body temperatures of adults are normally distributed with a mean of 98.6 F and a standard deviation of

0.60 F. If 36 adults are randomly selected, find the probability that their mean body temperature is greater

than 98.4 F.

A) 0.9772 B) 0.0228 C) 0.8188 D) 0.9360

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

10) Assume that the salaries of elementary school teachers in the United States are normally distributed with a

mean of $32,000 and a standard deviation of $3000. If 100 teachers are randomly selected, find the probability

that their mean salary is less than $32,500.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

11) Assume that the salaries of elementary school teachers in the United States are normally distributed with a

mean of $32,000 and a standard deviation of $3000. If 100 teachers are randomly selected, find the probability

that their mean salary is greater than $32,500.

A) 0.0475 B) 0.9525 C) 0.3312 D) 0.1312

12) Assume that blood pressure readings are normally distributed with a mean of 125 and a standard deviation of

4.8. If 36 people are randomly selected, find the probability that their mean blood pressure will be less than 127.

A) 0.9938 B) 0.0062 C) 0.8819 D) 0.8615

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13) Assume that blood pressure readings are normally distributed with a mean of 120 and a standard deviation of

8. If 100 people are randomly selected, find the probability that their mean blood pressure will be greater than

122.

A) 0.0062 B) 0.9938 C) 0.8819 D) 0.8615

14) The average number of pounds of red meat a person consumes each year is 196 with a standard deviation of 22

pounds (Source: American Dietetic Association). If a sample of 50 individuals is randomly selected, find the

probability that the mean of the sample will be less than 200 pounds.

A) 0.9015 B) 0.0985 C) 0.7613 D) 0.8815

15) The average number of pounds of red meat a person consumes each year is 196 with a standard deviation of 22

pounds (Source: American Dietetic Association). If a sample of 50 individuals is randomly selected, find the

probability that the mean of the sample will be greater than 200 pounds.

A) 0.0985 B) 0.7613 C) 0.8815 D) 0.9015

16) A coffee machine dispenses normally distributed amounts of coffee with a mean of 12 ounces and a standard

deviation of 0.2 ounce. If a sample of 9 cups is selected, find the probability that the mean of the sample will be

less than 12.1 ounces. Find the probability if the sample is just 1 cup.

A) 0.9332; 0.6915 B) 0.9332; 0.1915 C) 0.4332; 0.1915 D) 0.4332; 0.6915

17) A coffee machine dispenses normally distributed amounts of coffee with a mean of 12 ounces and a standard

deviation of 0.2 ounce. If a sample of 9 cups is selected, find the probability that the mean of the sample will be

greater than 12.1 ounces. Find the probability if the sample is just 1 cup.

A) 0.0668; 0.3085 B) 0.0668; 0.8085 C) 0.5668; 0.8085 D) 0.5668; 0.3085

3 Use Central Limit Theorem to Find Mean/Std Error of Mean

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Use the Central Limit Theorem to find the mean and standard error of the mean of the indicated sampling distribution.

18) The amounts of time employees of a telecommunications company have worked for the company are normally

distributed with a mean of 5.1 years and a standard deviation of 2.0 years. Random samples of size 18 are

drawn from the population and the mean of each sample is determined.

A) 5.1 years, 0.47 years B) 5.1 years, 0.11 years

C) 1.2 years, 0.47 years D) 1.2 years, 2.0 years

19) The monthly rents for studio apartments in a certain city have a mean of $1040 and a standard deviation of

$170. Random samples of size 30 are drawn from the population and the mean of each sample is determined.

A) $1040, $31.04 B) $1040, $5.67 C) $189.88, $31.04 D) $189.88, $170

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Provide an appropriate response.

20) Scores on a test have a mean of 72 and a standard deviation of 12. Random samples of size 64 are drawn from

this population and the mean of each sample is determined. Use the Central Limit Theorem to find the mean

and standard error of the mean of the sampling distribution. Then sketch a graph of the sampling distribution.

A) 72, 1.5

67.5 69 70.5 72 73.5 75 76.5 z

B) 72, 12

36 48 60 72 84 96 108 z

C) 72, 9

z 45 54 63 72 81 90 99

D) 9, 1.5

z 4.5 6 7.5 9 10.5 12 13.5

21) The weights of people in a certain population are normally distributed with a mean of 155 lb and a standard

deviation of 20 lb. Find the mean and standard error of the mean for this sampling distribution when using

random samples of size 3.

A) 155, 11.55 B) 155, 20 C) 155, 3 D) 155, 6.67

22) The mean annual income for adult women in one city is $28,520 and the standard deviation of the incomes is

$6000. The distribution of incomes is skewed to the right. Find the mean and standard error of the mean for

this sampling distribution when using random samples of size 78. Round your answers to the nearest dollar.

A) $28,520, $679 B) $28,520, $6000 C) $28,520, $78 D) $28,520, $77

23) What happens to the mean and standard deviation of the distribution of sample means as the size of the

sample decreases?

A) The mean of the sample means stays constant and the standard error increases.

B) The mean of the sample means stays constant and the standard error decreases.

C) The mean of the sample means decreases and the standard error increases.

D) The mean of the sample means increases and the standard error stays constant.

24) If the sample size is multiplied by 4, what happens to the standard deviation of the distribution of sample

means?

A) The standard error is halved.

B) The standard error is decreased by a factor of 4.

C) The standard error is doubled.

D) The standard error is increased by a factor of 4.

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4 Determine if the Finite Correction Factor Should be Used

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Provide an appropriate response.

25) A study of 800 homeowners in a certain area showed that the average value of the homes is $182,000 and the

standard deviation is $15,000. If 50 homes are for sale, find the probability that the mean value of these homes

is less than $185,000. Remember: check to see if the finite correction factor applies.

26) A study of 800 homeowners in a certain area showed that the average value of the homes is $182,000 and the

standard deviation is $15,000. If 50 homes are for sale, find the probability that the mean value of these homes

is greater than $185,000. Remember: check to see if the finite correction factor applies.

5 Concepts

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Provide an appropriate response.

27) A soda machine dispenses normally distributed amounts of soda with a mean of 20 ounces and a standard

deviation of 0.2 ounce. Are you more likely to randomly select one bottle with more than 20.3 ounces or are

you more likely to select a sample of eight bottles with a mean amount of more than 20.3 ounces? Explain.

28) A soda machine dispenses normally distributed amounts of soda with a mean of 20 ounces and a standard

deviation of 0.2 ounce. Are you more likely to randomly select one bottle with an amount between 19.8 ounces

and 20.2 ounces or are you more likely to select a sample of eight bottles with a mean amount between 19.8

ounces and 20.2 ounces? Explain.

5.5 Normal Approximations to Binomial Distributions

1 Decide if the Normal Distribution Can be Used to Approximate the Binomial Distribution

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Provide an appropriate response.

1) In a recent survey, 80% of the community favored building a police substation in their neighborhood. You

randomly select 23 citizens and ask each if he or she thinks the community needs a police substation.

Decide whether you can use the normal distribution to approximate the binomial distribution. If so, find the

mean and standard deviation. If not, explain why.

2) A recent survey found that 70% of all adults over 50 wear glasses for driving. You randomly select 43 adults

over 50, and ask if he or she wears glasses. Decide whether you can use the normal distribution to approximate

the binomial distribution. If so, find the mean and standard deviation, If not, explain why.

3) According to government data, the probability than an adult was never married is 14%. You randomly select 58

adults and ask if he or she was ever married. Decide whether you can use the normal distribution to

approximate the binomial distribution, If so, find the mean and standard deviation, If not, explain why.

4) Decide if it is appropriate to use the normal distribution to approximate the random variable x for a binomial

experiment with sample size of n = 7 and probability of success p = 0.2.

5) Decide if it is appropriate to use the normal distribution to approximate the random variable x for a binomial

experiment with sample size of n = 38 and probability of success p = 0.7.

6) Decide if it is appropriate to use the normal distribution to approximate the random variable x for a binomial

experiment with sample size of n = 25 and probability of success p = 0.2

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2 Find the Correction for Continuity

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Provide an appropriate response.

7) Ten percent of the population is left-handed. A class of 100 students is selected. Convert the binomial

probability P(x > 12) to a normal probability by using the correction for continuity.

A) P(x > 12.5) B) P(x < 11.5) C) P(x 12.5) D) P(x 11.5)

8) Ten percent of the population is left-handed. A class of 2350 students is selected. Convert the binomial

probability P(x 17) to a normal probability by using the correction for continuity.

A) P(x 16.5) B) P(x < 16.5) C) P(x > 17.5) D) P(x 17.5)

9) Ten percent of the population is left-handed. A class of 100 students is selected. Convert the binomial

probability P(x < 12) to a normal probability by using the correction for continuity.

A) P(x < 11.5) B) P(x > 12.5) C) P(x 12.5) D) P(x 11.5)

10) Ten percent of the population is left-handed. A class of 8600 students is selected. Convert the binomial

probability P(x > 10) to a normal probability by using the correction for continuity.

A) P(x > 10.5) B) P(x < 9.5) C) P(x 10.5) D) P(x 9.5)

3 Find a Probability Using the Appropriate Distribution

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Provide an appropriate response.

11) The failure rate in a statistics class is 20%. In a class of 30 students, find the probability that exactly five students

will fail. Use the normal distribution to approximate the binomial distribution.

12) A local motel has 50 rooms. The occupancy rate for the winter months is 60%. Find the probability that in a

given winter month at least 35 rooms will be rented. Use the normal distribution to approximate the binomial

distribution.

13) A local motel has 200 rooms. The occupancy rate for the winter months is 60%. Find the probability that in a

given winter month fewer than 140 rooms will be rented. Use the normal distribution to approximate the

binomial distribution.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

14) A student answers all 48 questions on a multiple-choice test by guessing. Each question has four possible

answers, only one of which is correct. Find the probability that the student gets exactly 15 correct answers. Use

the normal distribution to approximate the binomial distribution.

A) 0.0823 B) 0.8577 C) 0.7967 D) 0.0606

15) If the probability of a newborn child being female is 0.5, find the probability that in 100 births, 55 or more will

be female. Use the normal distribution to approximate the binomial distribution.

A) 0.1841 B) 0.7967 C) 0.8159 D) 0.0606

16) An airline reports that it has been experiencing a 15% rate of no-shows on advanced reservations. Among 150

advanced reservations, find the probability that there will be fewer than 20 no-shows.

A) 0.2451 B) 0.7549 C) 0.7967 D) 0.3187

17) Find the probability that in 200 tosses of a fair six-sided die, a five will be obtained at least 40 times.

A) 0.1210 B) 0.8810 C) 0.0871 D) 0.3875

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18) Find the probability that in 200 tosses of a fair six-sided die, a five will be obtained at most 40 times.

A) 0.9131 B) 0.1190 C) 0.8810 D) 0.0853

19) A telemarketer found that there was a 1% chance of a sale from his phone solicitations. Find the probability of

getting 5 or more sales for 1000 telephone calls.

A) 0.9599 B) 0.0401 C) 0.8810 D) 0.0871

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

20) The author of a statistics book has trouble deciding whether to use the words he or she in the books

examples. To solve the problem, the author flips a coin each time the problem arises. If a head shows, the

author uses he and if a tail shows, the author uses she. If this problem occurs 100 times in the book, what is

the probability that she will be used 58 times?

4 Concepts

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Provide an appropriate response.

21) Match the binomial probability P(x < 23) with the correct statement.

A) P(there are fewer than 23 successes) B) P(there are at most 23 successes)

C) P(there are more than 23 successes) D) P(there are at least 23 successes)

22) Match the binomial probability P(x > 89) with the correct statement.

A) P(there are more than 89 successes) B) P(there are fewer than 89 successes)

C) P(there are at most 89 successes) D) P(there are at least 89 successes)

23) Match the binomial probability P(x 23) with the correct statement.

A) P(there are at most 23 successes) B) P(there are at least 23 successes)

C) P(there are more than 23 successes) D) P(there are fewer than 23 successes)

24) Match the binomial probability P(x > 23) with the correct statement.

A) P(there are more than 23 successes) B) P(there are at least 23 successes)

C) P(there are at most 23 successes) D) P(there are fewer than 23 successes)

25) Ten percent of the population is left-handed. In a class of 100 students, write the binomial probability for the

statement There are at least 12 left-handed students in the class.

A) P(x 12) B) P(x < 12) C) P(x > 12) D) P(x 12)

26) Ten percent of the population is left-handed. In a class of 100 students, write the binomial probability for the

statement There are more than 12 left-handed students in the class.

A) P(x > 12) B) P(x < 12) C) P(x 12) D) P(x = 12)

27) Ten percent of the population is left-handed. In a class of 100 students, write the binomial probability for the

statement There are at most 12 left-handed students in the class.

A) P(x 12) B) P(x < 12) C) P(x = 12) D) P(x 12)

28) Ten percent of the population is left-handed. In a class of 107 students, write the binomial probability for the

statement There are at least 17 left-handed students in the class.

A) P(x 17) B) P(x 17) C) P(x > 17) D) P(x < 17)

29) Ten percent of the population is left-handed. In a class of 100 students, write the binomial probability for the

statement There are exactly 12 left-handed students in the class.

A) P(x = 12) B) P(x 12) C) P(x 12) D) P(x > 12)

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Ch. 5 Normal Probability Distributions

Answer Key

5.1 Introduction to Normal Distributions and the Standard Normal Distribution

1 Find Areas Under the Standard Normal Curve

1) A

2) A

3) A

4) A

5) A

6) A

7) A

8) A

9) A

10) A

11) A

12) A

13) A

14) A

2 Find Probabilities Using the Standard Normal Curve

15) A

16) A

17) A

18) A

19) A

20) A

3 Find Probabilities Using the Standard Normal Distribution

21) A

22) A

23) A

24) A

25) A

26) A

27) A

28) A

5.2 Normal Distributions: Finding Probabilities

1 Find Probabilities for Normally Distributed Variables

1) A

2) A

3) A

4) A

5) A

6) A

7) A

8) A

9) A

10) A

11) A

12) A

13) A

14) A

15) A

16) A

17) A

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18) A

19) A

20) A

21) If x = 58, then z = -2.24 and P(x) = 0.0125. If x = 80, then z = 6.56 and P(x) = 0.9999.

P(58 < x < 80) = 0.9999 0.0125 = 0.9874.

2 Interpret Normal Distributions

22) About 6 pregnancies

23) About 3 teenage boys

24) About 43 weeks

25) About 197 women

26) About 19 men

5.3 Normal Distributions: Finding Values

1 Find a z-score Given the Area Under the Normal Curve

1) z = -0.58

2) z = -1.71

3) z = 0.42

4) z = 3.07

5) A

6) A

7) A

8) A

9) A

10) A

11) A

2 Find a z-score Given a Percentile

12) A

13) A

14) A

15) A

3 Use Normal Distributions to Answer Questions

16) A

17) A

18) A

19) z = (62 56)/9 = 0.667; z = (83 75)/15 = 0.533. The student with the score of 62 has the better score.

20) A

21) A

22) A

23) A

24) A

25) A

26) (102.0, 120.0)

27) x = + z = 32,000 + (1.28)(4000) = $37,120

28) A

29) x = + z = 64.5 + (0.675)(7.8) = 69.77

30) 67.0 inches, 74.2 inches

31) A

32) A

33) A

34) A

35) = 39, = 1.96

36) A

5.4 Sampling Distributions and the Central Limit Theorem

1 Interpret Sampling Distributions

1) A

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2 Find Probabilities

2) A

3) A

4) A

5) A

6) A

7) A

8) 0.9996

9) A

10) 0.9525

11) A

12) A

13) A

14) A

15) A

16) A

17) A

3 Use Central Limit Theorem to Find Mean/Std Error of Mean

18) A

19) A

20) A

21) A

22) A

23) A

24) A

4 Determine if the Finite Correction Factor Should be Used

25) 50/800 = 0.0625 = 6.25%, hence the finite correction factor applies; P(x < 185,000) = 0.9279

26) 50/800 = 0.0625 = 6.25%, hence the finite correction factor applies;

P(x > 185,000) = 1 0.9279 = 0.0721

5 Concepts

27) It is more likely to select one bottle with more than 20.3 ounces because a large percentage of the data is now closer to

the mean.

28) It is more likely to select a sample of eight bottles with an amount between 19.8 ounces and 20.2 ounces because a

large percentage of the data is now closer to the mean.

5.5 Normal Approximations to Binomial Distributions

1 Decide if the Normal Distribution Can be Used to Approximate the Binomial Distribution

1) cannot use normal distribution, nq = (23)(0.2) = 4.6 < 5

2) use normal distribution, = 30.1 and = 3.00.

3) use normal distribution, = 8.12 and = 2.64.

4) cannot use normal distribution

5) can use normal distribution

6) can use normal distribution

2 Find the Correction for Continuity

7) A

8) A

9) A

10) A

3 Find a Probability Using the Appropriate Distribution

11) P(4.5 < X < 5.5) = P(-0.68 < z <-0.23) = 0.409 0.2483 = 0.1607

12) P(x 34.5) = 0.0968

13) P(x < 139.5) = 0.9975

14) A

15) A

16) A

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17) A

18) A

19) A

20) P(57.5 < x < 58.5) = P(1.50 < z < 1.70) = 0.9554 0.9332 = 0.0222

4 Concepts

21) A

22) A

23) A

24) A

25) A

26) A

27) A

28) A

29) A

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Ch. 11 Nonparametric Tests

11.1 The Sign Test

1 Perform a Sign Test

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Provide an appropriate response.

1) A convenience store owner believes that the median number of lottery tickets sold per day is 49. A random

sample of 20 days yields the data below. Find the critical value to test the owners claim.

Use = 0.05.

32 48 59 64 31 55 70 27 33 38

47 54 54 44 44 49 49 59 54 38

A) 4 B) 2 C) 3 D) 5

2) A convenience store owner believes that the median number of lottery tickets sold per day is 73. A random

sample of 20 days yields the data below. Find the test statistic x to test the owners claim.

56 72 83 88 55 79 94 51 57 62

71 78 78 68 68 73 73 83 78 62

A) 8 B) 10 C) 2 D) 18

3) A real estate agent surmises that the median rent for a one-bedroom apartment in a beach community in

southern California is at least $2000 per month. The rents for a random sample of 15 one-bedroom apartments

are listed below. Find the critical value to test the agents claim. Use = 0.01.

$2300 $2250 $1700 $1875 $1735

$2750 $2175 $1670 $2390 $3000

$1995 $2000 $2075 $2000 $1780

A) 1 B) 2 C) 3 D) 4

4) A real estate agent surmises that the median rent for a one-bedroom apartment in a beach community in

southern California is at least $1500 per month. The rents for a random sample of 15 one-bedroom apartments

are listed below. Find the test statistic x to test the agents claim.

$1800 $1750 $1200 $1375 $1235

$2250 $1675 $1170 $1890 $2500

$1495 $1500 $1575 $1500 $1280

A) 6 B) 7 C) 13 D) 1

5) A club professional at a major golf course claims that the course is so tough that even professional golfers

rarely break par of 68. The scores from a random sample of 20 professional golfers are listed below. Find the

critical value to test the club professionals claim. Use = 0.05.

67 65 68 68 71 70 62 74 68 73

65 67 69 69 76 74 68 70 71 61

A) 4 B) 3 C) 2 D) 1

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6) A club professional at a major golf course claims that the course is so tough that even professional golfers

rarely break par of 65. The scores from a random sample of 20 professional golfers are listed below. Find the

test statistic x to test the club professionals claim.

64 62 65 65 68 67 59 71 65 70

62 64 66 66 73 71 65 67 68 58

A) 6 B) 4 C) 14 D) 10

7) A government agency claims that the median hourly wages for workers at fast food restaurants in the western

U.S. is $6.65. In a random sample of 100 workers, 68 were paid less than $ 6.65, 10 were paid $6.65, and the rest

more than $6.65. Find the critical values to test the governments claim. Use = 0.05.

A) -1.96 B) 1.96 C) -1.645 D) 2.575

8) A government agency claims that the median hourly wages for workers at fast food restaurants in the western

U.S. is $6.10. In a random sample of 100 workers, 68 were paid less than $6.10, 10 were paid

$6.10, and the rest more than $6.10. Find the test statistic z to test the governments claim.

A) -4.743 B) -3.912 C) -3.187 D) -2.386

9) A college researcher claims that the median hours worked by full time students is at least 11 hours per week. In

a random sample of 100 students, 63 worked more than 11 hours, 10 worked exactly 11 hours and the rest

worked less than 11 hours. Find the critical value to test the researchers claim. Use = 0.05.

A) -1.645 B) -1.96 C) -2.33 D) -2.575

10) A college researcher claims that the median hours worked by full time students is at least 13 hours per week. In

a random sample of 100 students, 64 worked more than 13 hours, 10 worked exactly 13 hours and the rest

worked less than 13 hours. Find the test statistic to test the researchers claim.

A) -3.9 B) -4.006 C) -4.7 D) -4.8

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Perform the indicated sign test. Be sure to do the following: Identify the claim mathematically and state the null and

alternative hypotheses. Determine the critical value and find the test statistic. Decide whether to reject or fail to reject

the null hypothesis and interpret the decision in the context of the original claim.

11) A convenience store owner believes that the median number of lottery tickets sold per day is 61. A random

sample of 20 days yields the data below. Test the owners claim. Use = 0.05.

44 60 71 76 43 67 82 39 45 50

59 66 66 56 56 61 61 71 66 50

12) A real estate agent surmises that the median rent for a one-bedroom apartment in a beach community in

southern California is at least $2000 per month. The rents for a random sample of 15 one-bedroom apartments

are listed below. Test the agents claim. Use = 0.01.

$2300 $2250 $1700 $1875 $1735

$2750 $2175 $1670 $2390 $3000

$1995 $2000 $2075 $2000 $1780

13) A club professional at a major golf course claims that the course is so tough that the median score is greater

than 70. The scores from a random sample of 20 professional golfers are listed below. Test the club

professionals claim. Use = 0.05.

69 67 70 70 73 72 64 76 70 75

67 69 71 71 78 76 70 72 73 63

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14) A car dealer claims that its new model car still gets at least 26 miles per gallon of gas. Ten cars are tested. The

results are given below. Test the dealers claim. Use = 0.05.

20.8 18.6 24.8 19.9 23 25.2 28.3 22.9 17.7 24

15) A government agency claims that the median hourly wages for workers at fast food restaurants in the western

U.S. is $6.50. In a random sample of 100 workers, 68 were paid less than $6.50, 10 were paid

$6.50, and the rest more than $6.50. Test the governments claim. Use = 0.05.

16) Test the hypothesis that the median age of statistics teachers is 41 years. A random sample of 60 statistics

teachers found 25 above 41 years and 35 below 41 years. Use = 0.01.

17) Nine students took the SAT test. Their scores are listed below. Later on, they took a test preparation course and

retook the SAT. Their new scores are listed below. Use the sign test to test the claim that the test preparation

has no effect on their scores. Use = 0.05.

Student 1 2 3 4 5 6 7 8 9

Before Score 1080 1180 860 840 970 1140 1160 930 1010

After Score 1100 1180 850 880 1000 1150 1150 970 1030

18) A weight-lifting coach claims that weight-lifters can increase strength by taking vitamin E. To test the theory,

the coach randomly selects 9 athletes and gives them a strength test using a bench press. Thirty days later, after

regular training supplemented by vitamin E, they are tested again. The results are listed below. Use the sign

test to test the claim that the vitamin E supplement is effective in increasing the athletes strength. Use = 0.05.

Athlete 1 2 3 4 5 6 7 8 9

Before 214 212 279 206 260 242 254 268 237

After 224 217 279 204 267 257 259 263 242

19) A pharmaceutical company wishes to test a new drug with the expectation of lowering cholesterol levels. Ten

subjects are randomly selected and their cholesterol levels are recorded. The results are listed below. The

subjects were placed on the drug for a period of 6 months, after which their cholesterol levels were tested

again. The results are listed below. (All units are milligrams per deciliter.) Use the sign test to test the

companys claim that the drug lowers cholesterol levels. Use = 0.01.

Subject 1 2 3 4 5 6 7 8 9 10

Before 188 208 194 264 202 173 269 245 259 193

After 173 203 202 254 197 173 239 227 257 178

20) In a study of the effectiveness of physical exercise in weight loss, 20 people were randomly selected to

participate in a program for 30 days. Use the sign test to test the claim that exercise has no effect on weight loss.

Use = 0.02.

Weight Before Program

(in Pounds) 178 210 156 188 193 225 190 165 168 200

Weight After Program

(in Pounds) 182 205 156 190 183 220 195 155 165 200

Weight Before Program

(in Pounds) 186 172 166 184 225 145 208 214 148 174

Weight After Program

(in Pounds) 180 173 165 186 240 138 203 203 142 170

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21) A local school district is concerned about the number of school days missed by its teachers due to illness. A

random sample of 10 teachers is selected. The numbers of absences in one year are listed below. An incentive

program is offered in an attempt to decrease the number of days absent. The numbers of absences in the year

after the incentive program are also listed. Use the sign test to test the claim that the incentive program reduces

the number of days missed by teachers. Use = 0.05.

Teacher 1 2 3 4 5 6 7 8 9 10

Days Absent

Before Incentive 5 10 0 6 7 8 0 7 9 6

Days Absent

After Incentive 4 9 0 4 6 6 1 6 7 6

22) A physicians group claims that a persons diastolic blood pressure can be lowered by listening to a relaxation

tape each evening. Ten subjects are randomly selected and their blood pressures are measured. The 10 patients

are given the tapes and told to listen to them each evening for one month. At the end of the month, their blood

pressures are measured again. The blood pressures in mm Hg are listed below. Use the sign test to test the

physicians claim. Use = 0.01.

Patient 1 2 3 4 5 6 7 8 9 10

Before 85 96 92 83 80 91 79 98 93 96

After 82 90 92 75 74 80 82 88 89 80

23) A college researcher claims that the median time worked by full time students is at least 13 hours per week. In

a random sample of 100 students, 62 worked more than 13 hours, 10 worked exactly 13 hours and the rest

worked less than 13 hours. Test the researchers claim. Use = 0.05.

24) Test the hypothesis that the median age of statistics teachers is less than 45 years. A random sample of 60

statistics teachers found 25 above 45 years, 33 below 45 years, and the rest exactly 45 years. Use = 0.01.

25) A company claims that the median monthly earnings of its farm workers is greater than $943. To test the claim,

100 workers are randomly selected and asked to provide their monthly earnings. The data is shown below. Test

the companys claim. Use = 0.05.

Weekly Earnings Number of Workers

Less than $943 41

$943 5

More than $943 54

26) One hundred people go on a special diet with the intention of losing weight. At the end of 6 weeks, 59 lost

weight, 27 gained weight and the rest remained the same. Test the hypothesis that the diet is effective in

reducing weight. Use = 0.05. (Note: The diet will be effective if at least 50% lose weight.)

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11.2 The Wilcoxon Tests

1 Perform a Wilcoxon Test

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Provide an appropriate response.

1) Nine students took the SAT test. Their scores are listed below. Later on, they took a test preparation course and

retook the SAT. Their new scores are listed below. Use the Wilcoxon signed-rank test to find the test statistic

ws to test the claim that the test preparation had no effect on their scores. Use = 0.05.

Student 1 2 3 4 5 6 7 8 9

Before Score 1020 1200 950 970 1060 820 1000 850 920

After Score 1040 1200 940 1010 1090 830 990 890 940

A) 4 B) -4 C) 41 D) -41

2) A weight-lifting coach claims that a weight-lifter can increase strength by taking vitamin E. To test the theory,

the coach randomly selects 9 athletes and gives them a strength test using a bench press. The results are listed

below. Thirty days later, after regular training supplemented by vitamin E, they are tested again. The new

results are listed below. Find the critical value for a Wilcoxon signed-rank test to test the claim that the vitamin

E supplement is effective in increasing the athletes strength. Use = 0.05.

Athlete 1 2 3 4 5 6 7 8 9

Befor 279 246 233 244 220 254 257 276 272

After 289 251 233 242 227 269 262 271 277

A) 6 B) 4 C) 2 D) 3

3) A pharmaceutical company wishes to test a new drug with the expectation of lowering cholesterol levels. Ten

subjects are randomly selected and their cholesterol levels are recorded. The results are listed below. The

subjects were placed on the drug for a period of 6 months, after which their cholesterol levels were tested

again. The results are listed below. (All units are milligrams per deciliter.) Use the Wilcoxon signed -rank test to

find the test statistic ws to test the companys claim that the drug lowers cholesterol levels.

Subject 1 2 3 4 5 6 7 8 9 10

Before 214 264 248 199 258 268 206 200 228 170

After 199 259 256 189 253 268 176 182 226 155

A) 4 B) 7.5 C) -2 D) 5.5

4) A physician claims that a persons diastolic blood pressure can be lowered, if, instead of taking a drug, the

person listens to a relaxation tape each evening. Ten subjects are randomly selected. Their blood pressures,

measured in millimeters of mercury, are listed below. The 10 patients are given the tapes and told to listen to

them each evening for one month. At the end of the month, their blood pressures are taken again. The data are

listed below. Find the critical value for a Wilcoxon signed-rank test to test the physicians claim. Use = 0.05.

Patient 1 2 3 4 5 6 7 8 9 10

Before 98 83 92 99 81 87 97 84 82 94

After 95 77 92 91 75 76 100 74 78 78

A) 8 B) 6 C) 2 D) 4

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Copyright 2012 Pearson Education, Inc.

5) Verbal SAT scores for students randomly selected from two different schools are listed below. Use the

Wilcoxon rank sum test to find R to test the claim that there is no difference in the scores from each school.

School 1 School 2

550 520 770

480 750 530

580 780 610

590 730 750

490 440 680

430 710 590

690 550 530

630 640 540

A) 128.5 B) 171.5 C) 75.5 D) 38.5

6) A researcher wants to know if the time spent in prison for a particular type of crime was the same for men and

women. A random sample of men and women were each asked to give the length of sentence received. The

data, in years, are listed below. What is the appropriate test to test the claim that there is no difference in the

sentence received by each sex?

Men 14 26 20 22 23 30

Women 13 16 13 18 30 16

Men 18 26 16 23 27 28

Women 38 12 14 17 21 31

A) Wilcoxon rank sum test B) Wilcoxon signed-rank test

C) Sign test D) t-test

7) A researcher wants to know if the time spent in prison for a particular type of crime was the same for men and

women. A random sample of men and women were each asked to give the length of sentence received. The

data, in years, are listed below. Use the Wilcoxon rank sum test to find R, the sum of the ranks for the smaller

sample, to test the claim that there is no difference in the sentence received by each gender.

Men 17 29 23 25 26 33

Women 16 19 16 21 33 19

Men 21 29 19 26 30 31

Women 41 15 17 20 24 34

A) 125.5 B) 115.5 C) 173.5 D) 155.5

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Perform the indicated Wilcoxon test. Be sure to do the following: Identify the claim mathematically and state the null

and alternative hypotheses. Determine the critical value and find the test statistic. Decide whether to reject or fail to

reject the null hypothesis and interpret the decision in the context of the original claim.

8) A random sample of nine students took the SAT test. and later on retook the test after taking a test preparation

course. Their scores are listed below. Use the Wilcoxon signed-rank test to test the claim that the test

preparation has no effect on their scores. Use = 0.05.

Student 1 2 3 4 5 6 7 8 9

Before Score 1020 990 850 1010 1200 1000 1030 970 950

After Score 1040 990 840 1050 1230 1010 1020 1010 970

Page 240

Copyright 2012 Pearson Education, Inc.

9) A weight-lifting coach claims that weight-lifters can increase their strength by taking vitamin E. To test the

theory, the coach randomly selects 9 athletes and gives them a strength test using a bench press. Thirty days

later, after regular training supplemented by vitamin E, they are tested again. The results are listed below. Use

the Wilcoxon signed-rank test to test the claim that the vitamin E supplement is effective in increasing athletes

strength. Use = 0.05.

Athlete 1 2 3 4 5 6 7 8 9

Before 195 180 208 253 226 210 202 276 229

After 205 185 208 251 233 225 207 271 234

10) A pharmaceutical company wishes to test a new drug with the expectation of lowering cholesterol levels. Ten

subjects are randomly selected and their cholesterol levels are recorded. The subjects were then placed on the

drug for a period of 6 months, after which their cholesterol levels were tested again. The results (in mg per

deciliter) are listed below. Use the Wilcoxon signed-rank test to test the companys claim that the drug lowers

cholesterol levels. Use = 0.05.

Subject 1 2 3 4 5 6 7 8 9 10

Before 208 230 184 209 255 249 259 185 227 234

After 193 225 192 199 250 249 229 167 225 219

11) A physician claims that a persons diastolic blood pressure can be lowered by listening to a relaxation tape each

evening. Ten subjects are randomly selected and their blood pressures are measured. The 10 patients then listen

to the tapes each evening for one month. At the end of the month, their blood pressures are measured again.

The data (in mm Hg) are listed below. Use the Wilcoxon signed-rank test to test the physicians claim. Use =

0.05.

Patient 1 2 3 4 5 6 7 8 9 10

Before 95 94 85 97 96 82 87 90 80 83

After 92 88 85 89 90 71 90 80 76 67

12) In a study of the effectiveness of physical exercise on weight loss, 20 people were randomly selected to

participate in a program for 30 days. Use the Wilcoxon signed-rank test to test the claim that exercise has no

effect on weight loss. Use = 0.02.

Weig

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