# mathematics

1. What is the probability a family will have 7 or more people in it? Show your work below.

12. What is the probability a family will have more than 2 people in it? Show your work below.

13. What is the probability a family will have 3 or 4 people in it? Show your work below.

14. What is the expected value of the number of family members? Show your work below and

write your answer in the blank.

15. A deck of cards contains 52 cards, of which 4 are aces. You are offered the following wager:

draw one card at random from the deck. You win $10 if the card drawn is an ace. Otherwise

you lose $1. If you make this wager very many times, what will be the mean outcome?

a)

b)

c)

d)

e)

about -$1, because you will lose most of the time.

about $9, because you will win $10 but lose only $1.

about -$0.15, that is, on the average you lose about 15 cents.

about $0.77, that is, on the average you win about 77 cents.

about $0 because the random draw gives you a fair bet.

16. A deck of 52 cards contains 13 hearts. Here is another wager: draw one card at

random from the deck. If the card drawn is a heart, you win $2. Otherwise, you lose $1.

Compare this wager (call it Wager 2) with that of the previous question (call it Wager 1).

Which one should you prefer?

a)

b)

c)

d)

e)

Wager 1, because it has a higher expected value.

Wager 2, because it has a higher expected value.

Wager 1, because it has a higher probability of winning.

Wager 2, because it has a higher probability of winning.

Both wagers are equally favorable.

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17. A psychologist thinks that listening to Bach may help people think. She gives subjects a set of

puzzles and measures how many they solve in 5 minutes while listening to Bach. From data on

many people, the psychologist determines a probability model for solving 1, 2, 3,4 , and 5

puzzles solved. The expected value she calculates from this probability model is 2.6. The law of

large numbers says

a) observe whether each of many subjects solves a puzzle. The proportion who solve a puzzle

will be close to the expected value.

b) if you observe five subjects in a row who solve only one puzzle, the next several subjects are

likely to solve three or four puzzles because the average must stay close to the expected

value.

c) the expected value is correct only in a randomized comparative experiment.

d) observe many subjects and record how many puzzles each solves. The average will be close

to the expected value.

The distribution of heights of adult men is approximately Normal with mean 69 inches and standard

deviation 2.5 inches. Show your work. Answers without correct work will not receive any credit.

18. What percent of all men are shorter than 64 inches?

19. How tall is a man whose standardized height is z = -0.3? Answer in inches.

20. What percent of all men are taller than a man whose height is at the 60th percentile?

21. How tall is a man who is in the 82nd percentile? Answer in inches.

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The casino game craps is based on rolling two dice. Here is the assignment of probabilities to the sum of

the numbers on the up faces when two dice are rolled:

Outcome

2

Probability 1/36

3

2/36

4

3/36

5

4/36

6

5/36

7

6/36

8

5/36

9

4/36

10

3/36

11

2/36

12

1/36

The most common bet in craps is the “pass line.” A pass line bettor wins immediately if either a 7 or an

11 comes up on the first roll. This is called a “natural.” Use this information to answer questions 11

through 14.

22. What is the probability of a natural?

a. 2/36

b. 6/36

c. 8/36

d. 12/36

e. 20/36

23. What is the probability you do not roll a 7?

a.

6/36

b. 28/36

c. 0

d. 30/36

e. 8/36

24. Gigi has rolled a natural on four straight tosses of the dice. This excites the gamblers standing

around the table. They should know that:

a.

b.

c.

d.

e.

Gigi has a hot hand, so she is more likely to roll another natural.

The law of averages says that Gigi is now less likely to roll another natural.

Rolls are independent, so the chance of rolling another natural has not changed.

Four straight naturals are almost impossible, so the dice are probably loaded.

They should not be surprised because the probability of four straight naturals is 2/36.

25. The table above shows a legitimate probability model because:

a.

b.

c.

d.

e.

All the probabilities are between 0 and 1.

All the probabilities are between -1 and 1.

The sum of all the probabilities is exactly 1.

Both A and C.

Both B and C.

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26. A game involving a pair of dice pays you $4 with probability 16/36, costs you $2 with probability

14/36, and costs you $6 with probability 6/36. What is the expected value of the amount of

money you win or lose after one play of the game? Show your work using 3 decimal places. An

answer without correct work will receive no credit.

Fewer US teens smoke, drink than European peers: study

Fewer teenagers in the United States smoke and drink compared to their European counterparts, but

more use drugs, according to a University of Michigan study released Friday.

Using data from 36 European countries plus the United States, researchers found that 27 percent of US

adolescents had consumed alcohol in the month prior to being quizzed by pollsters, compared to 57

percent of Europeans.

Twelve percent of American teens had smoked tobacco, compared to 20 percent for the Europeans,

according to the study, the fifth of its kind since 1995 with a total of 100,000 students aged 15 and 16

taking part.

“One of the reasons that smoking and drinking rates among adolescents are so much lower here than in

Europe is that both behaviors have been declining and have reached historically low levels in the United

States,” lead author Lloyd Johnston said.

“But even in the earlier years of the European surveys, drinking and smoking by American adolescents

was quite low by comparison,” he said, adding however that “use of illicit drugs is quite a different

matter.”

Eighteen percent of the Americans had used marijuana or hashish, a proportion exceeded in Europe only

in France (24 percent) and Monaco (21 percent).

On average, only seven percent of young Europeans had used either substance.

Relatively easy access to marijuana and little awareness of its dangers explain the figures, according to

the responses that researchers collected from survey participants.

The Americans were also the biggest users of all other drugs besides marijuana — such as LSD, ecstasy

and amphetamines — at 16 percent, compared to six percent across Europe.

“Clearly the United States has attained relatively low rates of use for cigarettes and alcohol, though not

as low as we would like,” Johnston said. “But the level of illicit drug use by adolescents is still exceptional

here.”

15,400 teenagers in the United States took part in the survey, along with at least 2,400 counterparts in

each of the 36 European nations, the University of Michigan said in a statement.

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27. What is the population of interest for this study?

28. Name the most likely type of error that could impact the findings in this study. Explain your

choice.

29. What was the sample size for individuals from the US for the current year?

30. Calculate a 90% confidence interval (CI) for the proportion of U.S. teenagers that had used

marijuana or hashish (again for the current year).

31. Suppose you were to change the confidence level in question 4 to 95% using the same sample.

How would the confidence interval change? No calculations necessary.

a.

b.

c.

d.

The confidence interval would be the same width but shifted to the left.

The confidence interval would be the same width but shifted to the right.

The confidence interval would have the same center but would be wider.

The confidence interval would have the same center but would be narrower.

32. What if the sample size was only 5000 people for the 90% confidence interval in #4. How would

the confidence interval change with this smaller sample size? No calculations necessary.

a.

b.

c.

d.

The confidence interval would be the same width but shifted to the left.

The confidence interval would be the same width but shifted to the right.

The confidence interval would have the same center but would be wider.

The confidence interval would have the same center but would be narrower.

33. A recent poll reported a confidence 95% confidence interval of 52% ± 3%. The poll

was carried out by telephone, so people without phones are always excluded from the

sample. Any errors in the final results due to excluding people without phones:

a.

b.

c.

d.

are included in the announced margin of error.

are in addition to the announced margin of error.

can be ignored, because these people are not part of the population.

can be ignored because this is a nonsampling error.

9

34. The phrase “95% confidence” means

a.

b.

c.

d.

our results are true for 9% of the population of all adults.

95% of the population falls within the margin of error we announce.

the probability is 0.95 that a randomly chosen adult falls in the margin of error we announce.

we got these results using a method that gives correct answers in 95% of all samples.

35. Gallup polled 1,523 adults and 501 teens on whether they generally approved of legal gambling.

63% of adults and 52% of teens said yes. The margin of error for a 95% confidence statement

about teens would be

a.

b.

c.

d.

e.

greater than for adults, because the teen sample is smaller.

less than for adults, because the teen sample is smaller.

less than for adults, because there are fewer teens in the population.

the same as for adults, because they both come from the same sample survey.

Can’t say, because it depends on what percent of each population was in the sample.

36. Which of the following methods would decrease the width of a confidence

interval for a mean, if all else stays the same. You may choose more than one answer for

this question.

a.

b.

c.

d.

Increase the level of confidence.

Increase the sample size.

Decrease the level of confidence.

Decrease the sample size.

37. A 95% confidence interval indicates that:

a. 95% of the possible sample means (same-size samples) will be included by the interval.

b. 95% of the intervals constructed using this process based on same-sized samples from this

population will include the sample mean.

c. 95% of the possible population means will be included by the interval.

d. 95% of the intervals constructed using this process based on same-sized samples from this

population will include the population mean.

38. A nationally distributed college newspaper conducts a survey among students nationwide every

year. This year, responses from a simple random sample of 204 college students to the question

“About how many CDs do you own?” resulted in a sample mean of 72.8. Based on data from

previous years, the editors of the newspaper will assume a population standard deviation of 7.2.

What is a 95% confidence interval for the population mean number of CDs owned by all college

students? Show your work.

a.

(65.6, 80.0)

b. (71.8, 73.8)

c. (72.0, 73.6)

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d. (72.3, 73.3)

39. The whole point of doing a confidence interval is:

a. To get information about the sample statistic.

b. To do a census.

c. To estimate the standard deviation.

d. To estimate the population parameter.

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Practice Exam Answers

1. D

2. C

3. C

4. 1

5. 0.04 and 0.10

6. 97.73%

7. 0.081

8. Show a Normal curve with 0.07 in the middle and 0.04 to 0.10 marked in even increments at

proper spacing on either side. A line should be drawn at 0.081, with shading to the right of that

line and the shading labeled with “top 15%.”

9. B

10. B

11. 0.02

12. 0.58

13. 0.44

14. 3.14

15. C

16. A

17. D

18. 2.5% (or 2.27% using the table)

19. 68.25

20. 40

21. 71.25

22. C

23. D

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24. C

25. D

26. 0

27. All US and European teens

28. Response error b/c people might lie (other answers are plausible, but will depend on the

rationale)

29. 15,400

30. Sample proportion is 18% with n = 15,400. The confidence interval is (0.1749, 0.1851) using Z* =

1.645, but either 1.64 or 1.65 could be used.

31. C

32. C

33. B

34. D

35. A

36. B, C

37. D

38. B

39. D