# MATHEMATICS

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Review and Practice Exam Questions for Exam 2

Learning Objectives:

Chapter 17: Thinking about chance

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Explain how random events behave in the short run and in the long run and how random and

haphazard are not the same thing.

Perform basic probability calculations using die rolls and coin tosses.

Define probability, and apply the rules for probability.

Explain whether the law of averages is true.

Explain how personal probability differs from a scientific or experimental probability.

Chapter 18: Probability models

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Define a probability model. Create a probability model for a particular story’s events.

Apply the basic rules of probability to a story problem.

Calculate probabilities using a probability model, including summing up probabilities or

subtracting probabilities from the total.

Define a sampling distribution.

Chapter 20: The house edge: expected values

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Define expected value, and calculate the expected value when given a probability model.

Define the law of large numbers, and explain how it is different from the mythical “law of

averages.”

Explain how casinos and insurance companies stay in business and make money.

Chapter 13: The Normal distribution

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Identify data that is Normally distributed.

Discuss how the shape/position of the Normal curve changes when the standard deviation

increases/decreases or when the mean increases/decreases.

Define the standardized value or Z-score. Calculate the Z-score, and use the Z-score to do

comparisons.

Calculate probabilities and cut-off values using the 68%-95%-99.7% (Empirical) Rule.

Identify the mean, standard deviation, cut-off value, probability, and Z-score on a Normal curve.

Use the Normal table to get percentiles (probabilities) for forward problems and to get Z-scores

in order to determine cut-offs for backward problems using both > and < in the inequalities.

Recognize whether a story is a forward or backward Normal distribution problem, and perform

the appropriate calculations showing correct notation, the initial probability expression, and all

necessary steps.

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Chapter 21: What is a confidence interval?

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Define statistical inference and explain when statistical inference is used.

Explain what the confidence interval means and whether the results refer to the population or

the sample.

Calculate the margin of error and identify the margin of error in a confidence statement.

Explain what type of error is covered in the margin of error.

Determine whether a story is better described with a proportion or a mean.

Use appropriate notation for proportions and means, both in the population and the sample.

Calculate a confidence interval for a proportion and for a mean.

Describe how increasing/decreasing the sample size or confidence level changes the margin of

error (width of the confidence interval).

Apply cautions for using confidence intervals.

o Need a simple random sample.

o Data must be collected correctly.

o No outliers.

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Practice Exam Questions

These practice exam questions are not meant to be an exhaustive list. These are simply examples of

questions that have been asked in the past. You definitely should try working through these examples

with your cheat sheet, but make sure that you use the Learning Objectives to create your cheat sheet

and to help you find other problems to practice. Your instructor uses the Learning Objectives when

writing the exams. The odd-numbered problems in your book have answers in the back, and the

StatsPortal website has many great resources to help your studying, especially the Learning Curve

quizzes and the video series. Remember that you are allowed to bring a 1-page (both sides ok),

handwritten-in-your-own-handwriting, 8 ½” x 11” cheat sheet with you to the exam. Your name should

be on the cheat sheet, and it will be worth 1 point on the exam when you turn it in.

The experiment/sampling design and ethics topics from Exam 1 are important throughout the whole

semester and may show up on later exams.

1. Before the 2008 presidential campaign, the Gallup Poll asked a sample of 1,000 people for

whom they would vote for president; 52% said Obama. The margin of error for a 95%

confidence interval announced by news reports of this poll was:

A) ±8%

B) ±6%

C) ±4%

D) ±3%

E) ±2%

Show your work:

2. “Margin of error” in this situation means that if there is no bias

A)

B)

C)

D)

E)

every sample the Gallup Poll takes will come at least this close to the truth.

about half of the samples the Gallup Poll takes will come at least this close to the truth.

about 95% of all samples the Gallup Poll takes will come at least this close to the truth.

the sampling method is biased—otherwise the poll would always give the correct answer.

there are serious nonsampling errors—otherwise the poll would always give the correct

answer.

3. A Gallup poll surveyed 3,112 voters. An AP poll surveyed 778 voters from the same population

on the exact same question. How does the margin of error for a 95% confidence interval

compare for the Gallup and AP polls?

a. The Gallup margin of error is bigger than the AP margin of error.

b. The Gallup margin of error is the same as the AP margin of error.

c. The Gallup margin of error is smaller than the AP margin of error.

4. The mean is 80 and the standard deviation is 10. What is the standard score for an observation

of 90? Show your work below.

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Suppose that the BAC of students who drink five beers varies from student to student according to a

Normal distribution with mean 0.07 and standard deviation 0.01. For questions 10 through 13, show

your work below the question and write your answer in the line.

5. The middle 99.7% of students who drink five beers have BAC between what two numbers?

6. What percent of students who drink five beers have BAC below 0.09?

7. What BAC do the highest 15% of students have after drinking five beers?

8. Sketch a Normal curve for the problem above with mean = 0.07 and standard deviation = 0.01.

Label your x-axis from 0.04 to 0.10 in 0.01 increments. Also show the how the top 15% of

students (from #7) would be represented on that Normal curve.

9. A poker player is dealt poor hands for several hours. He decides to bet heavily on

the last hand of the evening on the grounds that after many bad hands he is due for a

winner.

a)

b)

c)

d)

He’s right, because the winnings have to average out.

He’s wrong, because successive deals are independent of each other.

He’s right, because successive deals are independent of each other.

He’s wrong, because his expected winnings are $0 and he’s below that now.

10. The probability of an outcome of a random phenomenon is

a)

b)

c)

d)

either 0 or 1, depending on whether the phenomenon can actually occur.

the proportion of a very long series of repetitions on which the outcome occurs.

the mean plus or minus two standard deviations.

the confidence level.

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In government data, a family consists of two or more persons who live together and are related by blood

or marriage. Choose an American family at random and count the number of people it contains. Here is

the assignment of probabilities for your outcome:

# of persons

Probability

2

0.42

3

0.23

4

0.21

5

0.09

6

0.03

7 or more

???

11. 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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