# Mathematics

Records show that 13% of all college students are foreign students who also smoke. It is also known that 50% of all foreign college students smoke. What percent of the students at this university are foreign?

Percent of the students %

19.

Determine whether the following probabilities are best categorized as subjective, empirical, or classical probabilities.

a.

Before flipping a fair coin, Sunil assesses that he has a 50% chance of obtaining tails.

Subjective probability

Empirical probability

Classical probability

b.

At the beginning of the semester, John believes he has a 90% chance of receiving straight A’s.

Subjective probability

Empirical probability

Classical probability

c.

A political reporter announces that there is a 48% chance that the next person to come out of the conference room will be a Republican, since there are 85 Republicans and 91 Democrats in the room.

Subjective probability

Empirical probability

Classical probability

20.

A data set has a mean of 1,080 and a standard deviation of 80.

a.

Using Chebyshev’s theorem, what percentage of the observations fall between 760 and 1,400? (Do not round intermediate calculations. Round your answer to the nearest whole percent.)

Percentage of observations

b.

Using Chebyshev’s theorem, what percentage of the observations fall between 920 and 1,240? (Do not round intermediate calculations. Round your answer to the nearest whole percent.)

Percentage of observations

rev: 07_31_2013_QC_32713

21.

Let P(A) = 0.62, P(B) = 0.27, and P(A ∩ B) = 0.17.

a. Calculate P(A | B). (Round your answer to 2 decimal places.)

P(A | B)

b. Calculate P(A U B). (Round your answer to 2 decimal places.)

P(A U B)

c. Calculate P((A U B)c). (Round your answer to 2 decimal places.)

P((A U B)c)

rev: 08_06_2013_QC_32707

22.

Let P(A) = 0.51, P(B | A) = 0.36, and P(B | Ac) = 0.14. Use a probability tree to calculate the following probabilities: (Round your answers to 3 decimal places.)

a. P(Ac)

b. P(A ∩ B)

P(Ac ∩ B)

c. P(B)

d. P(A | B)

rev: 08_06_2013_QC_32707, 10_09_2014_QC_55407

23.

Consider the following observations from a population:

133 240 38 93 93 26 184 108 38

a. Calculate the mean and median. (Round “mean” to 2 decimal places.)

Mean

Median

b.

Select the mode. (You may select more than one answer. Single click the box with the question mark to produce a check mark for a correct answer and double click the box with the question mark to empty the box for a wrong answer.)

240

26

108

133

38

93

184

rev: 07_31_2013_QC_32713

24.

An analyst thinks that next year there is a 40% chance that the world economy will be good, a 10% chance that it will be neutral, and a 50% chance that it will be poor. She also predicts probabilities that the performance of a start-up firm, Creative Ideas, will be good, neutral, or poor for each of the economic states of the world economy. The following table presents probabilities for three states of the world economy and the corresponding conditional probabilities for Creative Ideas.

State of

the World

Economy Probability

of Economic

State Performance

of Creative

Ideas Conditional

Probability of

Creative Ideas

Good 0.40 Good 0.20

Neutral 0.30

Poor 0.50

Neutral 0.10 Good 0.40

Neutral 0.10

Poor 0.50

Poor 0.50 Good 0.40

Neutral 0.40

Poor 0.20

a.

What is the probability that the performance of the world economy will be neutral and that of creative ideas will be poor? (Round your answer to 2 decimal places.)

Probability

b.

What is the probability that the performance of Creative Ideas will be poor? (Round your answer to 2 decimal places.)

Probability

c.

The performance of Creative Ideas was poor. What is the probability that the performance of the world economy had also been poor? (Round your answer to 2 decimal places.)

Probability

rev: 08_06_2013_QC_32707

25.

Complete the following probability table. (Round Prior Probability answers to 2 decimal places and intermediate calculations and other answers to 4 decimal places.)

Prior

Probability Conditional Probability Joint

Probability Posterior

Probability

P(B) 0.53 P(A | B) 0.15 P(A ∩ B ) P(B | A)

P(Bc) P(A | Bc) 0.38 P(A ∩ Bc) P(Bc | A)

Total P(A) Total

rev: 08_06_2013_QC_32707

26.

Consider the following sample data:

x 8 10 7 5 2

y 11 2 7 4 8

a.

Calculate the covariance between the variables. (Round your intermediate calculations to 4 decimal places and final answer to 2 decimal places.)

Covariance

b-1.

Calculate the correlation coefficient. (Round your intermediate calculations to 4 decimal places and final answer to 2 decimal places.)

Correlation coefficient

b-2. Interpret the correlation coefficient.

There is

relationship between x and y.

rev: 07_31_2013_QC_32713

27.

India is the second most populous country in the world, with a population of over 1 billion people. Although the government has offered various incentives for population control, some argue that the birth rate, especially in rural India, is still too high to be sustainable. A demographer assumes the following probability distribution of the household size in India.

Household Size Probability

1 0.04

2 0.12

3 0.18

4 0.24

5 0.13

6 0.15

7 0.10

8 0.04

a.

What is the probability that there are less than 5 members in a typical household in India? (Round your answer to 2 decimal places.)

Probability

b.

What is the probability that there are 5 or more members in a typical household in India? (Round your answer to 2 decimal places.)

Probability

c.

What is the probability that the number of members in a typical household in India is greater than 4 and less than 7 members? (Round your answer to 2 decimal places.)

Probability

rev: 02_26_2014_QC_45094

28.

The State Police are trying to crack down on speeding on a particular portion of the Massachusetts Turnpike. To aid in this pursuit, they have purchased a new radar gun that promises greater consistency and reliability. Specifically, the gun advertises ± one-mile-per-hour accuracy 70% of the time; that is, there is a 0.70 probability that the gun will detect a speeder, if the driver is actually speeding. Assume there is a 2% chance that the gun erroneously detects a speeder even when the driver is below the speed limit. Suppose that 67% of the drivers drive below the speed limit on this stretch of the Massachusetts Turnpike.

a.

What is the probability that the gun detects speeding and the driver was speeding? (Round your answer to 4 decimal places.)

Probability

b.

What is the probability that the gun detects speeding and the driver was not speeding? (Round your answer to 4 decimal places.)

Probability

c.

Suppose the police stop a driver because the gun detects speeding. What is the probability that the driver was actually driving below the speed limit? (Round your answer to 4 decimal places.)

Probability

rev: 08_06_2013_QC_32707

29.

At a local bar in a small Midwestern town, beer and wine are the only two alcoholic options. The manager noted that of all male customers who visited over the weekend, 153 ordered beer, 46 ordered wine, and 17 asked for soft drinks. Of female customers, 37 ordered beer, 23 ordered wine, and 10 asked for soft drinks.

a.

Construct a contingency table that shows frequencies for the qualitative variables Gender (male or female) and Drink Choice (beer, wine, or soft drink).

Drink Choice

Gender Beer (B) Wine (W) Soft Drinks (D) Totals

Male (M)

Female (F)

Total

b. Find the probability that a customer orders wine. (Round your intermediate calculations and final answer to 4 decimal places.)

P(W)

c.

What is the probability that a male customer orders wine? (Round your intermediate calculations and final answer to 4 decimal places.)

P (W | M )

d. Are the events “Wine” and “Male” independent?

Yes because P(“Wine” | “Male”) = P(“Wine”).

Yes because P(“Wine” ∩ “Male”) = P(“Wine”).

No because P(“Wine” | “Male”) ≠ P(“Wine”).

No because P(“Wine” ∩ “Male”) ≠ P(“Wine”).

rev: 08_06_2013_QC_32707

30.

Consider the following frequency distribution.

Class Frequency

2 up to 4 21

4 up to 6 59

6 up to 8 81

8 up to 10 21

a.

Population mean

b.

Calculate the population variance and the population standard deviation. (Round your intermediate calculations to 4 decimal places and final answers to 2 decimal places.)

Population variance

Population standard deviation

rev: 07_31_2013_QC_32713

31.

Which of the following variables is not continuous?

Time of a flight between Atlanta and Chicago

Height of NBA players

The number of obtained heads when a fair coin is tossed 20 times

Average temperature in the month of July in Orlando

32.

The one-year return (in %) for 24 mutual funds is as follows:

–10.7 –1.4 0.9 6.1 –15.9 –7.5

21.5 –9.6 4.5 11.1 14.5 4.7

–8.4 –8.4 19.5 14.9 29.3 7.7

22.0 24.8 –0.4 11.1 5.0 –11.0

a.

Construct a frequency distribution using classes of –20 up to –10, –10 up to 0, etc.

Class (in %) Frequency

–20 up to –10

–10 up to 0

0 up to 10

10 up to 20

20 up to 30

Total

b.

Construct the relative frequency, the cumulative frequency, and the cumulative relative frequency distributions. (Round “relative frequency” and “cumulative relative frequency” answers to 3 decimal places.)

Class (in %) Relative

Frequency Cumulative

Frequency Cumulative

Relative Frequency

–20 up to –10

–10 up to 0

0 up to 10

10 up to 20

20 up to 30

Total

c-1. How many of the funds had returns of at least 20% but less than 30%?

Number of funds

c-2. How many of the funds had returns of 0% or more?

Number of funds

d-1.

What percent of the funds had returns of at least –10% but less than 0%? (Round your answer to 1 decimal place.)

Percent of funds

d-2.

What percent of the funds had returns less than 20%? (Round your answer to 1 decimal place.)

Percent of funds

rev: 06_24_2013_QC_31991, 07_05_2013_QC_32367

33.

Investment advisors recommend risk reduction through international diversification. International investing allows you to take advantage of the potential for growth in foreign economies, particularly in emerging markets. Janice Wong is considering investment in either Europe or Asia. She has studied these markets and believes that both markets will be influenced by the U.S. economy, which has a 16% chance for being good, a 57% chance for being fair, and a 27% chance for being poor. Probability distributions of the returns for these markets are given in the accompanying table.

State of the

U.S. Economy Returns

in Europe Returns

in Asia

Good 14% 28%

Fair 5% 7%

Poor −12% −10%

a.

Find the expected value and the standard deviation of returns in Europe and Asia. (Round your intermediate calculations to 4 decimal places and final answers to 2 decimal places.)

Europe Asia

Expected value % %

Standard deviation

b. What will Janice pick as an investment if she is risk neutral?

Investment in Europe

Investment in Asia

rev: 08_07_2013_QC_33420

34.

Consider the following probabilities: P(Ac) = 0.32, P(B) = 0.58, and P(A ∩ Bc) = 0.25.

a. Find P(A | Bc). (Do not round intermediate calculations. Round your answer to 2 decimal places.)

P(A | Bc)

b. Find P(Bc | A). (Do not round intermediate calculations. Round your answer to 3 decimal places.)

P(Bc | A)

c. Are A and B independent events?

Yes because P(A | Bc) = P(A).

Yes because P(A ∩ Bc) ≠ 0.

No because P(A | Bc) ≠ P(A).

No because P(A ∩ Bc) ≠ 0.

rev: 08_06_2013_QC_32707

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