homework

1) Let P(A) = 0.35, P(B) = 0.30, and P(A ∩ B) = 0.17.

a.

Are A and B independent events?

b.

Are A and B mutually exclusive events?

c.

What is the probability that neither A nor B takes place

(Use computer) Assume that X is a Poisson random variable with μ = 40. Calculate the following probabilities. (Round your intermediate calculations and final answers to 4 decimal places.)

a. P(X ≤ 29)

b. P(X = 33)

c. P(X > 36)

d. P(36 ≤ X ≤ 47)

Scores on the final in a statistics class are as follows.

61

23

62

50

64

68

66

80

76

48

72

78

46

58

56

52

74

53

70

54

Click here for the Excel Data File

a.

Calculate the 25th, 50th, and 75th percentiles. (Do not round intermediate calculations. Round your answers to 2 decimal places.)

25th percentile

50th percentile

75th percentile

b-1.

Calculate the IQR, lower limit and upper limit to detect outliers. (Negative value should be indicated by a minus sign. Round your intermediate calculations to 4 decimal places and final answers to 2 decimal places.)

IQR

Lower limit

Upper limit

b-2.

Are there any outliers?

Consider the following observations from a population:

124

231

29

84

84

17

175

99

29

Click here for the Excel Data File

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.)

Consider the following sample data:

x

14

22

24

19

27

y

13

18

20

23

25

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.

A a(a(42)) , perfect , weak, strong, or no

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 93% of the time; that is, there is a 0.93 probability that the gun will detect a speeder, if the driver is actually speeding. Assume there is a 1% chance that the gun erroneously detects a speeder even when the driver is below the speed limit. Suppose that 90% 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

Let P(A) = 0.47, P(B | A) = 0.32, and P(B | Ac) = 0.08. 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)

Use computer) A committee of 68 members consists of 51 men and 17 women. A subcommittee consisting of 8 randomly selected members will be formed.

a.

What are the expected number of men and women in the subcommittee?

Expected Number

Men

Women

b.

What is the probability that at least four of the members in the subcommittee will be women? (Round your answer to 4 decimal places.)

Probability

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

0.03

2

0.11

3

0.14

4

0.23

5

0.24

6

0.13

7

0.10

8

0.02

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 3 and less than 6 members? (Round your answer to 2 decimal places.)

Probability

Professor Sanchez has been teaching Principles of Economics for over 25 years. He uses the following scale for grading.

Grade

Numerical Score

Probability

A

4

0.080

B

3

0.240

C

2

0.350

D

1

0.195

F

0

0.135

b.

Convert the above probability distribution to a cumulative probability distribution. (Round your answers to 3 decimal places.)

Grade

P(X ≤ x)

F

D

C

B

A

c.

What is the probability of earning at least a B in Professor Sanchez’s course? (Round your answer to 3 decimal places.)

Probability

d.

What is the probability of passing Professor Sanchez’s course? (Round your answer to 3 decimal places.)

Probability

Consider the following joint probability table.

B1

B2

B3

B4

A

0.09

0.14

0.14

0.11

Ac

0.20

0.10

0.09

0.13

Click here for the Excel Data File

a.

What is the probability that A occurs? (Round your answer to 2 decimal places.)

Probability

b.

What is the probability that B2 occurs? (Round your answer to 2 decimal places.)

Probability

c.

What is the probability that Ac and B4 occur? (Round your answer to 2 decimal places.)

Probability

d.

What is the probability that A or B3 occurs? (Round your answer to 2 decimal places.)

Probability

e.

Given that B2 has occurred, what is the probability that A occurs? (Round your intermediate calculations and final answers to 4 decimal places.)

Probability

f.

Given that A has occurred, what is the probability that B4 occurs? (Round your intermediate calculations and final answers to 4 decimal places.)

Probability

Assume that X is a binomial random variable with n = 6 and p = 0.59. Calculate the following probabilities. (Round your intermediate and final answers to 4 decimal places.)

a. P(X = 5)

b. P(X = 4)

c. P(X ≥ 4)

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)

A data set has a mean of 1,500 and a standard deviation of 80.

a.

Using Chebyshev's theorem, what percentage of the observations fall between 1,340 and 1,660? (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 1,180 and 1,820? (Do not round intermediate calculations. Round your answer to the nearest whole percent.)

Percentage of observations

Consider the following cumulative relative frequency distribution.

Class

Cumulative Relative Frequency

150 up to 200

0.23

200 up to 250

0.31

250 up to 300

0.46

300 up to 350

1.00

a-1.

Construct a relative frequency distribution. (Round your answers to 2 decimal places.)

Class

Relative Frequency

150 up to 200

200 up to 250

250 up to 300

300 up to 350

Total

a-2.

What percent of the observations are at least 300 but less than 350?

Percent of observations

Consider the following returns for two investments, A and B, over the past four years:

Investment 1:

9%

10%

–7%

15%

Investment 2:

7%

9%

–16%

14%

a-1.

Calculate the mean for each investment. (Round your answers to 2 decimal places.)

Mean

Investment 1

percent

Investment 2

percent

a-2.

Which investment provides the higher return?

b-1.

Calculate the standard deviation for each investment. (Round your answers to 2 decimal places.)

Standard Deviation

Investment 1

Investment 2

b-2.

Which investment provides less risk?

c-1.

Given a risk-free rate of 1.2%, calculate the Sharpe ratio for each investment. (Do not round intermediate calculations. Round your answers to 2 decimal places.)

Sharpe Ratio

Investment 1

Investment 2

c-2.

Which investment has performed better?

Consider the following frequency distribution.

Class

Frequency

2 up to 4

12

4 up to 6

68

6 up to 8

72

8 up to 10

12

a.

Calculate the population mean. (Round your answer to 2 decimal places.)

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

Records show that 5% of all college students are foreign students who also smoke. It is also known that 25% of all foreign college students smoke. What percent of the students at this university are foreign?

Percent of the students

%

A manager of a local retail store analyzes the relationship between advertising and sales by reviewing the store’s data for the previous six months.

Advertising (in $100s)

Sales (in $1,000s)

236

160

61

49

60

48

59

47

238

162

198

168

Click here for the Excel Data File

a.

Calculate the mean of advertising and the mean of sales. (Round your answers to 2 decimal places.)

Mean

Advertising

Sales

b.

Calculate the standard deviation of advertising and the standard deviation of sales. (Round your answers to 2 decimal places.)

Standard Deviation

Advertising

Sales

c-1.

Calculate the covariance between advertising and sales. (Round your answer to 2 decimal places.)

Covariance

c-2.

Interpret the covariance between advertising and sales.

d-1.

Calculate the correlation coefficient between advertising and sales. (Round your answer to 2 decimal places.)

Correlation coefficient

d-2.

Interpret the correlation coefficient between advertising and sales.

Consider the following probabilities: P(Ac) = 0.81, P(B) = 0.56, and P(A ∩ Bc) = 0.04.

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?

A basketball player is fouled while attempting to make a basket and receives two free throws. The opposing coach believes there is a 62% chance that the player will miss both shots, a 14% chance that he will make one of the shots, and a 24% chance that he will make both shots.

a.

Construct the appropriate probability distribution. (Round your answers to 2 decimal places.)

x

P(X = x)

0

1

2

b.

What is the probability that he makes no more than one of the shots? (Round your answer to 2 decimal places.)

Probability

c.

What is the probability that he makes at least one of the shots? (Round your answer to 2 decimal places.)

Probability

A researcher conducts a mileage economy test involving 116 cars. The frequency distribution describing average miles per gallon (mpg) appears in the following table.

Average mpg

Frequency

15 up to 20

11

20 up to 25

12

25 up to 30

28

30 up to 35

29

35 up to 40

13

40 up to 45

23

a.

Construct the corresponding relative frequency, cumulative frequency, and cumulative relative frequency distributions. (Round "relative frequency" and "cumulative relative frequency" to 4 decimal places.)

Average mpg

Relative Frequency

Cumulative Frequency

Cumulative Relative Frequency

15 up to 20

20 up to 25

25 up to 30

30 up to 35

35 up to 40

40 up to 45

Total

b-1.

How many of the cars got less than 25 mpg?

Number of cars

b-2.

What percent of the cars got at least 30 but less than 35 mpg? (Round your answer to 2 decimal places.)

Percentage of cars

b-3.

What percent of the cars got less than 20 mpg? (Round your answer to 2 decimal places.)

Percentage of cars

b-4.

What percent got 20 mpg or more? (Round your answer to 2 decimal places.)

Percentage of cars

An analyst thinks that next year there is a 60% chance that the world economy will be good, a 10% chance that it will be neutral, and a 30% 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

0.60

Good

0.20

Neutral

0.30

Poor

0.50

Neutral

0.10

Good

0.30

Neutral

0.60

Poor

0.10

Poor

0.30

Good

0.40

Neutral

0.40

Poor

0.20

Click here for the Excel Data File

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

Consider the following data set:

3

8

12

6

11

6

10

11

14

15

8

8

4

3

3

3

9

3

13

7

10

11

8

8

3

11

13

1

5

9

13

-5

27

16

-6

8

9

6

10

-5

34

-5

1

7

20

27

6

34

32

1

-7

-8

-2

12

-6

35

22

12

15

19

7

8

-3

9

26

29

11

2

-3

5

a-1.

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

Classes

Frequency

–10 up to 0

0 up to 10

10 up to 20

20 up to 30

30 up to 40

Total

a-2.

How many of the observations are at least 10 but less than 20?

Number of observations

b-1.

Construct a relative frequency distribution and a cumulative relative frequency distribution. (Round "relative frequency" and "cumulative relative frequency" to 3 decimal places.)

Class

Relative Frequency

Cumulative Relative Frequency

–10 up to 0

0 up to 10

10 up to 20

20 up to 30

30 up to 40

Total

b-2.

What percent of the observations are at least 10 but less than 20? (Round your answer to 1 decimal place.)

Percent of observations

%

b-3.

What percent of the observations are less than 20? (Round your answer to 1 decimal place.)

Percent of observations

%

c.

Is the distribution symmetric? If not, then how is it skewed?

Consider the following contingency table.

B

Bc

A

22

24

Ac

32

22

a.

Convert the contingency table into a joint probability table. (Round your intermediate calculations and final answers to 4 decimal places.)

B

Bc

Total

A

Ac

Total

b.

What is the probability that A occurs? (Round your intermediate calculations and final answer to 4 decimal places.)

Probability

c.

What is the probability that A and B occur? (Round your intermediate calculations and final answer to 4 decimal places.)

Probability

d.

Given that B has occurred, what is the probability that A occurs? (Round your intermediate calculations and final answer to 4 decimal places.)

Probability

e.

Given that Ac has occurred, what is the probability that B occurs? (Round your intermediate calculations and final answer to 4 decimal places.)

Probability

f.

Are A and B mutually exclusive events?

g.

Are A and B independent events?

(Use computer) Assume that X is a hypergeometric random variable with N = 40, S = 19, and n = 9. Calculate the following probabilities. (Round your answers to 4 decimal places.)

a. P(X = 4)

b. P(X ≥ 2)

c. P(X ≤ 6)

Consider the following population data:

44

52

21

18

30

a.

Calculate the range.

Range

b.

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

MAD

c.

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

Population variance

d.

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

Population standard deviation

The probabilities that stock A will rise in price is 0.40 and that stock B will rise in price is 0.60. Further, if stock B rises in price, the probability that stock A will also rise in price is 0.80.

a.

What is the probability that at least one of the stocks will rise in price? (Round your answer to 2 decimal places.)

Probability

b.

Are events A and B mutually exclusive?

c.

Are events A and B independent?

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

–2.4

6.5

8.8

14.0

–8.0

0.6

21.2

–9.9

4.2

15.2

–11.9

4.2

–0.1

–0.3

2.1

2.1

–11.0

–5.9

24.0

23.1

17.0

9.2

11.7

8.4

Click here for the Excel Data File

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 0% but less than 10%?

Number of funds

c-2.

How many of the funds had returns of 10% or more?

Number of funds

d-1.

What percent of the funds had returns of at least 10% but less than 20%? (Round your answer to 1 decimal place.)

Percent of funds

d-2.

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

Percent of funds

A professor has learned that six students in her class of 19 will cheat on the exam. She decides to focus her attention on eight randomly chosen students during the exam.

a.

What is the probability that she finds at least one of the students cheating? (Round your intermediate calculations and final answers to 4 decimal places.)

Probability

b.

What is the probability that she finds at least one of the students cheating if she focuses on nine randomly chosen students? (Round your intermediate calculations and final answers to 4 decimal places.)

Probability

Market observers are quite uncertain whether the stock market has bottomed out from the economic meltdown that began in 2008. In an interview on March 8, 2009, CNBC interviewed two prominent economists who offered differing views on whether the U.S. economy was getting stronger or weaker. An investor not wanting to miss out on possible investment opportunities considers investing $17,000 in the stock market. He believes that the probability is 0.21 that the market will improve, 0.40 that it will stay the same, and 0.39 that it will deteriorate. Further, if the economy improves, he expects his investment to grow to $21,000, but it can also go down to $14,000 if the economy deteriorates. If the economy stays the same, his investment will stay at $17,000.

a.

What is the expected value of his investment?

Expected value

$

b.

What should the investor do if he is risk neutral?

Investor invest the $17,000.

c.

Is the decision clear-cut if he is risk averse?

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)

0.48

P(A | B)

0.10

P(A ∩ B )

P(B | A)

P(Bc)

P(A | Bc)

0.43

P(A ∩ Bc)

P(Bc | A)

Total

P(A)

Total

Consider the following frequency distribution:

Class

Frequency

10 up to 20

21

20 up to 30

22

30 up to 40

33

40 up to 50

12

a.

Construct a relative frequency distribution. (Round your answers to 3 decimal places.)

Class

Relative Frequency

10 up to 20

20 up to 30

30 up to 40

40 up to 50

Total

b.

Construct a cumulative frequency distribution and a cumulative relative frequency distribution. (Round "cumulative relative frequency" to 3 decimal places.)

Class

Cumulative Frequency

Cumulative Relative Frequency

10 up to 20

20 up to 30

30 up to 40

40 up to 50

c-1.

What percent of the observations are at least 20 but less than 30? (Round your answer to 1 decimal place.)

Percent of observations

c-2.

What percent of the observations are less than 20? (Round your answer to 1 decimal place.)

Percent of observations

A 2010 poll conducted by NBC asked respondents who would win Super Bowl XLV in 2011. The responses by 21,175 people are summarized in the following table.

Team

Number of Votes

Atlanta Falcons

4,055

New Orleans Saints

1,895

Houston Texans

1,815

Dallas Cowboys

1,646

Minnesota Vikings

1,453

Indianapolis Colts

1,164

Pittsburgh Steelers

1,156

New England Patriots

1,110

Green Bay Packers

1,091

Others

a.

How many responses were for “Others”?

Number of responses

b.

The Green Bay Packers won Super Bowl XLV, defeating the Pittsburgh Steelers by the score of 31–25. What proportion of respondents felt that the Green Bay Packers would win? (Round your answer to 3 decimal places.)

Proportion of respondents

The following relative frequency distribution was constructed from a population of 300. Calculate the population mean, the population variance, and the population standard deviation. (Round your intermediate calculations to 4 decimal places and final answers to 2 decimal places.)

Class

Relative Frequency

−20 up to −10

0.25

−10 up to 0

0.20

0 up to 10

0.40

10 up to 20

0.15

Population mean

Population variance

Population standard deviation