# Mathematics

1.

A local restaurant is committed to providing its patrons with the best dining experience possible. On a recent survey, the restaurant asked patrons to rate the quality of their entrées. The responses ranged from 1 to 5, where 1 indicated a disappointing entrée and 5 indicated an exceptional entrée.

The results of the survey are as follows:

2 5 1 5 1 5 4 3 3 3 1 2

1 2 2 3 1 4 4 1 2 3 1 1

4 5 1 1 1 3 1 2 1 4 2 2

PictureClick here for the Excel Data File

a.

Construct frequency and relative frequency distributions that summarize the survey’s results. (Do not round intermediate calculations. Round “relative frequency” to 3 decimal places.)

Rating Frequency Relative

Frequency

5

4

3

2

1

Total

b.

Are patrons generally satisfied with the quality of their entrées?

No

Yes

rev: 07_05_2013_QC_32367, 03_04_2014_QC_44527

2.

Consider the following data set:

1 10 5 6 8 8 10 12 15 12

8 11 8 4 3 9 12 3 10 8

8 12 4 4 4 12 10 6 11 6

7 -6 31 16 -3 9 13 6 5 -4

29 -3 5 3 24 24 10 23 32 2

-5 -4 -2 14 -2 35 26 10 18 28

5 3 -6 7 28 36 16 3 -4 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?

Not symmetric, skewed to right

Symmetric or Skewed to left

rev: 07_05_2013_QC_32367

3.

Assume that X is a binomial random variable with n = 16 and p = 0.66. Calculate the following probabilities. (Round your intermediate and final answers to 4 decimal places.)

a. P(X = 15)

b. P(X = 14)

c. P(X ≥ 14)

rev: 04_26_2013_QC_29765; rev: 08_07_20

4.

A professor of management has heard that twelve students in his class of 52 have landed an internship for the summer. Suppose he runs into two of his students in the corridor.

a.

Find the probability that neither of these students has landed an internship. (Round your intermediate calculations and final answer to 4 decimal places.)

formula176.mml

b.

Find the probability that both of these students have landed an internship. (Round your intermediate calculations and final answer to 4 decimal places.)

P(T1 ∩ T2)

rev: 08_06_2013_QC_32707

5.

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 $15,000 in the stock market. He believes that the probability is 0.25 that the market will improve, 0.42 that it will stay the same, and 0.33 that it will deteriorate. Further, if the economy improves, he expects his investment to grow to $23,000, but it can also go down to $10,000 if the economy deteriorates. If the economy stays the same, his investment will stay at $15,000.

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 $15,000.

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

Yes

No

rev: 08_07_2013_QC_33420, 11_01_2013_QC_37895

6.

An investment strategy has an expected return of 12 percent and a standard deviation of 8 percent. Assume investment returns are bell shaped.

a.

How likely is it to earn a return between 4 percent and 20 percent? (Enter your response as decimal values (not percentages) rounded to 2 decimal places.)

Probability

b.

How likely is it to earn a return greater than 20 percent?(Enter your response as decimal values (not percentages) rounded to 2 decimal places.)

Probability

c.

How likely is it to earn a return below −4 percent?(Enter your response as decimal values (not percentages) rounded to 2 decimal places.)

Probability

rev: 02_26_2014_QC_44958, 07_12_2014_QC_51377

7.

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

rev: 07_05_2013_QC_32367, 08_12_2013_QC_33620

8.

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

68 24 70 56 72 76 74 116 87 55

82 88 54 66 64 58 84 60 79 62

PictureClick 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?

Yes

No

rev: 07_31_2013_QC_32713, 09_13_2013_QC_34880, 10_31_2013_QC_38175, 03_03_2014_QC_44705, 09_24_2014_QC_54188

9.

The estimation of which of the following requires sampling?

Total rainfall in Phoenix, Arizona, in 2010

The average SAT score of incoming freshmen at a university

U.S. unemployment rate

The Cleveland Indians’ hitting percentage in 2010

10.

A researcher conducts a mileage economy test involving 79 cars. The frequency distribution describing average miles per gallon (mpg) appears in the following table.

Average mpg Frequency

15 up to 20 7

20 up to 25 15

25 up to 30 14

30 up to 35 27

35 up to 40 12

40 up to 45 4

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 20 mpg?

Number of cars

b-2.

What percent of the cars got at least 25 but less than 30 mpg? (Round your answer to 2 decimal places.)

Percentage of cars

b-3.

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

Percentage of cars

b-4. What percent got 30 mpg or more? (Round your answer to 2 decimal places.)

Percentage of cars

rev: 07_05_2013_QC_32367

11.

Consider the following joint probability table.

B1 B2 B3 B4

A 0.14 0.10 0.15 0.09

Ac 0.15 0.17 0.10 0.10

PictureClick 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

rev: 08_06_2013_QC_32707

12.

Consider the following cumulative relative frequency distribution.

Class Cumulative

Relative

Frequency

150 up to 200 0.19

200 up to 250 0.26

250 up to 300 0.55

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 250 but less than 300?

Percent of observations

13.

Christine has always been weak in mathematics. Based on her performance prior to the final exam in Calculus, there is a 53% chance that she will fail the course if she does not have a tutor. With a tutor, her probability of failing decreases to 23%. There is only a 63% chance that she will find a tutor at such short notice.

a.

What is the probability that Christine fails the course? (Round your answer to 4 decimal places.)

Probability

b.

Christine ends up failing the course. What is the probability that she had found a tutor? (Round your answer to 4 decimal places.)

Probability

rev: 08_06_2013_QC_32707

14.

A 2010 poll conducted by NBC asked respondents who would win Super Bowl XLV in 2011. The responses by 20,925 people are summarized in the following table.

Team Number of Votes

Atlanta Falcons 4,100

New Orleans Saints 1,860

Houston Texans 1,900

Dallas Cowboys 1,641

Minnesota Vikings 1,500

Indianapolis Colts 1,159

Pittsburgh Steelers 1,155

New England Patriots 1,106

Green Bay Packers 1,087

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

rev: 07_05_2013_QC_32367

15.

Consider the following population data:

37 41 14 11 23

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

rev: 07_31_2013_QC_32713

16.

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 0.100

B 3 0.240

C 2 0.430

D 1 0.125

F 0 0.105

Part (a) omitted

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

rev: 02_28_2014_QC_45290

17.

A basketball player is fouled while attempting to make a basket and receives two free throws. The opposing coach believes there is a 55% chance that the player will miss both shots, a 25% chance that he will make one of the shots, and a 20% 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

rev: 09_13_2013_QC_35141