## David R. Anderson; Dennis J. Sweeney; Thomas A. Williams; Jeffrey D. Camm; James J. Cochran

## Chapter 3

## Descriptive Statistics: Numerical Measures - all with Video Answers

## Educators

Chapter Questions

Consider a sample with data values of $10,20,12,17$, and 16 . Compute the mean and median.

Hubert Agamasu

Numerade Educator

Consider a sample with data values of $10,20,21,17,16$, and 12 . Compute the mean and median.

Hubert Agamasu

Numerade Educator

Consider the following data and corresponding weights.

a. Compute the weighted mean.

b. Compute the sample mean of the four data values without weighting. Note the difference in the results provided by the two computations.

Nick Johnson

Numerade Educator

Consider the following data.

What is the mean growth rate over these five periods?

Jason Gerber

Numerade Educator

Consider a sample with data values of $27,25,20,15,30,34,28$, and 25 . Compute the 20 th, 25 th, 65 th, and 75 th percentiles.

Jason Gerber

Numerade Educator

Consider a sample with data values of $53,55,70,58,64,57,53,69,57,68$, and 53 . Compute the mean, median, and mode.

Megha Nayar

Numerade Educator

eICU Waiting Times. There is a severe shortage of critical care doctors and nurses to provide intensive-care services in hospitals. To offset this shortage, many hospitals, such as Emory Hospital in Atlanta, are using electronic intensive-care units (eICUs) to help provide this care to patients (Emory University News Center), elCUs use electronic monitoring took and twoway communication through video and audio so that a centralized staff of specially trained doctors and nurses-who can be located as far away as Australia-can provide critical care services to patients located in remote hospitals without fully staffed ICUs. One of the most important metrics tracked by these eICUs is the time that a patient must wait for the first video interaction between the patient and the eICU staff. Consider the following sample of 40 patient waiting times until their first video interaction with the eICU staff.

$$

\begin{array}{cccc}

\begin{array}{c}

\text { Wait Time } \\

\text { (minutes) }

\end{array} & \begin{array}{c}

\text { Wait Time } \\

\text { (minutes) }

\end{array} & \begin{array}{c}

\text { Wait Time } \\

\text { (minutes) }

\end{array} & \begin{array}{l}

\text { Wait Time } \\

\text { (minutes) }

\end{array} \\

40 & 46 & 49 & 44 \\

45 & 45 & 38 & 51 \\

42 & 46 & 41 & 45 \\

49 & 41 & 48 & 42 \\

49 & 40 & 42 & 43 \\

43 & 42 & 41 & 41 \\

55 & 43 & 42 & 40 \\

42 & 40 & 49 & 43 \\

44 & 45 & 61 & 37 \\

40 & 37 & 39 & 43

\end{array}

$$

a. Compute the mean waiting time for these 40 patients.

b. Compare the mean waiting time.

c. Compute the mode.

d. Compute the first and third quartiles.

Jason Gerber

Numerade Educator

Middle-Level Manager Salaries. Suppose that an independent study of middle-level managers employed at companies located in Atlanta, Georgia, was conducted to compare the salaries of managers working at firms in Atlanta to the salaries of middle-level managers across the nation. The following data show the salary, in thousands of dollars, for a sample of 15 middle-level managers employed at companies in the Atlanta area.

$$

\begin{array}{lllllllllllllll}

108 & 83 & 106 & 73 & 53 & 85 & 80 & 63 & 67 & 75 & 124 & 55 & 93 & 118 & 77

\end{array}

$$

a. Compute the median salary for the sample of 15 middle-level managers. Suppose the median salary of middle-level managers employed at companies located across the nation is $\$ 85,000$. How does the median salary for middle-level managers in the Atlanta area compare to the median for managers across the nation?

b. Compute the mean annual salary for managers in the Atlanta area and discuss how and why it differs from the median computed in part (a) for Atlanta area managers.

c. Compute the first and third quartiles for the salaries of middle-level managers in the Allanta area.

Jason Gerber

Numerade Educator

Advertising Spending. Which companies spend the most money on advertising?

Business Insider maintains a list of the top-spending companies. In 2014, Procter & Gamble spent more than any other company, a whopping $$\$ 5$$ billion. In second place was Comcast, which spent $$\$ 3.08$$ billion (Business Insider website). The top 12 companies and the amount each spent on advertising in billions of dollars are as follows.

$$

\begin{array}{lclc}

\text { Company } & \begin{array}{l}

\text { Advertising } \\

\text { (\$billions) }

\end{array} & \text { Company } & \begin{array}{c}

\text { Advertising } \\

\text { (\$billions) }

\end{array} \\

\text { Procter \& Gamble } & \$ 5.00 & \text { American Express } & \$ 2.19 \\

\text { Comcast } & 3.08 & \text { General Motors } & 2.15 \\

\text { AT\&T } & 2.91 & \text { Toyota } & 2.09 \\

\text { Ford } & 2.56 & \text { Fiat Chrysler } & 1.97 \\

\text { Verizon } & 2.44 & \text { Walt Disney Company } & 1.96 \\

\text { L'Oreal } & 2.34 & \text { J.P. Morgan } & 1.88

\end{array}

$$

a. What is the mean amount spent on advertising?

b. What is the median amount spent on advertising?

c. What are the first and third quartiles?

Jason Gerber

Numerade Educator

Hardshell Jacket Ratings. OutdorGearLab is an organization that tests outdoor gear used for climbing, camping, mountaineering, and backpacking. Suppose that the following data show the ratings of hardshell jackets based on the breathability, durability, versatility, features, mobility, and weight of each jacket. The ratings range from 0 (lowest) to 100 (highest).

\begin{tabular}{llllllllll}

42 & 66 & 67 & 71 & 78 & 62 & 61 & 76 & 71 & 67 \\

61 & 64 & 61 & 54 & 83 & 63 & 68 & 69 & 81 & 53

\end{tabular}

a. Compute the mean, median, and mode.

b. Compute the first and third quartiles.

c. Compute and interpret the 90 th percentile.

Jason Gerber

Numerade Educator

Time Spent Watching Traditional TV. Nielsen tracks the amount of time that people spend consuming media content across different platforms (digital, audio, television) in the United States. Nielsen has found that traditional television viewing habits vary based on the age of the consumer as an increasing number of people consume media through streaming devices (Nielsen website). The following data represent the weekly traditional TV viewing hours in 2016 for a sample of 14 people aged $18-34$ and 12 people aged $35-49$.

$$

\begin{array}{ll}

\text { Viewers aged } & 18-34: 24.2,21.0,17.8,19.6,23.4,19.1,14.6,27.1,19.2,18.3, \\

& 22.9,23.4,17.3,20.5 \\

\text { Viewers aged } & 35-49: 24.9,34.9,35.8,31.9,35.4,29.9,30.9,36.7,36.2,33.8, \\

& 29.5,30.8

\end{array}

$$

a. Compute the mean and median weekly hours of traditional TV viewed by those aged $18-34$.

b. Compute the mean and median weekly hours of traditional TV viewed by those aged $35-49$.

c. Compare the mean and median viewing hours for each age group. Which group watches more traditional TV per week?

Jason Gerber

Numerade Educator

Online Multiplayer Game Downloads. The creator of a new online multiplayer survival game has been tracking the monthly downloads of the newest game. The following table shows the monthly downloads (in thousands) for each month of the current and previous year.

$$

\begin{array}{lclc}

\begin{array}{lcc}

\text { Month } \\

\text { (previous year) }

\end{array} & \begin{array}{c}

\text { Downloads } \\

\text { (thousands) }

\end{array} & \begin{array}{l}

\text { Month } \\

\text { (current year) }

\end{array} & \begin{array}{c}

\text { Downloads } \\

\text { (thousands) }

\end{array} \\

\text { February } & 33.0 & \text { January } & 37.0 \\

\text { March } & 34.0 & \text { February } & 37.0 \\

\text { April } & 34.0 & \text { March } & 37.0 \\

\text { May } & 32.0 & \text { April } & 38.0 \\

\text { June } & 32.0 & \text { May } & 37.0 \\

\text { July } & 35.0 & \text { June } & 36.0 \\

\text { August } & 34.0 & \text { July } & 37.0 \\

\text { September } & 37.0 & \text { August } & 35.0 \\

\text { October } & 37.0 & \text { September } & 33.0 \\

\text { November } & 35.0 & \text { October } & 32.0 \\

\text { December } & 33.0 & &

\end{array}

$$

a. Compute the mean, median, and mode for number of downloads in the previous year.

b. Compute the mean, median, and mode for number of downloads in the current year.

c. Compute the first and third quartiles for downloads in the previous year.

d. Compute the first and third quartiles for downloads in the current year.

e. Compare the values calculated in parts a through d for the previous and current years. What does this tell you about the downloads of the game in the current year compared to the previous year?

Jason Gerber

Numerade Educator

Automobile Fuel Efficiencies. In automobile mileage and gasoline-consumption testing, 13 automobiles were road tested for 300 miles in both city and highway driving conditions. The following data were recorded for miles-per-gallon performance.

Highway: $\quad \begin{array}{lllllllllllllllllll}19.4 & 20.6 & 18.3 & 18.6 & 19.2 & 17.4 & 17.2 & 18.6 & 19.0 & 21.1 & 19.4 & 18.5 & 18.7\end{array}$

Use the mean, median, and mode to make a statement about the difference in performance for city and highway driving.

Jason Gerber

Numerade Educator

Unemployment Rates by State. The U.S. Bureau of Labor Statistics collects data on unemployment rates in each state. The data contained in the file UnemploymentRates show the unemployment rate for every state and the District of Columbia over two consecutive years. To compare unemployment rates for the previous year with unemployment rates for the current year, compute the first quartile, the median, and the third quartile for the previous year unemployment data and the current year unemployment data. What do these statistics suggest about the change in unemployment rates across the states over these two years?

Lucas Finney

Numerade Educator

Motor Oil Prices. Martinez Auto Supplies has retail stores located in eight cities in California. The price they charge for a particular product in each city varies because of differing competitive conditions. For instance, the price they charge for a case of a popular brand of motor oil in each city follows. Also shown are the number of cases that Martinez Auto sold last quarter in each city.

$$

\begin{array}{lcc}

\text { City } & \text { Price (\$) } & \text { Sales (cases) } \\

\text { Bakersfield } & 34.99 & 501 \\

\text { Los Angeles } & 38.99 & 1425 \\

\text { Modesto } & 36.00 & 294 \\

\text { Oakland } & 33.59 & 882 \\

\text { Sacramento } & 40.99 & 715 \\

\text { San Diego } & 38.59 & 1088 \\

\text { San Francisco } & 39.59 & 1644 \\

\text { San Jose } & 37.99 & 819 \\

& &

\end{array}

$$

Compute the average sales price per case for this product during the last quarter.

Jason Gerber

Numerade Educator

Calculating Grade Point Averages. The grade point average for college students is based on a weighted mean computation. For most colleges, the grades are given the following data values: A (4), B (3), C (2), D (1), and F (0). After 60 credit hours of course work, a student at State University carned 9 credit hours of A, 15 credit hours of B, 33 credit hours of $C$, and 3 credit bours of $D$.

a. Compute the student's grade point average.

b. Students at State University must maintain a 2.5 grade point average for their first 60 credit hours of course work in order to be admitted to the business college. Will this student be admitted?

Jason Gerber

Numerade Educator

Mutual Fund Rate of Return. The following table shows the total return and the number of funds for four categories of mutual funds.

$$

\begin{array}{lcc}

\text { Type of Fund } & \text { Number of Funds } & \text { Total Return (\%) } \\

\text { Domestic Equity } & 9191 & 4.65 \\

\text { Intornational Equity } & 2621 & 18.15 \\

\text { Specialty Stock } & 1419 & 11.36 \\

\text { Hybrid } & 2900 & 6.75

\end{array}

$$

a. Using the number of funds as weights, compute the weighted average total return for these mutual funds.

b. Is there any difficulty associated with using the "number of funds" as the weights in computing the weighted average total return in part (a)? Discuss. What else might be used for weights?

c. Suppose you invested $$\$ 10,000$$ in this group of mutual funds and diversified the investment by placing $$\$ 2000$$ in Domestic Equity funds, $$\$ 4000$$ in International Equity funds, $$\$ 3000$$ in Specialty Stock funds, and $$\$ 1000$$ in Hybrid funds. What is the expected return on the portfolio?

Jason Gerber

Numerade Educator

Business School Ranking. Based on a survey of master's programs in business administration, magazines such as U.S. News & World Report rank U.S. business schools. These types of rankings are based in part on surveys of business school deans and corporate recruiters. Each survey respondent is asked to rate the overall academic quality of the master's program on a scale from I "marginal" to 5 "outstanding." Use the sample of responses shown below to compute the weighted mean score for the business school deans and the corporate recruiters. Discuss.

$$

\begin{array}{ccc}

\text { Quality Assessment } & \text { Business School Deans } & \text { Corporate Recruiters } \\

5 & 44 & 31 \\

4 & 66 & 34 \\

3 & 60 & 43 \\

2 & 10 & 12 \\

1 & 0 & 0

\end{array}

$$

Jason Gerber

Numerade Educator

Revenue Growth Rate. Annual revenue for Corning Supplies grew by $5.5 \%$ in 2014, $1.1 \%$ in $2015,-3.5 \%$ in 2016, $-1.1 \%$ in 2017 , and $1.8 \%$ in 2018 . What is the mean growth annual rate over this period?

Jason Gerber

Numerade Educator

Mutual Fund Comparison. Suppose that at the beginning of Year 1 you invested $$\$ 10,000$$ in the Stivers mutual fund and $$\$ 5000$$ in the Trippi mutual fund. The value of each investment at the end of each subsequent year is provided in the table below. Which mutual fund performed better?

$$

\begin{array}{ccr}

\text { Year } & \text { Stivers } & \text { Trippi } \\

1 & 11,000 & 5,600 \\

2 & 12,000 & 6,300 \\

3 & 13,000 & 6,900 \\

4 & 14,000 & 7,600 \\

5 & 15,000 & 8,500 \\

6 & 16,000 & 9,200 \\

7 & 17,000 & 9,900 \\

8 & 18,000 & 10,600

\end{array}

$$

Jason Gerber

Numerade Educator

Asset Growth Rate. If an asset declines in value from $$\$ 5000$$ to $$\$ 3500$$ over nine years, what is the mean annual growth rate in the asset's value over these nine years?

Jason Gerber

Numerade Educator

Company Value Growth Rate. The current value of a company is $$\$ 25$$ million. If the value of the company six year ago was $$\$ 10$$ million, what is the company's mean annual growth rate over the past six years?

Jason Gerber

Numerade Educator

Consider a sample with data values of $10,20,12,17$, and 16 . Compute the range and interquartile range.

N E

Numerade Educator

Consider a sample with data values of $10,20,12,17$, and 16. Compute the variance and standard deviation.

Hubert Agamasu

Numerade Educator

Consider a sample with data values of $27,25,20,15,30,34,28$, and 25 . Compute the range, interquartile range, variance, and standard deviation.

Paul A.

California State Polytechnic University, Pomona

Price of Unleaded Gasoline. Data collected by the Oil Price Information Service from more than 90,000 gasoline and convenience stores throughout the U.S. showed that the average price for a gallon of unleaded gasoline was $\$ 3.28$ (MSN Auto website). The following data show the price per gallon (\$) for a sample of 20 gasoline and convenience stores located in San Francisco.

$$

\begin{array}{llllllllll}

3.59 & 3.59 & 4.79 & 3.56 & 3.55 & 3.71 & 3.65 & 3.60 & 3.75 & 3.56 \\

3.57 & 3.59 & 3.55 & 3.99 & 4.15 & 3.66 & 3.63 & 3.73 & 3.61 & 3.57

\end{array}

$$

a. Use the sample data to estimate the mean price for a gallon of unleaded gasoline in San Francisco.

b. Compute the sample standard deviation.

c. Compare the mean price per gallon for the sample data to the national average price. What conclusions can you draw about the cost living in San Francisco?

Jason Gerber

Numerade Educator

Round-Trip Flight Prices. The following table displays round-trip flight prices from 14 major U.S. cities to Atlanta and Salt Lake City.

$$

\begin{array}{lcc}

\text { Departure City } & \text { Atlanta } & \text { Salt Lake City } \\

\text { Cincinnati } & 340.10 & 570.10 \\

\text { New York } & 321.60 & 354.60 \\

\text { Chicago } & 291.60 & 465.60 \\

\text { Denver } & 339.60 & 219.60 \\

\text { Los Angeles } & 359.60 & 311.60 \\

\text { Seattle } & 384.60 & 297.60 \\

\text { Detroit } & 309.60 & 471.60 \\

\text { Philadelphia } & 415.60 & 618.40 \\

\text { Washington, D.C. } & 293.60 & 513.60 \\

\text { Miami } & 249.60 & 523.20 \\

\text { San Francisco } & 539.60 & 381.60 \\

\text { Las Vegas } & 455.60 & 159.60 \\

\text { Phoenix } & 359.60 & 267.60 \\

\text { Dallas } & 333.90 & 458.60

\end{array}

$$

a. Compute the mean price for a round-trip flight into Atlanta and the mean price for a round-trip flight into Salt Lake City. Is Atlanta less expensive to fly into than Salt Lake City? If so, what could explain this difference?

b. Compute the range, variance, and standard deviation for the two samples. What does this information tell you about the prices for flights into these two cities?

Jason Gerber

Numerade Educator

Annual Sales Amounts. Varatta Enterprises sells industrial plumbing valves. The following table lists the annual sales amounts for the different salespeople in the organization for the most recent fiscal year.

$$

\begin{array}{lclc}

\text { Salesperson } & \text { Sales Amount (\$1000) } & \text { Salesperson } & \text { Sales Amount (\$1000) } \\

\text { Joseph } & 147 & \text { Wei } & 465 \\

\text { Jennifer } & 232 & \text { Samantha } & 410 \\

\text { Phillip } & 547 & \text { Erin } & 298 \\

\text { Stanley } & 328 & \text { Dominic } & 321 \\

\text { Luke } & 295 & \text { Charlie } & 190 \\

\text { Lexie } & 194 & \text { Amol } & 211 \\

\text { Margaret } & 368 & \text { Lenisa } & 413

\end{array}

$$

a. Compute the mean, variance, and standard deviation for these annual sales values.

b. In the previous fiscal year, the average annual sales amount was $$\$ 300,000$$ with a standard deviation of $$\$ 95,000$$. Discuss any differences you observe between the annual sales amount in the most recent and previous fiscal years.

Jason Gerber

Numerade Educator

Air Quality Index. The Los Angeles Times regularly reports the air quality index for various areas of Southern California. A sample of air quality index values for Pomona provided the following data: $28,42,58,48,45,55,60,49$, and 50 .

a. Compute the range and interquartile range.

b. Compute the sample variance and sample standard deviation.

c. A sample of air quality index readings for Anaheim provided a sample mean of 48.5 , a sample variance of 136 , and a sample standard deviation of 11.66. What comparisons can you make between the air quality in Pomona and that in Anaheim on the basis of these descriptive statistics?

Nick Johnson

Numerade Educator

Reliability of Delivery Service. The following data were used to construct the histograms of the number of days required to fill orders for Dawson Supply, Inc., and J.C. Clark Distributors (see Figure 3.2).

Dawson Supply Days for Delivery: $\quad \begin{array}{lllllllllll}11 & 10 & 9 & 10 & 11 & 11 & 10 & 11 & 10 & 10\end{array}$

$\begin{array}{lllllllllllll}\text { Clark Distributors Days for Delivery: } & 8 & 10 & 13 & 7 & 10 & 11 & 10 & 7 & 15 & 12\end{array}$

Use the range and standard deviation to support the previous observation that Dawson Supply provides the more consistent and reliable delivery times.

Jason Gerber

Numerade Educator

Cellular Phone Spending. According to the 2016 Consumer Expenditure Survey, Americans spend an average of $$\$ 1124$$ on cellular phone service annually (U.S. Bureau of Labor Statistics website). Suppose that we wish to determine if there are differences in cellular phone expenditures by age group. Therefore, samples of 10 consumers were selected for three age groups ( $18-34,35-44,45$ and older). The annual expenditure for each person in the sample is provided in the table below.

$$

\begin{array}{ccc}

18-34 & \mathbf{3 5 - 4 4} & \mathbf{4 5} \text { and Older } \\

1355 & 969 & 1135 \\

115 & 434 & 956 \\

1456 & 1792 & 400 \\

2045 & 1500 & 1374 \\

1621 & 1277 & 1244 \\

994 & 1056 & 825 \\

1937 & 1922 & 763 \\

1200 & 1350 & 1192 \\

1567 & 1586 & 1305 \\

1390 & 1415 & 1510

\end{array}

$$

a. Compute the mean, variance, and standard deviation for the each of these three samples.

b. What observations can be made based on these data?

Jason Gerber

Numerade Educator

Advertising Spend by Companies. Advertising Age annually compiles a list of the 100 companies that spend the most on advertising. Consumer-goods company Procter \& Gamble has often topped the list, spending billions of dollars annually. Consider the data found in the file Advertising. It contains annual advertising expenditures for a sample of 20 companies in the automotive sector and 20 companies in the department store sector.

a. What is the mean advertising spent for each sector?

b. What is the standard deviation for each sector?

c. What is the range of advertising spent for each sector?

d. What is the interquartile range for each sector?

e. Based on this sample and your answers to parts (a) to (d), comment on any differences in the advertising spending in the automotive companies versus the department store companies.

Jason Gerber

Numerade Educator

Amateur Golfer Scores. Scores turned in by an amateur golfer at the Bonita Fairways Golf Course in Bonita Springs, Florida, during 2017 and 2018 are as follows:

$$

\begin{array}{lllllllll}

\text { 2017 Season: } & 74 & 78 & 79 & 77 & 75 & 73 & 75 & 77 \\

\text { 2018 Season: } & 71 & 70 & 75 & 77 & 85 & 80 & 71 & 79

\end{array}

$$

a. Use the mean and standard deviation to evaluate the golfer's performance over the two-year period.

b. What is the primary difference in performance between 2017 and 2018? What improvement, if any, can be seen in the 2018 scores?

Jason Gerber

Numerade Educator

Consistency of Running Times. The following times were recorded by the quarter-mile and mile runners of a university track team (times are in minutes).

$$

\begin{array}{lrrrrr}

\text { Quarter-Mile Times: } & .92 & .98 & 1.04 & .90 & .99 \\

\text { Mile Times: } & 4.52 & 4.35 & 4.60 & 4.70 & 4.50

\end{array}

$$

After viewing this sample of running times, one of the coaches commented that the quarter-milers turned in the more consistent times. Use the standard deviation and the coefficient of variation to summarize the variability in the data. Does the use of the coefficient of variation indicate that the coach's statement should be qualified?

Jason Gerber

Numerade Educator

Consider a sample with data values of $10,20,12,17$, and 16 . Compute the $z$-score for each of the five observations.

Hubert Agamasu

Numerade Educator

Consider a sample with a mean of 500 and a standard deviation of 100 . What are the z-scores for the following data values: $520,650,500,450$, and 280 ?

Breanna Ollech

Numerade Educator

Consider a sample with a mean of 30 and a standard deviation of 5 . Use Chebyshev's theorem to determine the percentage of the data within each of the following ranges:

a. 20 to 40

b. 15 to 45

c. 22 to 38

d. 18 to 42

c. 12 to 48

Nick Johnson

Numerade Educator

Suppose the data have a bell-shaped distribution with a mean of 30 and a standard deviation of 5 . Use the empirical rule to determine the percentage of data within each of the following ranges:

a. 20 to 40

b. 15 to 45

c. 25 to 35

Nick Johnson

Numerade Educator

Amount of Sleep per Night. The results of a national survey showed that on average, adults sleep 6.9 hours per night. Suppose that the standard deviation is 1.2 hours.

a. Use Chebyshev's theorem to calculate the percentage of individuals who sleep between 4.5 and 9.3 hours.

b. Use Chebyshev's theorem to calculate the percentage of individuals who sleep between 3.9 and 9.9 hours.

c. Assume that the number of hours of sleep follows a bell-shaped distribution. Use the empirical rule to calculate the percentage of individuals who sleep between 4.5 and 9.3 hours per day. How does this result compare to the value that you obtained using Chebyshev's theorem in part (a)?

Jason Gerber

Numerade Educator

Price per Gallon of Gasoline. Suppose that the mean retail price per gallon of regular grade gasoline in the United States is $$\$ 3.43$$ with a standard deviation of $$\$ .10$$ and that the retail price per gallon has a bell-shaped distribution.

a. What percentage of regular grade gasoline sold between $$\$ 3.33$$ and $$\$ 3.53$$ per gallon?

b. What percentage of regular grade gasoline sold between $$\$ 3.33$$ and $$\$ 3.63$$ per gallon?

c. What percentage of regular grade gasoline sold for more than $$\$ 3.63$$ per gallon?

Jason Gerber

Numerade Educator

GMAT Exam Scores. The Graduate Management Admission Test (GMAT) is a standardized exam used by many universities as part of the assessment for admission to graduate study in business. The average GMAT score is 547 (Magoosh website). Assume that GMAT scores are bell-shaped with a standard deviation of 100.

a. What percentage of GMAT scores are 647 or higher?

b. What percentage of GMAT scores are 747 or higher?

c. What percentage of GMAT scores are between 447 and 547 ?

d. What percentage of GMAT scores are between 347 and 647 ?

Jason Gerber

Numerade Educator

Cost of Backyard Structure. Many families in California are using backyard structures for home offices, art studios, and hobby areas as well as for additional storage. Suppose that the mean price for a customized wooden, shingled backyard structure is $$\$ 3100$$. Assume that the standard deviation is $$\$ 1200$$.

a. What is the z-score for a backyard structure costing $$\$ 2300$$ ?

b. What is the $z$-score for a backyard structure costing $$\$ 4900$$ ?

c. Interpret the $z$-scores in parts (a) and (b). Comment on whether either should be considered an outlier.

d. If the cost for a backyard shed-office combination built in Albany, California, is $$\$ 13,000$$, should this structure be considered an outlier? Explain.

Jason Gerber

Numerade Educator

Best Places to Live. Each year Money magazine publishes a list of "Best Places to Live in the United States." These listings are based on affordability, educational performance, convenience, safety, and livability. The list below shows the median household income of Money magazine's top city in each U.S. state for 2017 (Money magazine website).

$$

\begin{array}{|c|c|c|c|}

\hline \text { City } & \begin{array}{l}

\text { Median Household } \\

\text { Income (\$) }

\end{array} & \text { City } & \begin{array}{l}

\text { Median Household } \\

\text { Income (\$) }

\end{array} \\

\hline \text { Pelham, AL } & 66,772 & \text { Bozeman, MT } & 49,303 \\

\hline \text { Juneau, AK } & 84,101 & \text { Papillion, NE } & 79,131 \\

\hline \text { Paradise Valley, AZ } & 138,192 & \text { Sparks, NV } & 54,230 \\

\hline \text { Fayetteville, AR } & 40,835 & \text { Nashua, NH } & 66,872 \\

\hline \text { Monterey Park, CA } & 57,419 & \text { North Adlington, NJ } & 73,885 \\

\hline \text { Lone Tree, CO } & 116,761 & \text { Rio Rancho, NM } & 58,982 \\

\hline \text { Manchester, CT } & 64,828 & \text { Valley Stream, NY } & 88,693 \\

\hline \text { Hockessin, DE } & 115,124 & \text { Concord, NC } & 54,579 \\

\hline \text { St. Augustine, FL } & 47,748 & \text { Dickinson, ND } & 71,866 \\

\hline \text { Vinings, GA } & 73,103 & \text { Wooster, } \mathrm{OH} & 43,054 \\

\hline \text { Kapaa, HI } & 62,546 & \text { Mustang. OK } & 66,714 \\

\hline \text { Meridian, } 10 & 62,899 & \text { Beaverton, OR } & 58,785 \\

\hline \text { Schaumburg, IL } & 73,824 & \text { Lower Merion, PA } & 117,438 \\

\hline \text { Fishers, } \mathbb{I N} & 87,043 & \text { Warwick, RI } & 63,414 \\

\hline \text { Council Bluffs, LA } & 46,844 & \text { Mauldin, SC } & 57,480 \\

\hline \text { Lenexa, KS } & 76,505 & \text { Rapid City, SD } & 47,788 \\

\hline \text { Georgetown, } \mathrm{KY} & 58,709 & \text { Franklin, TN } & 82,334 \\

\hline \text { Bossier City, LA } & 47,051 & \text { Allen, TX } & 104,524 \\

\hline \text { South Portland, ME } & 56,472 & \text { Orem, UT } & 54,515 \\

\hline \text { Rockville, MD } & 100,158 & \text { Colchester, VT } & 69,181 \\

\hline \text { Waltham, MA } & 75,106 & \text { Reston, VA } & 112,722 \\

\hline \text { Farmington Hills, MI } & 71,154 & \text { Mercer Island, WA } & 128,484 \\

\hline \text { Woodbury, MN } & 99,657 & \text { Morgantown, WV } & 38,060 \\

\hline \text { Olive Branch, MS } & 62,958 & \text { New Berlin, WI } & 74,983 \\

\hline \text { St. Peters, MO } & 57,728 & \text { Cheyenne, WY } & 56,593 \\

\hline

\end{array}

$$

a. Compute the mean and median for these houschold income data.

b. Compare the mean and median values for these data. What does this indicate about the distribution of household income data?

c. Compute the range and standard deviation for these household income data.

d. Compute the first and third quartiles for these household income data.

c. Are there any outliers in these data? What does this suggest about the data?

Jason Gerber

Numerade Educator

NCAA Basketball Game Scores. A sample of 10 NCAA college basketball game scores provided the following data.

$$

\begin{array}{lclcc}

\text { Winning Team } & \text { Points } & \text { Losing Team } & \text { Points } & \begin{array}{c}

\text { Winning } \\

\text { Margin }

\end{array} \\

\text { Arizona } & 90 & \text { Oregon } & 66 & 24 \\

\text { Duke } & 85 & \text { Georgetown } & 66 & 19 \\

\text { Florida State } & 75 & \text { Wake Forest } & 70 & 5 \\

\text { Kansas } & 78 & \text { Colorado } & 57 & 21 \\

\text { Kentucky } & 71 & \text { Notre Dame } & 63 & 8 \\

\text { Louisville } & 65 & \text { Tennessee } & 62 & 3 \\

\text { Oklahoma State } & 72 & \text { Texas } & 66 & 6 \\

\text { Purdue } & 76 & \text { Michigan State } & 70 & 6 \\

\text { Stanford } & 77 & \text { Southem Cal } & 67 & 10 \\

\text { Wisconsin } & 76 & \text { Illinois } & 56 & 20

\end{array}

$$

a. Compute the mean and standard deviation for the points scored by the winning teams.

b. Assume that the points scored by the winning teams for all NCAA games follow a bell-shaped distribution. Using the mean and standard deviation found in part (a), estimate the percentage of all NCAA games in which the winning team scores 84 or more points. Estimate the percentage of NCAA games in which the winning team scores more than 90 points.

c. Compute the mean and standard deviation for the winning margin. Do the data contain outliers? Explain.

Jason Gerber

Numerade Educator

Apple iPads in Schools. The New York Times reported that Apple has unveiled a new iPad marketed specifically to school districts for use by students (The New York Times website). The 9.7 -inch iPads will have faster processors and a cheaper price point in an effort to take market share away from Google Chromebooks in public school districts. Suppose that the following data represent the percentages of students currently using Apple iPads for a sample of 18 U.S. public school districts.

$$

\begin{array}{lllllllllllllll}

15 & 22 & 12 & 21 & 26 & 18 & 42 & 29 & 64 & 20 & 15 & 22 & 18 & 24 & 27 \\

24 & 26 & 19 & & & & & & & & & & & &

\end{array}

$$

a. Compute the mean and median percentage of students currently using Apple iPads.

b. Compare the first and third quartiles for these data.

c. Compute the range and interquartile range for these data.

d. Compute the variance and standard deviation for these data.

e. Are there any outliers in these data?

f. Based on your calculated values, what can we say about the percentage of students using iPads in public school districts?

Jason Gerber

Numerade Educator

Consider a sample with data values of $27,25,20,15,30,34,28$, and 25 . Provide the five-number summary for the data.

Nick Johnson

Numerade Educator

Show the boxplot for the data in exercise 46 .

Jason Gerber

Numerade Educator

Show the five-number summary and the boxplot for the following data: $5,15,18,10$, $8,12,16,10,6$

Zach Steedman

Numerade Educator

A data set has a first quartile of 42 and a third quartile of 50 . Compute the lower and upper limits for the corresponding boxplot. Should a data value of 65 be considered an outlier?

Jason Gerber

Numerade Educator

Naples Half-Marathon Times. Naples, Florida, hosts a half-marathon (13.1-mile race) in January each year. The event attracts top runners from throughout the United States as well as from around the world. In the race results shown below 22 men and 31 women entered the 19-24 age class. Finish times in minutes are as follows. Times are shown in order of finish.

$$

\begin{array}{ccccccccc}

\text { Finish } & \text { Men } & \text { Women } & \text { Finish } & \text { Men } & \text { Women } & \text { Finish } & \text { Men } & \text { Women } \\

1 & 65.30 & 109.03 & 11 & 109.05 & 123.88 & 21 & 143.83 & 136.75 \\

2 & 66.27 & 111.22 & 12 & 110.23 & 125.78 & 22 & 148.70 & 138.20 \\

3 & 66.52 & 111.65 & 13 & 112.90 & 129.52 & 23 & & 139.00 \\

4 & 66.85 & 111.93 & 14 & 113.52 & 129.87 & 24 & & 147.18 \\

5 & 70.87 & 114.38 & 15 & 120.95 & 130.72 & 25 & & 147.35 \\

6 & 87.18 & 118.33 & 16 & 127.98 & 131.67 & 26 & & 147.50 \\

7 & 96.45 & 121.25 & 17 & 128.40 & 132.03 & 27 & 147.75 \\

8 & 98.52 & 122.08 & 18 & 130.90 & 133.20 & 28 & 153.88 \\

9 & 100.52 & 122.48 & 19 & 131.80 & 133.50 & 29 & 154.83 \\

10 & 108.18 & 122.62 & 20 & 138.63 & 136.57 & 30 & 189.27 \\

& & & & & & 31 & 189.28

\end{array}

$$

a. George Towett of Marietta, Georgia, finished in first place for the men and Lauren Wald of Gainesville, Florida, finished in first place for the women. Compare the first-place finish times for men and women. If the 53 men and women runners had competed as one group, in what place would Lauren have finished?

b. What is the median time for men and women runners? Compare men and women runners based on their median times.

c. Provide a five-number summary for both the men and the women.

d. Are there outliers in either group?

e. Show the boxplots for the two groups. Did men or women have the most variation in finish times? Explain.

Jason Gerber

Numerade Educator

Pharmaceutical Company Sales. Annual sales, in millions of dollars, for 21 pharmaceutical companies follow.

$$

\begin{array}{rrrrrr}

8408 & 1374 & 1872 & 8879 & 2459 & 11413 \\

608 & 14138 & 6452 & 1850 & 2818 & 1356 \\

10498 & 7478 & 4019 & 4341 & 739 & 2127 \\

3653 & 5794 & 8305 & & &

\end{array}

$$

a. Provide a five-number summary.

b. Compute the lower and upper limits.

c. Do the data contain any outliers?

d. Johnson & Johnson's sales are the largest on the list at $$\$ 14,138$$ million. Suppose a data entry error (a transposition) had been made and the sales had been entered as $$\$ 41,138$$ million. Would the method of detecting outliers in part (c) identify this problem and allow for correction of the data entry error?

c. Show a boxplot.

Jason Gerber

Numerade Educator

Cell Phone Companies Customer Satisfaction. Consumer Reports provides overall customer satisfaction scores for AT\&T, Sprint, T-Mobile, and Verizon cell-phone services in major metropolitan areas throughout the United States. The rating for each service reflects the overall customer satisfaction considering a variety of factors such as cost, connectivity problems, dropped calls, static interference, and customer support. A satisfaction scale from 0 to 100 is used with 0 indicating completely dissatisfied and 100 indicating completely satisfied. Suppose that the ratings for the four cell-phone services in 20 metropolitan areas are as shown below.

$$

\begin{array}{lcccc}

\text { Metropolitan Area } & \text { AT\&T } & \text { Sprint } & \text { T-Mobile } & \text { Verizon } \\

\text { Atlanta } & 70 & 66 & 71 & 79 \\

\text { Boston } & 69 & 64 & 74 & 76 \\

\text { Chicago } & 71 & 65 & 70 & 77 \\

\text { Dallas } & 75 & 65 & 74 & 78 \\

\text { Denver } & 71 & 67 & 73 & 77 \\

\text { Dotroit } & 73 & 65 & 77 & 79 \\

\text { Jacksonville } & 73 & 64 & 75 & 81 \\

\text { Las Vegas } & 72 & 68 & 74 & 81 \\

\text { Los Angeles } & 66 & 65 & 68 & 78 \\

\text { Miami } & 68 & 69 & 73 & 80 \\

\text { Minneapolis } & 68 & 66 & 75 & 77 \\

\text { Philadelphia } & 72 & 66 & 71 & 78 \\

\text { Phoenix } & 68 & 66 & 76 & 81 \\

\text { San Antonio } & 75 & 65 & 75 & 80 \\

\text { San Diego } & 69 & 68 & 72 & 79 \\

\text { San Francisco } & 66 & 69 & 73 & 75 \\

\text { Seattlo } & 68 & 67 & 74 & 77 \\

\text { St. Louis } & 74 & 66 & 74 & 79 \\

\text { Tampa } & 73 & 63 & 73 & 79 \\

\text { Washington } & 72 & 68 & 71 & 76

\end{array}

$$

a. Consider T-Mobile first. What is the median rating?

b. Develop a five-number summary for the T-Mobile service.

c. Are there outliers for T-Mobile? Explain.

d. Repeat parts (b) and (c) for the other three cell-phone services.

e. Show the boxplots for the four cell-phone services on one graph. Discuss what a comparison of the boxplots tells about the four services. Which service does Consumer Reports recommend as being best in terms of overall customer satisfaction?

Jason Gerber

Numerade Educator

Most Admired Companies. Fortune magazine's list of the world's most admired companies for 2014 is provided in the data contained in the file AdmiredCompanies (Fortune magazine website). The data in the column labeled "Return" shows the one-year total return (\%) for the top ranked 50 companies. For the same time period the S&P average return was $18.4 \%$.

a. Compute the median return for the top ranked 50 companies.

b. What percentage of the top-ranked 50 companies had a one-year return greater than the S&P average return?

c. Develop the five-number summary for the data.

d. Are there any outliers?

e. Develop a boxplot for the one-year total return.

Alexander Cheng

Numerade Educator

U.S. Border Crossings. The Bureau of Transportation Statistics keeps track of all border crossings through ports of entry along the U.S.-Canadian and U.S.-Mexican borders. The data contained in the file BorderCrossings show the most recently published figures for the number of personal vehicle crossings (rounded to the nearest 1000 ) at the 50 busiest ports of entry during the month of August (U.S. Department of Transportation website).

a. What are the mean and median numbers of crossings for these ports of entry?

b. What are the first and third quartiles?

c. Provide a five-number summary.

d. Do the data contain any outliers? Show a boxplot.

Andy Wong

Numerade Educator

Five observations taken for two variables follow.

$$

\begin{array}{r|rrrrr}

x_i & 4 & 6 & 11 & 3 & 16 \\

\hline y_i & 50 & 50 & 40 & 60 & 30

\end{array}

$$

a. Develop a scatter diagram with $x$ on the horizontal axis.

b. What does the scatter diagram developed in part (a) indicate about the relationship between the two variables?

c. Compute and interpret the sample covariance.

d. Compute and interpret the sample correlation coefficient.

Nick Johnson

Numerade Educator

Five observations taken for two variables follow.

$$

\begin{array}{r|rrrrr}

x_i & 6 & 11 & 15 & 21 & 27 \\

\hline y_i & 6 & 9 & 6 & 17 & 12

\end{array}

$$

a. Develop a scatter diagram for these data.

b. What does the scatter diagram indicate about a relationship between $x$ and $y$ ?

c. Compute and interpret the sample covariance.

d. Compute and interpret the sample correlation coefficient.

Nick Johnson

Numerade Educator

Stock Price Comparison. The file StockComparison contains monthly adjusted stock prices for technology company Apple, Inc., and consumer-goods company Procter \& Gamble (P\&G) from 2013-2018.

a. Develop a scatter diagram with Apple stock price on the horizontal axis and P\&G stock price on the vertical axis.

b. What appears to be the relationship between these two stock prices?

c. Compute and interpret the sample covariance.

d. Compute the sample correlation coefficient. What does this value indicate about the relationship between the stock price of Apple and the stock price of P\&G?

Paul A.

California State Polytechnic University, Pomona

Driving Speed and Fuel Efficiency. A department of transportation's study on driving speed and miles per gallon for midsize automobiles resulted in the following data:

$$

\begin{array}{l|llllllllll}

\text { Speed (Miles per Hour) } & 30 & 50 & 40 & 55 & 30 & 25 & 60 & 25 & 50 & 55 \\

\hline \text { Miles per Gallon } & 28 & 25 & 25 & 23 & 30 & 32 & 21 & 35 & 26 & 25

\end{array}

$$

Compute and interpret the sample correlation coefficient.

Jason Gerber

Numerade Educator

Smoke Detector Use and Death Rates. Over the past 40 years, the percentage of homes in the United States with smoke detectors has risen steadily and has plateaued at about $96 \%$ as of 2015 (National Fire Protection Association website). With this increase in the use of home smoke detectors, what has happened to the death rate from home fires? The file SmokeDetectors contains 17 years of data on the estimated percentage of homes with smoke detectors and the estimated home fire deaths per million of population.

a. Do you expect a positive or negative relationship between smoke detector use and deaths from home fires? Why or why not?

b. Compute and report the correlation coefficient. Is there a positive or negative correlation between smoke detector use and deaths from home fires? Comment.

c. Show a scatter plot of the death rate per million of population and the percentage of homes with smoke detectors.

Dominador Tan

Numerade Educator

Stock Market Indexes Comparison. The Russell 1000 is a stock market index consisting of the largest U.S. companies. The Dow Jones Industrial Average is based on 30 large companies. The file Russell gives the annual percentage returns for each of these stock indexes for the years 1988 to 2012 (1Stock1 website).

a. Plot these percentage returns using a scatter plot.

b. Compute the sample mean and standard deviation for each index.

c. Compute the sample correlation.

d. Discuss similarities and differences in these two indexes.

Dominador Tan

Numerade Educator

Best Private Colleges. A random sample of 30 colleges from Kiplinger's list of the best values in private college provided the data shown in the file BestPrivateColleges (Kiplinger website). The variable named Admit Rate (\%) shows the percentage of students that applied to the college and were admitted, and the variable named $4-\mathrm{yr}$ Grad. Rate (\%) shows the percentage of students that were admitted and graduated in four years.

a. Develop a scatter diagram with Admit Rate (\%) as the independent variable. What does the scatter diagram indicate about the relationship between the two variables?

b. Compute the sample correlation coefficient. What does the value of the sample correlation coefficient indicate about the relationship between the Admit Rate (\%) and the 4-yr Grad. Rate (\%)?

Shu Naito

Numerade Educator

Americans Dining Out. Americans tend to dine out multiple times per week. The number of times a sample of 20 families dined out last week provides the following data.

$$

\begin{array}{llllllllll}

6 & 1 & 5 & 3 & 7 & 3 & 0 & 3 & 1 & 3 \\

4 & 1 & 2 & 4 & 1 & 0 & 5 & 6 & 3 & 1

\end{array}

$$

a. Compute the mean and median.

b. Compute the first and third quartiles.

c. Compute the range and interquartile range.

d. Compute the variance and standard deviation.

e. The skewness measure for these data is 34 . Comment on the shape of this distribution. Is it the shape you would expect? Why or why not?

f. Do the data contain outliers?

Jason Gerber

Numerade Educator

NCAA Football Coaches Salaries. A 2017 USA Today article reports that NCAA football coaches' salaries have continued to increase in recent years (USA Today). The annual base salaries for the previous head football coach and the new head football coach at 23 schools are given in the file Coaches.

a. Determine the median annual salary for a previous head football coach and a new head football coach.

b. Compute the range for salaries for both previous and new head football coaches.

c. Compute the standard deviation for salaries for both previous and new head football coaches.

d. Based on your answers to (a) to (c), comment on any differences between the annual base salary a school pays a new head football coach compared to what it paid its previous head football coach.

John Long

Numerade Educator

Physician Office Waiting Times. The average waiting time for a patient at an El Paso physician's office is just over 29 minutes, well above the national average of 21 minutes. In order to address the issue of long patient wait times, some physician's offices are using wait tracking systems to notify patients of expected wait times. Patients can adjust their arrival times based on this information and spend less time in waiting rooms. The following data show wait times (minutes) for a sample of patients at offices that do not have an office tracking system and wait times for a sample of patients at offices with an office tracking system.

$$

\begin{array}{cc}

\begin{array}{lc}

\text { Without Wait } \\

\text { Tracking System }

\end{array} & \begin{array}{l}

\text { With Wait } \\

\text { Tracking System }

\end{array} \\

24 & 31 \\

67 & 11 \\

17 & 14 \\

20 & 18 \\

31 & 12 \\

44 & 37 \\

12 & 9 \\

23 & 13 \\

16 & 12 \\

37 & 15

\end{array}

$$

a. What are the mean and median patient wait times for offices with a wait tracking system? What are the mean and median patient wait times for offices without a wait tracking system?

b. What are the variance and standard deviation of patient wait times for offices with a wait tracking system? What are the variance and standard deviation of patient wait times for visits to offices without a wait tracking system?

c. Do offices with a wait tracking system have shorter patient wait times than offices without a wait tracking system? Explain.

d. Considering only offices without a wait tracking system, what is the z-score for the tenth patient in the sample?

e. Considering only offices with a wait tracking system, what is the $z$-score for the sixth patient in the sample? How does this $z$-score compare with the $z$-score you calculated for part (d)?

f. Based on $z$-scores, do the data for offices without a wait tracking system contain any outliers? Based on $z$-scores, do the data for offices with a wait tracking system contain any outliers?

Jason Gerber

Numerade Educator

Worker Productivity and Insomnia. U.S. companies lose $$\$ 63.2$$ billion per year from workers with insomnia. According to a 2013 article in the Wall Street Journal, workers lose an average of 7.8 days of productivity per year due to lack of sleep. The following data show the number of hours of sleep attained during a recent night for a sample of 20 workers.

$$

\begin{array}{rccccccccc}

6 & 5 & 10 & 5 & 6 & 9 & 9 & 5 & 9 & 5 \\

8 & 7 & 8 & 6 & 9 & 8 & 9 & 6 & 10 & 8

\end{array}

$$

a. What is the mean number of hours of sleep for this sample?

b. What is the variance? Standard deviation?

Jason Gerber

Numerade Educator

Smartphone Use. Smartphones have become ubiquitous for most people and have become the predominant means of communication among people. Consider the following data indicating the number of minutes in a month spent interacting with others via a smartphone for a sample of 50 smartphone users.

$$

\begin{array}{lllll}

353 & 458 & 404 & 394 & 416 \\

437 & 430 & 369 & 448 & 430 \\

431 & 469 & 446 & 387 & 445 \\

354 & 468 & 422 & 402 & 360 \\

444 & 424 & 441 & 357 & 435 \\

461 & 407 & 470 & 413 & 351 \\

464 & 374 & 417 & 460 & 352 \\

445 & 387 & 468 & 368 & 430 \\

384 & 367 & 436 & 390 & 464 \\

405 & 372 & 401 & 388 & 367

\end{array}

$$

a. What is the mean number of minutes spent interacting with others for this sample? How does it compare to the mean reported in the study?

b. What is the standard deviation for this sample?

c. Are there any outliers in this sample?

Jason Gerber

Numerade Educator

Work Commuting Methods. Public transportation and the automobile are two methods an employee can use to get to work each day. Samples of times recorded for each method are shown. Times are in minutes.

$$

\begin{array}{lllllllllll}

\text { Public Transportation: } & 28 & 29 & 32 & 37 & 33 & 25 & 29 & 32 & 41 & 34 \\

\text { Automobile: } & 29 & 31 & 33 & 32 & 34 & 30 & 31 & 32 & 35 & 33

\end{array}

$$

a. Compute the sample mean time to get to work for each method.

b. Compute the sample standard deviation for each method.

c. On the basis of your results from parts (a) and (b), which method of transportation should be preferred? Explain.

d. Develop a box plot for each method. Does a comparison of the box plots support your conclusion in part (c)?

Jason Gerber

Numerade Educator

Household Incomes. The following data represent a sample of 14 household incomes $$(\$ 1000 \mathrm{~s})$$. $$

\begin{array}{lllllll}

49.4 & 52.4 & 53.4 & 51.3 & 52.1 & 48.7 & 52.1 \\

52.2 & 64.5 & 51.6 & 46.5 & 52.9 & 52.5 & 51.2

\end{array}

$$

a. What is the median household income for these sample data?

b. According to a previous survey, the median annual household income five years ago was $\$ 55,000$. Based on the sample data above, estimate the percentage change in the median houschold income from five years ago to today.

c. Compute the first and third quartiles.

d. Provide a five-number summary.

e. Using the $\tau$-score approach, do the data contain any outliers? Does the approach that uses the values of the first and third quartiles and the interquartile range to detect outliers provide the same results?

Jason Gerber

Numerade Educator

Restaurant Chains' Sales per Store. The data contained in the file FoodIndustry show the company/chain name, the average sales per store ( $$\$ 1000 \mathrm{~s}$$ ), and the food segment industry for 47 restaurant chains (Quick Service Restaurant Magazine website).

a. What was the mean U.S. sales per store for the 47 restaurant chains?

b. What are the first and third quartiles? What is your interpretation of the quartiles?

c. Show a boxplot for the level of sales and discuss if there are any outliers in terms of sales that would skew the results.

d. Develop a frequency distribution showing the average sales per store for each segment. Comment on the results obtained.

Jason Gerber

Numerade Educator

Best Hotels. Travel + Leisure magazine provides an annual list of the 500 best hotels in the world. The magazine provides a rating for each hotel along with a brief description that includes the size of the hotel, amenities, and the cost per night for a double room. A sample of 12 of the top-rated hotels in the United States follows.

$$

\begin{array}{llcc}

\text { Hotel } & \text { Location } & \text { Rooms } & \text { Cost/Night } \\

\text { Boulders Resort \& Spa } & \text { Phoenix, AZ } & 220 & 499 \\

\text { Disney's Wilderness Lodge } & \text { Orlando, FL } & 727 & 340 \\

\text { Four Seasons Hotel Beverly Hills } & \text { Los Angeles, CA } & 285 & 585 \\

\text { Four Seasons Hotel } & \text { Boston, MA } & 273 & 495 \\

\text { Hay-Adams } & \text { Washington, DC } & 145 & 495 \\

\text { Inn on Biltmore Estate } & \text { Asheville, NC } & 213 & 279 \\

\text { Loews Ventana Canyon Resort } & \text { Phoenix, AZ } & 398 & 279 \\

\text { Mauna Lani Bay Hotel } & \text { Island of Hawail } & 343 & 455 \\

\text { Montage Laguna Beach } & \text { Laguna Beach, CA } & 250 & 595 \\

\text { Sofitel Water Tower } & \text { Chicago, IL } & 414 & 367 \\

\text { St. Regis Monarch Beach } & \text { Dana Point, CA } & 400 & 675 \\

\text { The Broadmoor } & \text { Colorado Springs, CO } & 700 & 420

\end{array}

$$

a. What is the mean number of rooms?

b. What is the mean cost per night for a double room?

c. Develop a scatter diagram with the number of rooms on the horizontal axis and the cost per night on the vertical axis. Does there appear to be a relationship between the number of rooms and the cost per night? Discuss.

d. What is the sample correlation coefficient? What does it tell you about the relationship between the number of rooms and the cost per night for a double room? Does this appear reasonable? Discuss.

Jason Gerber

Numerade Educator

NFL Teams Worth. In 2014, the 32 teams in the National Football League (NFL) were worth, on average, $\$ 1.17$ billion, $5 \%$ more than in 2013 . The following data show the annual revenue ( $\$$ millions) and the estimated team value ( $\$$ millions) for the 32 NFL teams in 2014 (Forbes website).

a. Develop a scatter diagram with Revenue on the horizontal axis and Value on the vertical axis. Does there appear that there is any relationship between the two variables?

b. What is the sample correlation coefficient? What can you say about the strength of the relationship between Revenue and Value?

Jason Gerber

Numerade Educator

MLB Team Winning Percentages. Does a major league baseball team's record during spring training indicate how the team will play during the regular season? Over a six-year period, the correlation coefficient between a team's winning percentage in spring training and its winning percentage in the regular season is . 18 . Shown are the winning percentages for the 14 American League teams during a previous season.

a. What is the correlation coefficient between the spring training and the regular season winning percentages?

b. What is your conclusion about a team's record during spring training indicating how the team will play during the regular season? What are some of the reasons why this occurs? Discuss.

Jason Gerber

Numerade Educator

Money Market Funds Days to Maturity. The days to maturity for a sample of five money market funds are shown here. The dollar amounts invested in the funds are provided. Use the weighted mean to determine the mean number of days to maturity for dollars invested in these five money market funds

Jason Gerber

Numerade Educator

Automobile Speeds. Automobiles traveling on a road with a posted speed limit of 55 miles per hour are checked for speed by a state police radar system. Following is a frequency distribution of speed.

a. What is the mean speed of the automobiles traveling on this road?

b. Compute the variance and the standard deviation.

Jason Gerber

Numerade Educator

Annual Returns for Panama Railroad Company Stock. The Panama Railroad Company was established in 1850 to construct a railroad across the isthmus that would allow fast and easy access between the Atlantic and Pacific Oceans. The following table provides annual returns for Panama Railroad stock from 1853 through 1880.

a. Create a graph of the annual returns on the stock. The New York Stock Exchange eamed an annual average return of $8.4 \%$ from 1853 through 1880. Can you tell from the graph if the Panama Railroad Company stock outperformed the New York Stock Exchange?

b. Calculate the mean annual return on Panama Railroad Company stock from 1853 through 1880. Did the stock outperform the New York Stock Exchange over the same period?

Jason Gerber

Numerade Educator