Statistics for Business and Economics: Global Edition

Newbold P., Carlson W.L., Thorne B.M.

Chapter 2

Using Numerical Measures to Describe Data - all with Video Answers

Educators

Chapter Questions

Problem 1

A random sample of 5 weeks showed that a cruise agency received the following number of weekly specials to the Caribbean:
$\begin{array}{lllll}20 & 73 & 75 & 80 & 82\end{array}$
a. Compute the mean, median, and mode.
b. Which measure of central tendency best describes the data?

Sheryl Ezze

Sheryl Ezze

Numerade Educator

Problem 2

A department-store manager is interested in the number of complaints received by the customer-service department about the quality of electrical products sold by the store. Records over a 5-week period show the following number of complaints for each week:
$\begin{array}{lllll}13 & 15 & 8 & 16 & 8\end{array}$
a. Compute the mean number of weekly complaints.
b. Calculate the median number of weekly complaints.
c. Find the mode.

Sheryl Ezze

Numerade Educator

Problem 3

Ten economists were asked to predict the percentage growth in the Consumer Price Index over the next year. Their forecasts were as follows:
$\begin{array}{lllll}3.6 & 3.1 & 3.9 & 3.7 & 3.5\end{array}$
$\begin{array}{lllll}3.7 & 3.4 & 3.0 & 3.7 & 3.4\end{array}$
a. Compute the sample mean.
b. Compute the sample median.
c. Find the mode.

Sheryl Ezze

Numerade Educator

Problem 4

A department-store chain randomly sampled 10 stores in a state. After a review of sales records, it was found that, compared with the same period last year, the following percentage increases in dollar sales had been achieved over the Christmas period this year:
$\begin{array}{rrrrr}10.2 & 3.1 & 5.9 & 7.0 & 3.7 \\ 2.9 & 6.8 & 7.3 & 8.2 & 4.3\end{array}$
a. Calculate the mean percentage increase in dollar sales.
b. Calculate the median.

Sheryl Ezze

Numerade Educator

Problem 5

A sample of 12 senior executives found the following results for percentage of total compensation derived from bonus payments:
$\begin{array}{llllll}15.8 & 17.3 & 28.4 & 18.2 & 15.0 & 24.7\end{array}$
$\begin{array}{lllllll}13.1 & 10.2 & 29.3 & 34.7 & 16.9 & 25.3\end{array}$
a. Compute the sample median.
b. Compute the sample mean.

Sheryl Ezze

Numerade Educator

Problem 6

During the last 3 years Consolidated Oil Company expanded its gasoline stations into convenience food stores (CFSs) in an attempt to increase total sales revenue. The daily sales (in hundreds of dollars) from a random sample of 10 weekdays from one of its stores are:
$\begin{array}{llllllllll}6 & 8 & 10 & 12 & 14 & 9 & 11 & 7 & 13 & 11\end{array}$
a. Find the mean, median and mode for this store.
b. Find the five-number summary.

Sheryl Ezze

Numerade Educator

Problem 7

A textile manufacturer obtained a sample of 50 bolts of cloth from a day's output. Each bolt is carefully inspected and the number of imperfections is recorded as follows:
$$
\begin{array}{lrrrr}
\hline \text { Number of imperfections } & 0 & 1 & 2 & 3 \\
\hline \text { Number of bolts } & 35 & 10 & 3 & 2 \\
\hline
\end{array}
$$
Find the mean, median, and mode for these sample data.

Sheryl Ezze

Numerade Educator

Problem 8

The ages of a sample of 12 students enrolled in an online macroeconomics course are as follows:
$\begin{array}{llllll}21 & 22 & 27 & 36 & 18 & 19 \\ 22 & 23 & 22 & 28 & 36 & 33\end{array}$
a. What is the mean age for this sample?
b. Find the median age.
c. What is the value of the modal age?

Sheryl Ezze

Numerade Educator

Problem 9

A random sample of 156 grade point averages for students at one university is stored in the data file Grade Point Averages.
a. Compute the first and third quartiles.
b. Calculate the 30 th percentile.
c. Calculate the 80 th percentile.

Lucas Finney

Numerade Educator

Problem 10

A sample of 33 accounting students recorded the number of hours spent studying the course material during the week before the final exam. The data are stored in the data file Study.
a. Compute the sample mean.
b. Compute the sample median.
c. Comment on symmetry or skewness.
d. Find the five-number summary for this data.

Jon Southam

Numerade Educator

Problem 11

The data file Sun contains the volumes for a random sample of 100 bottles $(237 \mathrm{~mL})$ of a new suntan lotion.
a. Find and interpret the mean volume.
b. Determine the median volume.
c. Are the data symmetric or skewed? Explain.
d. Find the five-number summary for this data.

Julie Miller

Numerade Educator

Problem 12

Compute the variance and standard deviation of the following sample data:
$$
\begin{array}{llllllll}
6 & 8 & 7 & 10 & 3 & 5 & 9 & 8
\end{array}
$$

Sheryl Ezze

Numerade Educator

Problem 13

Compute the variance and standard deviation of the following sample data:
$$
\begin{array}{llllll}
3 & 0 & -2 & -1 & 5 & 10
\end{array}
$$

Sheryl Ezze

Numerade Educator

Problem 14

Calculate the coefficient of variation for the following sample data:
$$
\begin{array}{lllll}
10 & 8 & 11 & 7 & 9
\end{array}
$$

Sheryl Ezze

Numerade Educator

Problem 15

The ages of a random sample of people who attended a recent soccer match are as follows:
$\begin{array}{lllllll}23 & 35 & 14 & 37 & 38 & 15 & 45 \\ 12 & 40 & 27 & 13 & 18 & 19 & 23 \\ 37 & 20 & 29 & 49 & 40 & 65 & 53 \\ 18 & 17 & 23 & 27 & 29 & 31 & 42 \\ 35 & 38 & 22 & 20 & 15 & 17 & 21\end{array}$
a. Find the mean age.
b. Find the standard deviation.
c. Find the coefficient of variation.

Sheryl Ezze

Numerade Educator

Problem 16

Construct a stem-and-leaf display of the ages of a random sample of people who attended a recent soccer match given in Exercise 2.15. Then find the interquartile range.

Sheryl Ezze

Numerade Educator

Problem 17

A random sample of data has a mean of 75 and a variance of 25.
a. Use Chebyshev's theorem to determine the percent of observations between 65 and 85 .
b. If the data are mounded, use the empirical rule to find the approximate percent of observations between 65 and 85 .

Sheryl Ezze

Numerade Educator

Problem 18

If the mean of a population is 250 and its standard deviation is 20 , approximately what proportion of observations is in the interval between each pair of values?
a. 190 and 310
b. 210 and 290

Sheryl Ezze

Numerade Educator

Problem 19

A set of data is mounded, with a mean of 450 and a variance of 625. Approximately what proportion of the observations is
a. greater than 425 ?
b. less than 500 ?
c. greater than 525 ?

Sheryl Ezze

Numerade Educator

Problem 20

The annual percentage returns on common stocks over a 7-year period were as follows:
$$
\begin{array}{lllllll}
4.0 \% & 14.3 \% & 19.0 \% & -14.7 \% & -26.5 \% & 37.2 \% & 23.8 \%
\end{array}
$$
Over the same period the annual percentage returns on U.S. Treasury Bills were as follows:
$\begin{array}{lllllll}6.5 \% & 4.4 \% & 3.8 \% & 6.9 \% & 8.0 \% & 5.8 \% & 5.1 \%\end{array}$
a. Compare the means of these two population distributions.
b. Compare the standard deviations of these two population distributions.

Sheryl Ezze

Numerade Educator

Problem 21

How much time do corporate executives exercise daily? Training programs exist to help executives improve their health so that they can think more clearly and make better business decisions. Suppose that we randomly sample ten executives and obtain the following daily exercise times (in minutes):
$\begin{array}{llll}20 &35 &28 &22 &10 &40 &23 &32 &28 &30\end{array}$
a. Find the mean daily exercise time.
b. Calculate the standard deviation using Equation 2.13.
c. Calculate the standard deviation using Equation 2.14.
d. Calculate the standard deviation using Equation 2.15.
e. Find the coefficient of variation.

Emily Himsel

Numerade Educator

Problem 22

The operations manager at a plant that bottles natural spring water wants to be sure that the filling process for 1-gallon bottles (1 gallon is approximately 3.785 liters) is operating properly. A random sample of 75 bottles is selected and the contents are measured. The volume of each bottle is contained in the data file.
a. Find the range, variance, and standard deviation of the volumes.
b. Find and interpret the interquartile range for the data.
c. Find the value of the coefficient of variation.

Emily Himsel

Numerade Educator

Problem 23

In Chapter 1 we described graphically, with a frequency distribution and histogram, the time (in seconds) for a random sample of $n=110$ employees to complete a particular task. Describe the data in Table 1.6 numerically. The data are stored in the data file Completion Times.
a. Find the mean.
b. Find the variance and standard deviation.
c. Find the coefficient of variation.

Emily Himsel

Numerade Educator

Problem 24

The assessment rates (in percentages) assigned to a random sample of 40 commercially zoned parcels of land in the year 2012 are stored in the data file.
a. What is the standard deviation in the assessment rates?
b. Approximately what proportion of the rates will be within \pm 2 standard deviations of the mean?

Emily Himsel

Numerade Educator

Problem 25

Calculate the mean dollar amount and the standard deviation for the dollar amounts charged to a Visa account at Florin's Flower Shop. Data are stored in the data file.

Emily Himsel

Numerade Educator

Problem 26

Consider the following sample of five values and corresponding weights:
$$
\begin{array}{cc}
\hline x_i & w_i \\
\hline 4.6 & 8 \\
3.2 & 3 \\
5.4 & 6 \\
2.6 & 2 \\
5.2 & 5 \\
\hline
\end{array}
$$
a. Calculate the arithmetic mean of the $x_i$ values without weights.
b. Calculate the weighted mean of the $x_i$ values.

Sheryl Ezze

Numerade Educator

Problem 27

Consider the following frequency distribution for a sample of 40 observations:
$$
\begin{array}{cc}
\hline \text { Class } & \text { Frequency } \\
\hline 0-4 & 5 \\
5-9 & 8 \\
10-14 & 11 \\
15-19 & 9 \\
20-24 & 7 \\
\hline
\end{array}
$$
a. Calculate the sample mean.
b. Calculate the sample variance and sample standard deviation.

Sheryl Ezze

Numerade Educator

Problem 28

An online pharmaceutical company obtained the following frequency distribution of shipping times (number of hours between the time an order is placed and the time the order is shipped) for a random sample of 40 orders. (Be sure to complete all appropriate columns and show your work).
$$
\begin{array}{rr}
\hline \text { Number of Hours } & f_i \\
\hline 4<10 & 8 \\
10<16 & 15 \\
16<22 & 10 \\
22<28 & 7 \\
\hline
\end{array}
$$
a. What is the approximate mean shipping time?
b. What is the approximate variance and standard deviation?

Sheryl Ezze

Numerade Educator

Problem 29

A manufacturer of portable radios obtained a sample of 50 radios from a week's output. The radios were checked and the numbers of defects were recorded as follows.
$$
\begin{array}{lcccc}
\hline \text { Number of defects } & 0 & 1 & 2 & 3 \\
\hline \text { Number of radios } & 12 & 15 & 17 & 6 \\
\hline
\end{array}
$$
Calculate the standard deviation.

Sheryl Ezze

Numerade Educator

Problem 30

A random sample of 50 personal property insurance policies showed the following number of claims over the past 2 years.
$$
\begin{array}{lccccccc}
\hline \text { Number of claims } & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\
\hline \text { Number of policies } & 21 & 13 & 5 & 4 & 2 & 3 & 2 \\
\hline
\end{array}
$$
a. Find the mean number of claims per policy.
b. Find the sample variance and standard deviation.

Sheryl Ezze

Numerade Educator

Problem 31

For a random sample of 25 students from a very large university, the accompanying table shows the amount of time (in hours) spent studying for final exams.
$$
\begin{array}{lccccc}
\hline \text { Study time } & 0<4 & 4<8 & 8<12 & 12<16 & 16<20 \\
\hline \begin{array}{l}
\text { Number of } \\
\text { students }
\end{array} & 3 & 7 & 8 & 5 & 2 \\
\hline
\end{array}
$$
a. Estimate the sample mean study time.
b. Estimate the sample standard deviation.

Sheryl Ezze

Numerade Educator

Problem 32

A sample of 20 financial analysts was asked to provide forecasts of earnings per share of a corporation for next year. The results are summarized in the following table:
$$
\begin{array}{lc}
\hline \text { Forecast (\$ per share) } & \text { Number of Analysts } \\
\hline \$ 9.95 \text { to under } \$ 10.45 & 2 \\
\$ 10.45 \text { to under } \$ 10.95 & 8 \\
\$ 10.95 \text { to under } \$ 11.45 & 6 \\
\$ 11.45 \text { to under } \$ 11.95 & 3 \\
\$ 11.95 \text { to under } \$ 12.45 & 1 \\
\hline
\end{array}
$$
a. Estimate the sample mean forecast.
b. Estimate the sample standard deviation.

Sheryl Ezze

Numerade Educator

Problem 33

A publisher receives a copy of a 500-page textbook from a printer. The page proofs are carefully read and the number of errors on each page is recorded, producing the data in the following table:
$$
\begin{array}{ccccccc}
\hline \text { Number of errors } & 0 & 1 & 2 & 3 & 4 & 5 \\
\hline \text { Number of pages } & 102 & 138 & 140 & 79 & 33 & 8 \\
\hline
\end{array}
$$
Find the mean and standard deviation in number of errors per page.

Sheryl Ezze

Numerade Educator

Problem 34

In Chapter 1, we described graphically using a frequency distribution table and a histogram the time (in seconds) for a random sample of $n=110$ employees to complete a particular task. Describe the data numerically based on the frequency distribution given in Table 1.7. The data is stored in the data file.
a. Compute the mean using Equation 2.21.
b. Compute the variance using Equation 2.22.
c. Compare your answers to the mean and variance calculated in Exercise 2.23.

Emily Himsel

Numerade Educator

Problem 35

Following is a random sample of seven $(x, y)$ pairs of data points:
$$
(1,5)(3,7)(4,6)(5,8)(7,9)(3,6)(5,7)
$$
a. Compute the covariance.
b. Compute the correlation coefficient.

Sheryl Ezze

Numerade Educator

Problem 36

Following is a random sample of five $(x, y)$ pairs of data points:
$$
(12,200)(30,600)(15,270)(24,500)(14,210)
$$
a. Compute the covariance.
b. Compute the correlation coefficient.

Sheryl Ezze

Numerade Educator

Problem 37

Following is a random sample of price per piece of plywood, $X$, and quantity sold, $Y$ (in thousands):
$$
\begin{array}{cc}
\hline \text { Price per Piece }(x) & \text { Thousands of Pieces Sold }(y) \\
\hline \$ 6 & 80 \\
7 & 60 \\
8 & 70 \\
9 & 40 \\
10 & 0 \\
\hline
\end{array}
$$
a. Compute the covariance.
b. Compute the correlation coefficient.

Sheryl Ezze

Numerade Educator

Problem 38

River Hills Hospital is interested in determining the effectiveness of a new drug for reducing the time required for complete recovery from knee surgery. Complete recovery is measured by a series of strength tests that compare the treated knee with the untreated knee. The drug was given in varying amounts to 18 patients over a 6-month period. For each patient the number of drug units, $X$, and the days for complete recovery, $Y$, are given by the following $(x, y)$ data:
$\begin{array}{llllll}(5,53) & (21,65) & (14,48) & (11,66) & (9,46) & (4,56) \\ (7,53) & (21,57) & (17,49) & (14,66) & (9,54) & (7,56) \\ (9,53) & (21,52) & (13,49) & (14,56) & (9,59) & (4,56)\end{array}$
a. Compute the covariance.
b. Compute the correlation coefficient.
c. Briefly discuss the relationship between the number of drug units and the recovery time. What dosage might we recommend based on this initial analysis?

Sheryl Ezze

Numerade Educator

Problem 39

A Hong Kong snack-food vendor offers 3 types of boxed "lunches to go," priced at $$\$3$$, $$\$5$$, and $$\$10$$, respectively. The vendor would like to establish whether there is a relationship between the price of the boxed lunch and the number of sales achieved per hour. Consequently, over a 15-day period the vendor records the number of sales made for each of the 3 types of boxed lunches. The following data show the boxedlunch price $(x)$ and the number sold (y) during each of the 15 lunch hours.
$(3,7),(5,5),(10,2),(3,9),(5,6),(10,5),(3,6),(5,6)$, $(10,1),(3,10),(5,7),(10,4),(3,5),(5,6),(10,4)$
a. Describe the data numerically with their covariance and correlation.
b. Discuss the relationship between the price and number of boxed lunches sold.

Sheryl Ezze

Numerade Educator

Problem 40

The following data give $X$, the price charged for a particular item, and $Y$, the quantity of that item sold (in thousands):
$$
\begin{array}{cc}
\hline \text { Price per Piece }(X) & \text { Hundreds of Pieces Sold }(Y) \\
\hline \$ 5 & 55 \\
6 & 53 \\
7 & 45 \\
8 & 40 \\
9 & 20 \\
\hline
\end{array}
$$
a. Compute the covariance.
b. Compute the correlation coefficient.

Sheryl Ezze

Numerade Educator

Problem 41

Snappy Lawn Care, a growing business in central Florida, keeps records of the temperature (in degrees Fahrenheit) and the time (in hours) required to complete a contract. A random sample of temperatures and time for $n=11$ contracts is stored in the data file.
a. Compute the covariance.
b. Compute the correlation coefficient.

Emily Himsel

Numerade Educator

Problem 42

A consumer goods company has been studying the effect of advertising on total profits. As part of this study, data on advertising expenditures (in thousands of dollars) and total sales (in thousands of dollars) were collected for a 5 -month period and are as follows:
$$
(10,100)(15,200)(7,80)(12,120)(14,150)
$$
The first number is advertising expenditures and the second is total sales. Plot the data and compute the correlation coefficient.

Sheryl Ezze

Numerade Educator

Problem 43

The president of Floor Coverings Unlimited wants information concerning the relationship between retail experience (years) and weekly sales (in hundreds of dollars). He obtained the following random sample on experience and weekly sales:
$$
(2,5)(4,10)(3,8)(6,18)(3,6)(5,15)(6,20)(2,4)
$$
The first number for each observation is years of experience, and the second number is weekly sales. Compute the covariance and the correlation coefficient.

Sheryl Ezze

Numerade Educator

Problem 44

A major airport recently hired consultant John Cadariu to study the problem of air traffic delays. He recorded the number of minutes planes were late for a sample of flights in the following table:
$$
\begin{array}{lcccccc}
\hline \begin{array}{l}
\text { Minutes } \\
\text { late }
\end{array} & 0<10 & 10<20 & 20<30 & 30<40 & 40<50 & 50<60 \\
\hline \begin{array}{l}
\text { Number } \\
\text { of flights }
\end{array} & 30 & 25 & 13 & 6 & 5 & 4 \\
\hline
\end{array}
$$
a. Estimate the mean number of minutes late.
b. Estimate the sample variance and standard deviation.

Sheryl Ezze

Numerade Educator

Problem 45

Snappy Lawn Care, a growing business in central Florida, keeps records of charges for its professional lawn care services. A random sample of $n=$ 50 charges is stored in the data file Snappy Lawn Care. Describe the data numerically.
a. Compute the mean charge.
b. Compute the standard deviation.
c. Compute the five-number summary.

Emily Himsel

Numerade Educator

Problem 46

In Example 2.9 we calculated the variance and standard deviation for Location 1 of Gilotti's Pizzeria restaurants. Use the data in the data file Gilotti's Pizzeria to find the variance and the standard deviation for Location 2, Location 3, and Location 4.

Emily Himsel

Numerade Educator

Problem 47

Describe the following data numerically:
$$
\begin{array}{llllll}
(4,53) & (10,65) & (15,48) & (10,66) & (8,46) & (5,56) \\
(7,60) & (11,57) & (12,49) & (14,70) & (10,54) & (7,56) \\
(9,50) & (8,52) & (11,59) & (10,66) & (8,49) & (5,50)
\end{array}
$$

Sheryl Ezze

Numerade Educator

Problem 48

Only 67 students in the data file Student GPA have SAT verbal scores.
a. Construct the scatter plot of GPAs and SAT scores for these 67 students.
b. Calculate the correlation between GPAs and SAT scores for these 67 students.

MA

Melissa A

Numerade Educator

Problem 49

Consider the following four populations:
• $1,2,3,4,5,6,7,8$
• $1,1,1,1,8,8,8,8$
• $1,1,4,4,5,5,8,8$,
• $-6,-3,0,3,6,9,12,15$
All these populations have the same mean. Without doing the calculations, arrange the populations according to the magnitudes of their variances, from smallest to largest. Then calculate each of the variances manually.

Sheryl Ezze

Numerade Educator

Problem 50

An auditor finds that the values of a corporation's accounts receivable have a mean of $$\$ 295$$ and a standard deviation of $$\$ 63$$.
a. It can be guaranteed that $60 \%$ of these values will be in what interval?
b. It can be guaranteed that $84 \%$ of these values will be in what interval?

Sheryl Ezze

Numerade Educator

Problem 51

In one year, earnings growth of the 500 largest U.S. corporations averaged $9.2 \%$; the standard deviation was $3.5 \%$.
a. It can be guaranteed that $84 \%$ of these earnings growth figures will be in what interval?
b. Using the empirical rule, it can be estimated that approximately $68 \%$ of these earnings growth figures will be in what interval?

Sheryl Ezze

Numerade Educator

Problem 52

Tires of a particular brand have a lifetime mean of 29,000 miles and a standard deviation of 3,000 miles.
a. It can be guaranteed that $75 \%$ of the lifetimes of tires of this brand will be in what interval?
b. Using the empirical rule, it can be estimated that approximately $95 \%$ of the lifetimes of tires of this brand will be in what interval?

Khalida Dawar

Numerade Educator

Problem 53

The supervisor of a very large plant obtained the time (in seconds) for a random sample of $n=110$ employees to complete a particular task. The data is stored in the data file Completion Times.
a. Find and interpret the IQR.
b. Find the five-number summary.

Robin Corrigan

Numerade Educator

Problem 54

How much time (in minutes) do people spend on a typical visit to a local mall? A random sample of $n=104$ shoppers was timed and the results (in minutes) are stored in the data file Shopping Times. You were asked to describe graphically the shape of the distribution of shopping times in Exercise 1.72 (Chapter 1 ). Now describe the shape of the distribution numerically.
a. Find the mean shopping time.
b. Find the variance and standard deviation in shopping times.
c. Find the 95th percentile.
d. Find the five-number summary.
e. Find the coefficient of variation.
f. Ninety percent of the shoppers completed their shopping within approximately how many minutes?

Robin Corrigan

Numerade Educator

Problem 55

A random sample for five exam scores produced the following (hours of study, grade) data values:
$$
\begin{array}{cc}
\hline \text { Hours Studied }(x) & \text { Test Grade }(y) \\
\hline 3.5 & 88 \\
2.4 & 76 \\
4 & 92 \\
5 & 85 \\
1.1 & 60 \\
\hline
\end{array}
$$
a. Compute the covariance.
b. Compute the correlation coefficient

Sheryl Ezze

Numerade Educator

Problem 56

A corporation administers an aptitude test to all new sales representatives. Management is interested in the extent to which this test is able to predict weekly sales of new representatives. Aptitude test scores range from 0 to 30 with greater scores indicating a higher aptitude. Weekly sales are recorded in hundreds of dollars for a random sample of 10 representatives. Test scores and weekly sales are as follows:
$$
\begin{array}{lllllllllll}
\hline \text { Test Score, } x & 12 & 30 & 15 & 24 & 14 & 18 & 28 & 26 & 19 & 27 \\
\hline \text { Weekly Sales, } y & 20 & 60 & 27 & 50 & 21 & 30 & 61 & 54 & 32 & 57 \\
\hline
\end{array}
$$
a. Compute the covariance between test score and weekly sales.
b. Compute the correlation between test score and weekly sales.

Sheryl Ezze

Numerade Educator