Elementary Statistics

Robert Johnson, Patricia Kuby

Chapter 3

Descriptive Analysis and Presentation of Bivariate Data - all with Video Answers

Educators

Section 1

Bivariate Data

Problem 1

a. Is there a relationship (pattern) between the two variables length of a rainbow trout and weight of a rainbow trout? Explain why or why not.
b. Do you think it is reasonable (or possible) to predict the weight of a rainbow trout based on the length of the rainbow trout? Explain why or why not.

James York

James York

Numerade Educator

Problem 2

a. Is there a relationship between a person's height and shoe size as he or she grows from an infant to age $16 ?$ As one variable gets larger, does the other also get larger? Explain your answers.
b. Is there a relationship between height and shoe size for people who are older than 16 years of age? Do taller people wear larger shoes? Explain your answers.

James York

Numerade Educator

Problem 3

In a national survey of 500 business and 500 leisure travelers, each was asked where he or she would most like "more space."a. Express the table as percentages of the total.
b. Express the table as percentages of the row totals. Why might one prefer the table to be expressed this way?
c. Express the table as percentages of the column totals. Why might one prefer the table to be expressed this way?

James York

Numerade Educator

Problem 4

The "In the eye of the beholder" graphic shows two circle graphs, each with four sections. This same information could be represented in the form of a $2 \times 4$ contingency table of two qualitative variables.
a. Identify the population and name the two variables.
b. Construct the contingency table using entries of percentages based on row totals.

James York

Numerade Educator

Problem 5

The perfect age" graphic shows the results from a $9 \times 2$ contingency table for one qualitative and one quantitative variable.a. Identify the population and name the qualitative and quantitative variables.
b. Construct a bar graph showing the two distributions side by side.
c. Does there seem to be a big difference between the genders on this subject?

James York

Numerade Educator

Problem 6

The National Highway System Designation Act of 1995 allows states to set their own highway speed limits. Most of the states have raised the limits. The November 2008 maximum speed limits on interstate highways (rural) for cars and trucks by each state are given in the following table (in miles per hour):a. Build a cross-tabulation of the two variables vehicle type and maximum speed limit on interstate highways. Express the results in frequencies, showing marginal totals.
b. Express the contingency table you derived in part a in percentages based on the grand total.
c. Draw a bar graph showing the results from part b.
d. Express the contingency table you derived in part a in percentages based on the marginal total for speed limit.
e. Draw a bar graph showing the results from part d.
If you are using a computer or a calculator, try the cross-tabulation table commands on page 124.

Lucas Finney

Numerade Educator

Problem 7

A statewide survey was conducted to investigate the relationship between viewers' preferences for ABC, CBS, NBC, PBS, or FOX for news information and their political party affiliation. The results are shown in tabular form:How many viewers were surveyed?b. Why are these bivariate data? Name the two variables. What type of variable is each one?
c. How many viewers preferred to watch CBS?
d. What percentage of the survey was Republican?
e. What percentage of the Democrats preferred ABC?
f. What percentage of the viewers were Republican and preferred PBS?

James York

Numerade Educator

Problem 8

Consider the accompanying contingency table, which presents the results of an advertising survey about the use of credit by Martan Oil Company customers.
a. How many customers were surveyed?
b. Why are these bivariate data? What type of variable is each one?
c. How many customers preferred to use an oilcompany card?
d. How many customers made 20 or more purchases last year?
e. How many customers preferred to use an oilcompany card and made between five and nine purchases last year?
f. What does the 80 in the fourth cell in the second row mean?

James York

Numerade Educator

Problem 9

The June 2009 unemployment rates for eastern and western U.S. states were as follows:$$\begin{array}{lcccccc}\hline \text { Eastern } & 8.0 & 10.6 & 10.1 & 7.3 & 9.2 & 11.0 & 12.1 & 7.2 \\\text { Western } & 8.7 & 11.6 & 8.4 & 6.4 & 12.0 & 12.2 & 5.7 & 9.3 \\\hline\end{array}$$.Display these rates as two dotplots using the same scale; compare means and medians.

Jason Gerber

Numerade Educator

Problem 10

What effect does the minimum amount have on the interest rate being offered on 3 -month certificates of deposit (CDs)? The following are advertised rates of return, $y,$ for a minimum deposit of $\$ 500, \$ 1000$ $\$ 2500, \$ 5000,$ or $\$ 10,000, x .$ (Note that $x$ is in $\$ 100$ and $y$ is annual percentage rate of return.)$$\begin{array}{cc|cc|cc}
\text { Min Deposit } & \text { Rate } & \text { Min Deposit } & \text { Rate } & \text { Min Deposit } & \text { Rate } \\\hline 100 & 0.95 & 25 & 1.00 & 25 & 0.75 \\100 & 1.24 & 50 & 1.00 & 10 & 0.75 \\10 & 1.24 & 100 & 1.00 & 100 & 0.70 \\10 & 1.15 & 5 & 1.00 & 5 & 0.64 \\100 & 1.10 & 10 & 1.00 & 10 & 0.50 \\50 & 1.09 & 10 & 0.80 & 100 & 0.35 \\100 & 1.07 & 10 & 0.75 & 25 & 0.35 \\
5 & 1.00 & 10 & 0.75 & 5 & 0.99 \\25 & 0.75 & & & & \\\hline\end{array}$$.a. Prepare a dotplot of the five sets of data using a common scale.
b. Prepare a 5 -number summary and a boxplot of the five sets of data. Use the same scale for the boxplots.
c. Describe any differences you see among the three sets of data.

Alison Rodriguez

Alison Rodriguez

Numerade Educator

Problem 11

Can a woman's height be predicted from her mother's height? The heights of some mother-daughter pairs are listed; $x$ is the mother's height and $y$ is the daughter's height.$$\begin{array}{l|llllllllll}\hline x & 63 & 63 & 67 & 65 & 61 & 63 & 61 & 64 & 62 & 63 \\y & 63 & 65 & 65 & 65 & 64 & 64 & 63 & 62 & 63 & 64 \\\hline\end{array}$$,$$\begin{array}{l|lllllllllll}
\hline \boldsymbol{x} & 64 & 63 & 64 & 64 & 63 & 67 & 61 & 65 & 64 & 65 & 66 \\
\boldsymbol{y} & 64 & 64 & 65 & 65 & 62 & 66 & 62 & 63 & 66 & 66 & 65 \\\hline\end{array}$$.
a. Draw two dotplots using the same scale and showing the two sets of data side by side.
b. What can you conclude from seeing the two sets of heights as separate sets in part a? Explain.
c. Draw a scatter diagram of these data as ordered pairs.
d. What can you conclude from seeing the data presented as ordered pairs? Explain.

James Kiss

Numerade Educator

Problem 12

The following tables list the ages, heights (in inches), and weights (in pounds) of the players on the 2009 roster for the National Hockey League teams Boston Bruins and Edmonton Oilers.a. Compare each of the three variables-height, weight, and age - using either a dotplot or a histogram (use the same scale).
b. $\quad$ Based on what you see in the graphs in part a, can you detect a substantial difference between the two teams in regard to these three variables? Explain.
c. Explain why the data, as used in part a, are not bivariate data.

Nick Johnson

Numerade Educator

Problem 13

Consider the two variables of a person's height and weight. Which variable, height or weight, would you use as the input variable when studying their relationship? Explain why.

James York

Numerade Educator

Problem 14

Draw a coordinate axis and plot the points (0,6) $(3,5),(3,2),$ and (5,0) to form a scatter diagram. Describe the pattern that the data show in this display.

James York

Numerade Educator

Problem 15

Does studying for an exam pay off?
a. Draw a scatter diagram of the number of hours studied, $x,$ compared with the exam grade received, $y$.$$\begin{array}{l|rrrrr}\hline \boldsymbol{x} & 2 & 5 & 1 & 4 & 2 \\
\boldsymbol{y} & 80 & 80 & 70 & 90 & 60 \\\hline\end{array}$$.b. Explain what you can conclude based on the pattern of data shown on the scatter diagram drawn in part a. (Retain these solutions to use in Exercise $3.55,$ p. 157 )

Nick Johnson

Numerade Educator

Problem 16

Americans Love Their Automobiles" (Applied Example 3.4 on p. 128) to answer the following questions:
a. Name the two variables used.
b. Does the scatter diagram suggest a relationship between the two variables? Explain.
c. What conclusion, if any, can you draw from the appearance of the scatter diagram?

James York

Numerade Educator

Problem 17

Growth charts are commonly used by a child's pediatrician to monitor a child's growth. Consider the growth chart that follows.a. What are the two variables shown in the graph?
b. What information does the ordered pair (3,87) represent?an
two
c. Describe how the pediatrician might use this chart and
when what types of conclusions might be based on the information displayed by it.

James York

Numerade Educator

Problem 18

Draw a scatter diagram showing height, $x,$ and weight, $y,$ for the Boston Bruins hockey team, using the data in Exercise 3.12 7d se b. Draw a scatter diagram showing height, $x,$ and
$\Rightarrow ? \quad$ weight, $y,$ for the Edmonton Oilers hockey team using the data in Exercise 3.12 5), c. Explain why the data, as used in parts a and b, are n. bivariate data.

Jameson Kuper

Numerade Educator

Problem 19

The accompanying data show the number of hours, $x,$ studied for an exam and the grade
50 received, $y(y \text { is measured in tens; that is, } y=8$ means that the grade, rounded to the nearest 10 points, is 80 ). Draw the scatter diagram. (Retain this solution to use in Exercise $3.37,$ p. $143 .$ )

James York

Numerade Educator

Problem 20

An experimental psychologist asserts that the older a child is, the fewer irrelevant answers he or she will give during a controlled experiment. To investigate this claim, the following data were collected. Draw a scatter diagram. (Retain this solution to use in Exercise $3.38, p .143 .)$.
$$\begin{array}{l|rrrrrrrrrr}\hline \text { Age, } x & 2 & 4 & 5 & 6 & 6 & 7 & 9 & 9 & 10 & 12 \\
\text { Irr Answers, } y & 12 & 13 & 9 & 7 & 12 & 8 & 6 & 9 & 7 & 5 \\\hline\end{array}$$.

James York

Numerade Educator

Problem 21

A sample of 15 upper-class students whe commute to classes was selected at registration. They were asked to estimate the distance $(x)$ and the time $(y)=$ required to commute each day to class (see the following table).$$\begin{array}{cc|cc}\begin{array}{c}\text { Distance, } x \\\text { (nearest mile) }\end{array} & \begin{array}{c}\text { Time, } y \\\text { (nearest }5 \text { minutes })
\end{array} & \begin{array}{c}\text { Distance, } x \\\text { (nearest mile) }\end{array} & \begin{array}{c}\text { Time, } y \\\text { (nearest } 5 \text { minutes }
\end{array} \\\hline 18 & 20 & 2 & 5 \\8 & 15 & 15 & 25 \\20 & 25 & 16 & 30 \\5 & 20 & 9 & 20 \\5 & 15 & 21 & 30 \\11 & 25 & 5 & 10 \\9 & 20 & 15 & 20 \\10 & 25 & & \\\hline
\end{array}$$.a. Do you expect to find a linear relationship between the two variables commute distance and commute time? If so, explain what relationship you expect.
b. Construct a scatter diagram depicting these data.
c. Does the scatter diagram in part b reinforce what you expected in part a?

James York

Numerade Educator

Problem 22

Refer to the 2009 4-wheel-drive, 6-cylinder SUVs chart in Applied Example 3.4 on page 128 and the two variables gas tank capacity, $x,$ and the cost to fill it, $y$
a. If you were to draw scatter diagrams of these two variables, on the same graph but separate, for the SUVs that use regular and premium gasoline, do you think the two sets of data would be distinguishable? Explain what you anticipate seeing.
b. Construct a scatter diagram of tank capacity, $x,$ and fill-up cost, $y,$ for the SUVs using regular gasoline.
c. Construct a scatter diagram of tank capacity, $x,$ and fill-up cost, $y,$ for the SUVs using premium gasoline on the scatter diagram for part b.
d. Are the two sets of data distinguishable?
e. How does your answer in part a compare to your answer in part d? Explain any difference.

James York

Numerade Educator

Problem 23

Baseball stadiums vary in age, style and size, and many other ways. Fans might think of the size of a stadium in terms of the number of seats, while players might measure the size of a stadium in terms of the distance from home plate to the centerfield fence.$$\begin{array}{ll|cc|cc}\text { Seats } & \text { CF } & \text { Seats } & \text { CF } & \text { Seats } & \text { CF } \\\hline 38,805 & 420 & 36,331 & 434 & 40,950 & 435 \\41,118 & 400 & 43,405 & 405 & 38,496 & 400 \\56,000 & 400 & 48,911 & 400 & 41,900 & 400 \\45,030 & 400 & 50,449 & 415 & 42,271&404 \\34,077 & 400 & 50,091 & 400 & 43,647 & 401 \\40,793 & 400 & 43,772 & 404 & 42,600 & 396 \\56,144 & 408 & 49,033 & 407 & 46,200 & 400 \\50,516 & 400 & 47,447 & 405 & 41,222 & 403 \\40,615 & 400 & 40,120 & 422 & 52,355 & 408 \\48,190 & 406 & 41,503 & 404 & 45,000 & 408 \\\hline\end{array}$$.Is there a relationship between these two measurements of the "size" of the 30 Major League Baseball stadiums?
a. What do you think you will find? Bigger fields have more seats? Smaller fields have more seats? No relationship between field size and number of seats?
A strong relationship between field size and number of seats? Explain.
b. Construct a scatter diagram.
c. Describe what the scatter diagram tells you, including a reaction to your answer in part a.

Wendi Obritz

Numerade Educator

Problem 24

Most adult Americans drive. But do you have any idea how many licensed drivers there are in each U.S. state? The table here lists the number of male and female drivers licensed in each of 15 randomly selected U.S. states during 2007 $$\begin{array}{lcccc}
\hline \text { Male } & \text { Femole } & \text { Male } & \text { Female } \\
\hline 17.92 & 17.10 & 59.07 & 54.62 \\5.18 & 5.10 & 2.38 & 2.33 \\21.24 & 21.85 & 15.01 & 16.26 \\10.03 & 10.15 & 75.98 & 75.86 \\14.52 & 14.82 & 8.32 & 8.20 \\15.91 & 15.59 & 25.26 & 23.53 \\3.74 & 3.62 & 2.05 & 1.93 \\6.77 & 6.89 & & \\\hline\end{array}$$.a. Do you expect to find a linear (straight-line) relationship between number of male and number of female licensed drivers per state? How strong do you anticipate this relationship to be? Describe.
b. Construct a scatter diagram using $x$ for the number of male drivers and $y$ for the number of female drivers.
c. Compare the scatter diagram to your expectations in part a. How did you do? Explain.
d. Are there data points that look like they are separate from the pattern created by the rest of ordered pairs? If they were removed from the dataset, would the results change? What caused these point(s) to be separate from the others, but yet still be part of the extended pattern? Explain.
e. Use the dataset for all 51 states to construct a scatter diagram. Compare the pattern of the sample of
15 to the pattern shown by all $51 .$ Describe in detail.
f. Did the sample provide enough information for you to understand the relationship between the two variables in this situation? Explain.

Vaidik Stats

Numerade Educator

Problem 25

Ronald Fisher, an English statistician $(1890-1962),$ collected measurements for a sample of 150 irises. Of concern were five variables: species, petal width (PW), petal length (PL), sepal width (SW), and sepal length (SL) (all in $\mathrm{mm}$ ). Sepals are the outermost leaves that encase the flower before it opens. The goal of Fisher's experiment was to produce a simple function that could be used to classify flowers correctly. A random sample of his complete data set is given in the accompanying table.a. Construct a scatter diagram of petal length, $x,$ and petal width, $y .$ Use different symbols to represent the three species..
b. Construct a scatter diagram of sepal length, $x,$ and sepal width, $y .$ Use different symbols to represent the three species.
c. Explain what the scatter diagrams in parts a and b portray.

Donald Albin

Numerade Educator

Problem 26

Total solar eclipses actually take place nearly as often as total lunar eclipses, but the former are visible over a much narrower path. Both the path width and the duration vary substantially from one eclipse to the next. The table below shows the duration (in seconds) and path width (in miles) of 44 total solar eclipses measured in the past and those projected to the year 2010 :a. Draw a scatter diagram showing duration, $y,$ and path width, $x,$ for the total solar eclipses.
b. How would you describe this diagram?
c. The durations and path widths for the years $2006-2009$ were projections. The recorded values were:Compare the recorded values to the projections. Comment on accuracy.
$$\begin{array}{lll}
\text { Year } & \text { Path Width } & \text { Duration } \\
\hline 2006 & 65 \text { miles } & 247 \text { sec } \\
2008 & 147 \text { miles } & 147 \text { sec } \\
2009 & 160 \text { miles } & 399 \text { sec }
\end{array}$$.Compare the recorded values to the projections. Comment on accuracy.

Yuou Sun

Numerade Educator