Stewart J. College Algebra: Concepts and Contexts

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4 CHAPTER 1

■

Data, Functions, and Models

In Example 2 the median seems to be a better indicator of “central tendency”

than the average. This is because one of the incomes in the data is much greater than

all the other incomes. In general, when the data have some “way out” numbers called

outliers, the median is a better measure of central tendency than the average is. For

this reason, for instance, the National Association of Realtors publishes yearly data

on the median price of a home, and not the average price. Some homes are valued at

tens of millions of dollars (like the house on the beach pictured here). It seems hardly

fair to average the price of such a house with that of a typical $250,000 home!

example

Median Price of a House

The table gives the selling prices of houses sold in 2007 in a neighborhood of

Austin, Texas.

(a) Find the average selling price of a house in this neighborhood.

(b) Find the median selling price of a house in this neighborhood.

IN CONTEXT ➤

Solution

(a) Since we are finding the average income of five college graduates, we find the

sum of all the incomes and divide by 5.

So the average income of the five graduates is $102,000.

(b) To find the median income, we first order the list of incomes:

Since there are five numbers in this list and the number five is odd, the median

is the middle number 59. So the median income of the five graduates is $59,000.

data point above the average, that is, 280.

(d) We can see from parts (a) and (b) that the average and the median can be very

different. The average income of the five graduates is $102,000, but this is not

the typical income. In fact, not one of the graduates had an income close to

$102,000. However, the median income of $59,000 is a more typical income

for these five graduates. So the median income is a better description of the

central tendency of the incomes in this case.

■ NOW TRY EXERCISE 7 ■

53, 56, 59, 62, 280

280 + 56 + 59 + 62 + 53

= 102

Ricardo Miguel/Shutterstock.com 2009

SECTION 1.1

■

Making Sense of Data 5

Selling price (ⴛ $1000)

159 195 167 172 169 216 169 172

Solution

(a) Since we are finding the average of the selling prices of 8 houses, we find the

sum of all the selling prices and divide by 8:

So in 2007 the average selling price was about $177,400.

(b) To find the median selling price, we first order the list of selling prices:

Since there are eight numbers in this list and the number eight is even, the me-

dian is the average of the middle two numbers

So the median selling price was $170,500.

previous example because there are no outliers.

■ NOW TRY EXERCISE 21 ■

169 + 172

= 170.5

159, 167, 169, 169, 172, 172, 195, 216

159 + 195 + 167 + 172 + 169 + 216 + 169 + 172

L 177.4

■ Analyzing Two-Variable Data

Most real-world data involve two varying quantities. For example, we might want to

match age with height, education with income, and so on. If we list the age and height

of each child in a preschool class, we obtain data about the two variable quantities age

and height. In general, data that involve two variables are called two-variable data.

Our goal is to discover any relationships that may exist between the two variables.

example

Age and Height

The ages and heights of children in a preschool class are given in the table.

(a) Does being older necessarily imply being taller?

(b) Is the oldest student the tallest?

Solution

(a) No. The table shows several cases in which older does not necessarily imply

taller. For example, there is a 2-year-old who is 36 in. tall and a 3-year-old

who is only 35 in. tall.

Age (yr) 2 2 2 3 3 3 4 4 4 4 4 5

Height (in.) 32 31 36 38 35 41 47 43 42 38 39 45

6 CHAPTER 1

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Data, Functions, and Models

example

Time and Temperature

The table below gives the temperature in two hour intervals in Chemainus, British

Columbia, on a pleasant June day.

(a) What is the highest temperature recorded?

(b) When was the lowest temperature recorded?

Solution

(a) From the table we see that the highest temperature is 68°F.

(b) From the table we see that the lowest temperature occurred at 2:00 P.M.

■ NOW TRY EXERCISE 27 ■

example

Depth and Pressure

A deep sea diver measures the water pressure at different ocean depths. The results

of her measurements are listed in the table.

(a) What is the pressure at a depth of 50 ft?

(b) At what depth is the pressure 28.2 lb/in

10 ft? From 10 ft to 20 ft? From 20 ft to 30 ft?

(d) What pattern or trend do you see in these data?

Solution

(a) The data indicate that at a depth of 50 ft the pressure is 37.2 lb/in

(b) The pressure is 28.2 lb/in

at a depth of 30 ft.

to 19.2 lb/in

, for a total increase of

lb/in

19.2 - 14.7 = 4.5

Time Hours since

6:00

A.M.

Temperature

(°F)

6:00 A.M. 0 59

8:00 A.M. 2 62

10:00 A.M. 4 68

12:00 P.M. 6 65

2:00 P.M. 8 58

4:00 P.M. 10 60

6:00 P.M. 12 62

Depth

(ft)

Pressure

(lb/in

)

0 14.7

10 19.2

20 23.7

30 28.2

40 32.7

50 37.2

60 41.7

(b) No. We see from the table that the oldest student is 5 years old and is 45 in. tall,

but there is a 4-year-old who is 47 in. tall, so the oldest student is not the tallest.

■ NOW TRY EXERCISE 25 ■

In Example 4(b) we found that a particular 2-year-old child may be taller than a

particular 3-year-old. But in general, the trend is that older children tend to be taller.

The process of finding such trends in data is discussed in the next section.

SECTION 1.1

■

Making Sense of Data 7

From a depth of 10 ft to a depth of 20 ft, the pressure changes from 19.2 lb/in

to 23.7 lb/in

, for a total increase of

lb/in

From a depth of 20 ft to a depth of 30 ft, the pressure changes from 23.7 lb/in

28.2 lb/in

, for a total increase of

lb/in

You can check that the increase in pressure is 4.5 lb/in

for each successive

10-ft increase in depth shown in the data.

(d) From the table we see that the pressure seems to increase steadily as the depth

increases.

■ NOW TRY EXERCISE 29 ■

28.2 - 23.7 = 4.5

23.7 - 19.2 = 4.5

1.1 Exercises

CONCEPTS

Fundamentals

1. (a) What are one-variable data? Give examples.

(b) What is the difference between one-variable data and two-variable data? Give

examples.

2. What is meant by central tendency for one-variable data? Measures of central tendency

for one-variable data include the average and the

_______.

3. To find the average of a list of n numbers, we first

_______ all the numbers and then

divide by

_______.

4. To find the median of a list of numbers, we first rearrange the numbers to put them in

_______. To find the median of a list of n ordered numbers, we do the following.

(a) If n is odd, the median is the

_______ number in the list.

(b) If n is even, the median is the average of the two

_______ numbers in the list.

Think About It

5. If you have a million data points (as may be available on the Internet), what technology

would you need to help you find the average or the median of the data?

6. Find several examples of two-variable data that you may be able to collect from your

classmates, for instance, height and shoe size.

7–10

■ A table of one-variable data is given.

(a) Find the average of the data.

(b) Find the median of the data.

median?

A 69 71 74 73 72 73 69 69

A 132 510 119 132 141 132 121

A 113 21 16 16 19 29 21

SKILLS

8 CHAPTER 1

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Data, Functions, and Models

CONTEXTS

10.

11–14

■ A table of two-variable data is given. What pattern or trend do you see in these data?

11.

12.

13. 14.

15. Basketball Stats The list below shows the number of points scored by Kobe Bryant

of the Los Angeles Lakers in each basketball game in which he played in February 2007.

What is the average number of points per game that Kobe Bryant scored in that month?

16. Lion Prides A wildlife biologist in the south Sahara Desert of Africa records the

number of lions in the different prides in her area. What is the average number of lions

in a pride?

17. Apgar Score Doctors use the Apgar score to assess the health of a newborn baby

immediately after the child is born. The list below shows the Apgar scores of babies

born in Memorial Hospital on March 11, 2007. What are the average and the median

Apgar scores for these babies?

18. Weights of Sextuplets The Hanselman sextuplets were born three months premature

on February 26, 2004, in Akron, Ohio, and they have all survived to this date. The list

below shows the birth weights of each child in pounds. What are the average and the

median birth weights of these sextuplets?

19. Quiz Average The table below shows Jordan’s scores on her first five algebra quizzes.

(a) What is her average quiz score?

(b) Jordan receives a score of 10 on her sixth quiz, so now what is her average quiz score?

20. Quiz Average The table below shows Chad’s scores on his first four geography

quizzes.

(a) What is his average quiz score?

(b) Chad receives a score of 5 on his fifth quiz, so now what is his average quiz score?

Score 7 9 8 9

Score 9 9 6 7 7

2.6250, 1.5625, 2.3750, 2.5000, 2.5000, 2.0625

9, 8, 10, 3, 5, 8, 10

18, 8, 14, 15, 16, 12, 17

46, 30, 6, 11, 36, 33, 31, 29, 23, 41, 17, 21, 30, 21, 33

A 0 1 2 3 4 5

B 100 50 25 12 10 10

A 0 1 2 3 4 5

B 3 10 25 26 25 25

A 0 1 2 3 4 5

B 100 95 89 82 76 71

A 0 1 2 3 4 5

B 20 40 80 250 600 1000

A 91 87 84 82 87 84 93 82

SECTION 1.1

■

Making Sense of Data 9

21. Investment Seminar The organizer of an investment seminar surveys the

participants on their yearly income. The table below shows the yearly income of the

participants.

(a) Find the average and median income of the participants.

(b) How many participants have an income above the average?

are the average income and median income, and how many participants have an

income above the average? Is the average or the median the better measure of

central tendency?

22. Dairy Farming A dairy farmer in Illinois records the weights of all his cows. The

table below shows his data.

(a) Find the average and median weight of the cows on the farm.

(b) How many cows have a weight above the average?

average weight and median weight of the cows on the farm, and how many cows

have a weight above the average? In this case, is the average or the median the

better measure of central tendency?

23. Home Sales A realty agency in Albuquerque, New Mexico, records the prices of

homes sold in one neighborhood.

(a) What are the average and median home prices in this neighborhood? Which is the

better indicator of central tendency?

(b) Is the house that sold for $329,000 above or below the average price? The

median price?

$2,860,000, so now what are the average and median home prices if we include the

latest price? Now what is the better indicator of central tendency?

24. Home Sales A realty agency in Napa Valley, California, records the prices of homes

sold in one neighborhood.

(a) What are the average and median home prices in this neighborhood? Which is the

better indicator of central tendency?

(b) Is the house that sold for $2,319,000 above or below the average price? The

median price?

$300,000, so now what are the average and median home prices if we include the

latest price? Now what is the better indicator of central tendency?

25. Pets per Household The home owners association of a small gated community

conducts a survey to determine trends in the number of pets in their neighborhood. The

Price ($) 299,000 329,000 355,000 316,000 330,000

Weight (lb) 880 970 930 890 980 920 900

Yearly income ($) 56,000 58,000 48,000 45,000 59,000 72,000 63,000

Price ($) 2,278,000 2,231,000 2,319,000 2,279,000 2,365,000 2,279,000 2,319,000

10 CHAPTER 1

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Data, Functions, and Models

number of people in a household and the number of pets they own are recorded. The

table below shows the results of the survey.

(a) What is the average number of pets in a household in this community?

(b) Does the largest family have the most pets?

26. Cars per Household The home owners association of Exercise 25 conducts another

survey to determine trends in the number of cars owned by members of their

community. The number of people in a household and the number of cars they own are

recorded. The table below shows the results of this survey.

(a) What is the average number of cars in a household in this community?

(b) Does the largest family own the most cars?

27. Snowfall The Sierra Nevada mountain range is well known for its large snowfalls,

especially in the area of Lake Tahoe. This area usually gets over 200 inches of snow a

year. The table below gives the snowfall for each month in 2006.

(a) What was the snowfall for the month of February 2006?

(b) What month had the highest snowfall? What month had 49 inches?

(d) Find the median monthly snowfall for 2006.

28. Precipitation The Olympic Peninsula in the state of Washington is home to three

temperate rain forests and receives 80 or more inches of rainfall a year. The table below

gives this area’s monthly precipitation in 2007.

(a) How many inches of rainfall were recorded in May?

(b) What month had the highest rainfall? What month had 2 inches of rain?

(d) Find the median monthly precipitation for 2007.

Month 1 2 3 4 5 6 7 8 9 10 11 12

Snowfall (in.) 49 12 83 42 2 0 0 0 0 0 8 14

Number of people 2 1 5 3 2 3

Number of cars 1 1 3 0 5 3

Number of people 2 1 5 3 2 3

Number of pets 5 11 2 0 15 1

Month 1 2 3 4 5 6 7 8 9 10 11 12

Rainfall (in.) 15.6 10.8 9.8 4.9 3.2 2.1 2.0 1.3 2.7 8.3 14.0 14.6

SECTION 1.1

■

Making Sense of Data 11

29. Falling Watermelons The fifth grade class at Wilson Elementary School performs

an experiment to determine the time it takes for a watermelon to fall to the ground.

They have access to a five-story building for their experiment. The class drops a

watermelon out of a window from each story of the building and records the time it

takes for the watermelon to fall to the ground. The table below shows the results of

their experiment.

(a) How long does it take for a watermelon to fall 5 ft from the first-story window?

(b) How much longer does it take a watermelon to fall from 15 ft than from 5 ft? From

45 ft than from 35 ft?

Story Distance (ft) Time (s)

First 5 0.60

Second 15 0.96

Third 25 1.26

Fourth 35 1.49

Fifth 45 1.68

30. Cooling Car Engine The operating temperature of Sofia’s car is about 180°F. When

Sofia returns home from work at 5:00

P.M., she parks her car in the garage. The table

shows the temperature of the engine at half-hour intervals.

(a) Find the temperature of the engine at 5:30

P.M. and 7:30 P.M.

(b) When does the temperature reach 100°F?

31. Algebra and Alcohol The data in the Prologue (page P2) give the blood alcohol

concentration following the consumption of different doses of alcohol. Consider the

data for consumption of 15 mL.

(a) Find the blood alcohol concentration (mg/mL) after 0.2, 0.5, and 1.5 hours.

(b) When does the blood alcohol concentration drop below 0.1?

Time

(

P.M.)

Hours since

5:00

P.M.

Temperature

(°F)

5:00 0.0 180

5:30 0.5 145

6:00 1.0 100

6:30 1.5 83

7:00 2.0 76

7:30 2.5 72

8:00 3.0 71

12 CHAPTER 1

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Data, Functions, and Models

1.2 Visualizing Relationships in Data

■

Relations: Input and Output

■

Graphing Two-Variable Data in a Coordinate Plane

■

Reading a Graph

IN THIS SECTION… we learn how to represent a set of two-variable data as a “relation.”

We also learn how graphs of two-variable data help us get information about the data.

GET READY… by reviewing how to plot points in a coordinate plane in Algebra

Toolkit D.1. Test your readiness by doing the Algebra Checkpoint at the end of this

section.

When business people, sociologists, or scientists analyze data, they look for a trend

or pattern from which to draw a conclusion about the process they are studying. It is

often hard to look at a list of numbers and see any kind of pattern; this is especially

true when the lists are huge. One of the best ways to reveal a hidden pattern in data

is to draw a graph, which we do in this section.

■ Relations: Input and Output

The following data give the height and weight of five college algebra students. We

can also represent the data as ordered pairs of numbers.

Data Data as ordered pairs

For example, the pair (72, 180) represents the student with height 72 inches and

weight 180 pounds. In general, if we let x stand for height and y stand for weight,

then the ordered pair (x, y) represents the student with height x and weight y. In

mathematics any collection of ordered pairs is called a relation.

Relation

Notice that by writing two numbers as an ordered pair, we are saying that the

two numbers are linked together, or related to each other. We say that the first num-

ber is an input and the second is an output of the relation. So for the ordered pair

(72, 180) the input is 72 and the output is 180. We can visualize this relation by draw-

ing a diagram as follows.

Height

(in.)

Weight

(lb)

72 180

60 204

60 120

63 145

70 184

A relation is any set of ordered pairs.

(height, weight)

(72, 180)

(60, 204)

(60, 120)

(63, 145)

(70, 184)

SECTION 1.2

■

Visualizing Relationships in Data 13

180

204

120

145

184

OutputInput

The domain of a relation is the set of all inputs, and the range is the set of all out-

puts. In this example, the domain is the set of heights and the range is the set of

weights:

Domain

Range

Notice that the domain and range are sets, so each number is listed only once. Even

though the number 60 occurs twice as an input, it is listed only once in the domain.

5120, 145, 180, 184, 2046

560, 63, 70, 726

example

Two-Variable Data as a Relation

The students in an algebra course gathered data on the final exam scores and the num-

ber of hours of sleep students had before the exam. The data are shown in the table.

(a) Express the data as a relation.

(b) Draw a diagram of the relation.

(d) Find the output(s) corresponding to the input 80.

(e) Find the domain and range of the relation.

Solution

(a) The set of ordered pairs that defines this relation is

(b) A diagram of the relation is shown below.

5192, 82, 180, 52, 170, 12, 182, 92, 191, 72, 180, 72, 170, 326

Score Hours of sleep

92 8

80 5

70 1

82 9

91 7

80 7

70 3

Out

utsIn

uts

hours of sleep before the exam.

(d) There are two outputs that correspond to the input 80; they are 5 and 7.

(e) The domain of this relation is the set , and the range is the

set .

■ NOW TRY EXERCISES 11 AND 35 ■

51, 3, 5, 7, 8, 96

570, 80, 82, 91, 926