Stewart J. College Algebra: Concepts and Contexts

Подождите немного. Документ загружается.

134 CHAPTER 1

Collecting and Analyzing Data

OBJECTIVE To experience the process of first collecting data, then getting

information from the data.

In many of the exercises in this book, data are given, and we are asked to analyze the

data. Sometimes we are asked to search for data on the Internet. But the process of

gathering real-life data can be quite difficult or tedious. For example, obtaining sci-

entific data often requires complicated equipment. The U.S. Census Bureau employs

hundreds of agents to survey thousands of citizens to obtain demographic data.

Financial data are collected from hundreds of sources in order to measure the eco-

nomic health of the country.

Sometimes data collection can be an adventure. For example, to determine how

closely the earth resembles a sphere, the French Academy sent an expedition to Peru

in 1735 to obtain data on the length of a degree at the equator. The expedition, headed

by Charles-Marie de La Condamine, included an Atlantic crossing by boat, traveling

by mule and on foot, and working at high altitudes. It took over eight years to obtain

the needed data.

In this exploration we get some experience in data collection by obtaining data

within the classroom and from our classmates.

I. Collecting Data

To collect data from our classmates, we make measurements and take a survey.

1. Let’s make some measurements.

(a) For each student in the class, make the following measurements: height,

hand span, hat size, shoe size. Make these measurements using the same

unit (either inches or centimeters).

Hand span Hat size Shoe size

EXPLORATIONS

■

EXPLORATIONS

■

EXPLORATIONS

■

EXPLORATIONS

A drawing from the 1735 La

Condamine expedition to Peru

French School, (18th century)/Bibliotheque de I’Institut de France,

Pris, France/Lauros/Giraudon/The Bridgeman Art Library

EXPLORATIONS 135

Keep a record of these data. You’ll

need them for several exercises in

Section 2.5.

Name Gender Height Hand span Hat size Shoe size

Adam

Betty

Cathy

David

2. Let’s make a survey.

(a) Make a survey that asks the following questions of your classmates. Note

that the survey is anonymous; students do not need to put their names on

the survey sheet.

EXPLORATIONS

■

EXPLORATIONS

■

EXPLORATIONS

■

EXPLORATIONS

Measurement Data

Survey

1. Gender: Male ❑ Female ❑

2. What is the value (in cents) of the coins in your pocket or purse? _______

3. How far is your daily commute to school (in miles)? _______

4. How many siblings do you have (including yourself)? _______

5. How many hours a week do you spend on the Internet? _______

6. How many hours a week do you spend on homework? _______

7. Rate your happiness.

① not happy ② happy ➂ very happy

8. Rate your satisfaction with your school work.

① not satisfied ② satisfied ➂ very satisfied

(b) Enter the information you obtained in a table as shown. Note that the

information for each individual is entered in a row of the table.

136 CHAPTER 1

(b) After the survey sheets have been collected, give each sheet an ID

number—for example, 1, 2, 3, . . . . Now enter the information you

obtained in a table like the following. For the column headings let’s use

short descriptions (instead of the question number) so that we can quickly

recognize what each column represents.

EXPLORATIONS

■

EXPLORATIONS

■

EXPLORATIONS

■

EXPLORATIONS

ID Gender Coins Commute Siblings Internet Homework Happy Satisfied

Survey Data

3. Think of other measurements or survey questions that would help us get

useful data from the class.

II. One-Variable Data Analysis

Each column in the data tables we completed in Questions 1(b) and 2(b) above is a

set of one-variable data. The methods we’ve learned for getting information from

such data are to calculate the mean or the median.

1. Consider the “Height” column in the measurement data table.

(a) Make two sets of data: one for the heights of male students and one for the

heights of female students.

(b) Find the mean of each set of data. Compare the mean heights of males and

females in your class. What conclusions can you make?

and females in your class. What conclusions can you make?

2. Consider the “Hand span” column in the measurement data table.

(a) Make two sets of data: one for the hand spans of male students and one for

the hand spans of the female students in the class.

(b) Find the mean of each set of data. Compare the mean hand spans of males

and females in your class. What conclusions can you make?

males and females in your class. What conclusions can you make?

3. Make an analysis similar to the one in Question 1 for one other set of one-

variable data from the measurement data table.

4. Consider the “Coins” column from the survey data table.

(a) Find the mean of the data.

(b) Do you suspect that there is a difference in the average value of the coins

carried by males and females? If you think there might be a difference, then

calculate the mean value of coins for men and women separately to find out.

EXPLORATIONS 137

EXPLORATIONS

■

EXPLORATIONS

■

EXPLORATIONS

■

EXPLORATIONS

5. Consider the “Happy” column in the survey data table.

(a) How do these data differ from the measurement data or the data in the first

five survey questions?

(b) Find the mean of the data. What does a mean of 1.5 represent? What does

the mean of your class tell you about the state of happiness of your class?

is the mode. The mode is the number that appears most often in the data.

So if there are more 2’s than 0’s or 1’s in the “Happy” column, then the

mode is 2. Find the mode for your class. Do you think the mode is a better

measure of central tendency than the mean in this case?

6. Make an analysis of one other set of one-variable data from the survey data

table.

III. Two-Variable Data

Each pair of columns in the tables we completed in Questions 1(b) and 2(b) above is

a set of two-variable data. For example, in the measurement data table let’s choose

the “Height” and “Hand span” columns. Then for each individual we form the or-

dered pair , where a is the height of the person and b is the hand span. In this

chapter we learned to get information from two-variable data by graphing.

1. Consider the “Height” and “Hand span” columns in the measurement data

table.

(a) Make a set of two-variable data for the height and hand span of male

students.

(b) Make a scatter plot of the data in part (a), with the horizontal axis repre-

senting height and the vertical axis representing hand span.

larger (or smaller) hand spans?

(d) Repeat parts (a)–(c) for the “Height” and “Hand span” data of female

students.

2. Make a graphical analysis of the “Height” and “Shoe size” data by following

the steps in Question 1.

3. Consider the “Internet” and “Happy” columns in the survey data table.

(a) Make a scatter plot of the data, with the horizontal axis representing

“Internet” and the vertical axis representing “Happy.”

(b) Do you see any trends in the data? Do heavy Internet users tend to be

happier?

4. Consider the survey data table.

(a) Do you think that there may be a relationship between the “Commute” and

“Coins” columns? How about the “Homework” and “Satisfied” columns?

Or the “Commute” and “Homework” columns? Make a research hypothesis

stating whether or not you think there is a relationship.

(b) Test any research hypotheses you made in part (a) by making a scatter plot

and observing any resulting trends.

1a, b 2

138 CHAPTER 1

EXPLORATIONS

■

EXPLORATIONS

■

EXPLORATIONS

■

EXPLORATIONS

Every Graph Tells a Story

OBJECTIVE To learn how to give a verbal summary of a situation described by a

graph.

If a picture is worth a thousand words, then a graph is worth at least a few sentences

of prose. In fact, a graph can sometimes tell a story much more quickly and effec-

tively than many words. The devastating impact of the stock market crash of 1929

is immediately evident from a graph of the Dow Jones index (DJIA). Such graphs

were printed in the newspapers of the time as an effective way of conveying the mag-

nitude of the crash. No words are needed to convey the message in the cartoon be-

low. The graph tells a simple story: Something is way down—perhaps sales, profits,

or productivity—and the responsible person is very very worried.

In this exploration we read the stories that graphs tell, and we make graphs that tell

stories.

I. Reading a Story from a Graph

1. Four graphs of temperature versus time (starting at 7:00 A.M.) are shown

below, followed by three stories.

(a) Match each of the stories with one of the graphs.

(b) Write a similar story for the graph that didn’t match any story.

100

200

300

400

DJIA

Date

1/6/1929

1/6/1930

1/6/1931

1/6/1932

1/6/1933

Graph A

Graph C

Graph B

Graph D

A.M. 5 P.M.Noon 7 A.M. 5 P.M.Noon 7 A.M. 5 P.M.Noon 7 A.M. 5 P.M.Noon

Aardvark Marketing

EXPLORATIONS 139

3. Make up a story involving any situation that would correspond to the graph

shown below.

I took a roast out of the freezer at noon and left it on the counter to thaw. Then

I cooked it in the oven when I got home.

I took a roast out of the freezer in the morning and left it on the counter to

thaw. Then I cooked it in the oven when I got home.

I took a roast out of the freezer in the morning and left it on the counter to

thaw. I forgot about it and went out for Chinese food on my way home from

work. I put the roast in the refrigerator when I finally got home.

2. Three runners compete in a 100-meter hurdle race. The graph shows the

distance run as a function of time for each runner. Describe what the graph

tells you about this race. Who won the race? Did each runner finish the race?

What do you think happened to Runner B?

t (sec)

(m)

100

AB C

EXPLORATIONS

■

EXPLORATIONS

■

EXPLORATIONS

■

EXPLORATIONS

Story 1

Story 2

Story 3

141

Global warming? Is the world getting hotter, or are we just having a

temporary warm spell? To answer this question, scientists collect huge

amounts of data on global temperature. A graph of the data can help to reveal

long term changes in temperature, but more precise algebra methods must be

used to detect any trend that is different from normal temperature fluctuations.

Significant global warming could have drastic consequences for the survival

of many species. For example, melting polar ice eliminates the ice paths polar

bears need to reach their feeding grounds, resulting in the bears’ starving or

possibly drowning. (See Section 2.5, Exercise 15, page 197.) Of course, any

changes in the global ecology also have implications for our own well-being.

2.1 Working with Functions:

Average Rate of Change

2.2 Linear Functions:

Constant Rate of Change

2.3 Equations of Lines:

Making Linear Models

2.4 Varying the Coefficients:

Direct Proportionality

2.5 Linear Regression:

Fitting Lines to Data

2.6 Linear Equations: Getting

Information from a Model

2.7 Linear Equations:

Where Lines Meet

EXPLORATIONS

When Rates of Change

Change

2 Linear Patterns

3 Bridge Science

4 Correlation and Causation

5 Fair Division of Assets

Linear Functions

and Models

Keith Levit/Shutterstock.com 2009

142 CHAPTER 2

■

Linear Functions and Models

2.1 Working with Functions: Average Rate of Change

■

Average Rate of Change of a Function

■

Average Speed of a Moving Object

■

Functions Defined by Algebraic Expressions

IN THIS SECTION… we consider the net change of a function on an interval. This leads

to the concept of average rate of change of a function.

GET READY… by reviewing the concept of net change from Section 1.5.

In Section 1.7 we used the graph of a function to determine where the function was

increasing and where it was decreasing. In this section we take a closer look at these

properties; namely, we ask, “How fast is the function increasing?” or “At what rate

is the function decreasing?” To answer these questions, we give a precise meaning

to the term how fast or at what rate.

■ Average Rate of Change of a Function

The rate at which a function increases or decreases can vary dramatically, as in the

case of the speed of a runner in a marathon or the rate at which books are added to a

library. However, a good approximation is the average rate of change of a function

over an interval.

A university library receives new books each month. The head librarian needs

to estimate how many books “on average” the library receives monthly. The data

listed in the margin give the number of books the library has on the last day of each

month in 2007.

Let’s find the average monthly increase in the number of books for the summer

months from the end of May to the end of August. First we find the net change (see

Section 1.5) in the number of books during these months:

net change in number of books ⫽ 127,754 ⫺ 127,187 ⫽ 567

To find the number of months over which this increase in books took place, we

subtract:

number of months ⫽ 8 ⫺ 5 ⫽ 3

So the average rate of change per month in the number of books is

/month

So from May to August the library added an average of 189 books per month.

567 books

3 months

= 189 books

average rate of change =

net change in number of books

change in time

Month

number

Total

number

of books

Jan. 1 126,554

Feb. 2 126,672

Mar. 3 126,823

Apr. 4 127,003

May 5 127,187

June 6 127,523

July 7 127,634

Aug. 8 127,754

Sept. 9 127,896

Oct. 10 128,214

Nov. 11 128,415

Dec. 12 128,729

SECTION 2.1

■

Working with Functions: Average Rate of Change 143

figure 1 Average rate of change

of a function f

The graph in Figure 1 shows that is the net change in the value of

f and is the change in the value of x. b - a

f 1b 2- f 1a2

example

Average Rate of Installation

Sima is installing new Italian ceramic flooring in her house. The table in the margin

gives the total number of tiles she has installed after working for x hours.

(a) Find the average rate of installation in the first hour.

(b) Find the average rate of installation for the first 4 hours.

(d) Draw a graph of f and use the graph to find the hour in which Sima had the

fastest average rate of installation.

Solution

(a) We find the average rate of change of the function f between and :

/hour

So the average rate of installation for the first hour is 21 tiles per hour.

(b) We find the average rate of change of the function f between and :

/hour

So the average rate of installation for the first 4 hours is about 47 tiles per hour.

/hour

So the average rate of installation for this time period is about 79 tiles per hour.

(d) A scatter plot is shown in Figure 2. We can see from the graph that the

steepest rise in the graph is during the seventh hour (from hour 6 to hour 7), so

the fastest average rate of installation occurred in the seventh hour.

■ NOW TRY EXERCISE 21 ■

average rate of change =

f 18 2- f 162

8 - 6

403 - 245

= 79 tiles

x = 8x = 6

average rate of change =

f 142- f 10 2

4 - 0

189 - 0

= 47.25 tiles

x = 4x = 0

average rate of change =

f 11 2- f 102

1 - 0

21 tiles - 0 tiles

1 hour

= 21 tiles

x = 1x = 0

f 1x 2

Hours

f(x)

Tiles

0 0

1 21

2 69

3 126

4 189

5 216

6 245

7 347

8 403

The average rate of change of the function between and

average rate of change =

net change in y

change in x

f 1b 2- f 1a2

b - a

x = b

x = ay = f 1x2

f(b)

f(a)

f(b)-f(a)

Net change

in y

b-a

Change in x

200

300

400

100

24681357

figure 2

In general, we find the average rate of change of a function by calculating the

net change in the function values and dividing by the net change in the x-values.

Average Rate of Change of a Function