KARL PEARSON: THE FOUNDING DEVELOPER OF CHI-SQUARE TESTS
14.1 Linear Equations with One Independent Variable
To understand linear regression, let’s first review linear equations with one independent variable. The general form of alinear equationwith one independent variable can be written as
y=b0+b1x,
whereb0andb1are constants (fixed numbers),xis the independent variable, andyis the dependent variable.†
The graph of a linear equation with one independent variable is astraight line,or simply aline;furthermore, any nonvertical line can be represented by such an equa- tion. Examples of linear equations with one independent variable are y=4+0.2x, y= −1.5−2x, andy= −3.4+1.8x. The graphs of these three linear equations are shown in Fig. 14.1.
FIGURE 14.1 Graphs of three linear equations
y = −3.4 + 1.8x y = 4 + 0.2x
−6
−5
−4
−3
−2
−1 1 2 3 4 5 6
x y
−6−5−4−3−2−1 1 2 3 4 5 6
y = −1.5 − 2x
†You may be familiar with the formy=mx+binstead of the formy=b0+b1x. Statisticians prefer the latter form because it allows a smoother transition to multiple regression, in which there is more than one independent variable. Material on multiple regression is provided in the chaptersMultiple Regression AnalysisandModel Building in Regressionon the WeissStats CD accompanying this book.
630 CHAPTER 14 Descriptive Methods in Regression and Correlation
Linear equations with one independent variable occur frequently in applications of mathematics to many different fields, including the management, life, and social sciences, as well as the physical and mathematical sciences.
EXAMPLE 14.1 Linear Equations
Word-Processing Costs CJ2Business Services offers its clients word processing at a rate of $20 per hour plus a $25 disk charge. The total cost to a customer depends, of course, on the number of hours needed to complete the job. Find the equation that expresses the total cost in terms of the number of hours needed to complete the job.
Solution Because the rate for word processing is $20 per hour, a job that takes x hours will cost $20x plus the $25 disk charge. Hence the total cost, y, of a job that takesxhours isy=25+20x.
The equation y=25+20x is linear; hereb0=25 and b1=20. This equation gives us the exact cost for a job if we know the number of hours required. For instance, a job that takes 5 hours will cost y=25+20ã5=$125; a job that takes 7.5 hours will costy=25+20ã7.5=$175. Table 14.1 displays these costs and a few others.
As we have mentioned, the graph of a linear equation, such as y=25+20x, is a line. To obtain the graph of y=25+20x, we first plot the points displayed in Table 14.1 and then connect them with a line, as shown in Fig. 14.2.
TABLE 14.1 Times and costs for five word-processing jobs Time (hr) Cost ($)
x y
5.0 125
7.5 175
15.0 325
20.0 425
22.5 475
FIGURE 14.2 Graph ofy= 25 + 20x, obtained from the points displayed in Table 14.1
Cost ($)
Time (hr)
5 10 15 20 25 x
y
100 200 300 400 500
0
y = 25 + 20x
The graph in Fig. 14.2 is useful for quickly estimating cost. For example, a glance at the graph shows that a 10-hour job will cost somewhere between $200 and $300.
The exact cost isy=25+20ã10=$225.
Exercise 14.5 on page 633
Intercept and Slope
For a linear equationy=b0+b1x, the numberb0is they-value of the point of inter- section of the line and they-axis. The numberb1measures the steepness of the line;
more precisely,b1indicates how much they-value changes when thex-value increases by 1 unit. Figure 14.3 illustrates these relationships.
FIGURE 14.3 Graph ofy=b0+b1x
y = b0+ b1x (0, b0) b0
1 unit increase
b1 units up
x y
The numbers b0 and b1 have special names that reflect these geometric inter- pretations.
DEFINITION 14.1 y-Intercept and Slope
For a linear equationy=b0+b1x, the numberb0is called they-intercept and the numberb1is called theslope.
? What Does It Mean?
They-intercept of a line is where it intersects they-axis.
The slope of a line measures its steepness.
In the next example, we apply the concepts of y-intercept and slope to the illus- tration of word-processing costs.
EXAMPLE 14.2 y -Intercept and Slope
Word-Processing Costs In Example 14.1, we found the linear equation that ex- presses the total cost,y, of a word-processing job in terms of the number of hours,x, required to complete the job. The equation isy=25+20x.
a. Determine they-intercept and slope of that linear equation.
b. Interpret they-intercept and slope in terms of the graph of the equation.
c. Interpret they-intercept and slope in terms of word-processing costs.
Solution
a. They-intercept for the equation isb0=25, and the slope isb1=20.
b. The y-interceptb0=25 is they-value where the line intersects the y-axis, as shown in Fig. 14.4. The slopeb1=20 indicates that the y-value increases by 20 units for every increase inx of 1 unit.
FIGURE 14.4 Graph ofy= 25 + 20x
y = 25 + 20x
Cost ($)
Time (hr)
x y
100 200 300 400
5 10 15 20 25
0 b0 = 25
500
632 CHAPTER 14 Descriptive Methods in Regression and Correlation
c. They-interceptb0=25 represents the total cost of a job that takes 0 hours. In other words, they-intercept of $25 is a fixed cost that is charged no matter how long the job takes. The slopeb1=20 represents the cost per hour of $20; it is the amount that the total cost goes up for every additional hour the job takes.
Exercise 14.9 on page 633
A line is determined by any two distinct points that lie on it. Thus, to draw the graph of a linear equation, first substitute two differentx-values into the equation to get two distinct points; then connect those two points with a line.
For example, to graph the linear equation y=5−3x, we can use the x-values 1 and 3 (or any other two x-values). The y-values corresponding to those two x-values are y=5−3ã1=2 and y=5−3ã3= −4, respectively. Therefore the graph ofy=5−3x is the line that passes through the two points(1,2)and(3,−4), as shown in Fig. 14.5.
FIGURE 14.5 Graph ofy= 5 – 3x
y = 5 − 3x
(3, −4) (1, 2)
x y
−4
−3
−2
−1 1 2 3 4 5 6
−5
−6
−4 −3 −2 −1 1 2 3 4 5 6
−5
−6
Note that the line in Fig. 14.5 slopes downward—the y-values decrease as x increases—because the slope of the line is negative: b1= −3<0. Now look at the line in Fig. 14.4 on page 631, the graph of the linear equationy=25+20x. That line slopes upward—the y-values increase as x increases—because the slope of the line is positive:b1=20>0.
KEY FACT 14.1 Graphical Interpretation of Slope
The graph of the linear equationy=b0+b1xslopes upward ifb1>0, slopes downward ifb1<0, and is horizontal ifb1=0, as shown in Fig. 14.6.
FIGURE 14.6 Graphical interpretation of slope
x y
x y
x y
b1 > 0 b1 < 0 b1 = 0
Exercises 14.1
Understanding the Concepts and Skills
14.1 Regarding linear equations with one independent variable, answer the following questions:
a. What is the general form of such an equation?
b. In your expression in part (a), which letters represent constants and which represent variables?
c. In your expression in part (a), which letter represents the inde- pendent variable and which represents the dependent variable?
14.2 Fill in the blank. The graph of a linear equation with one independent variable is a .
14.3 Consider the linear equationy=b0+b1x. a. Identify and give the geometric interpretation ofb0. b. Identify and give the geometric interpretation ofb1.
14.4 Answer true or false to each statement, and explain your answers.
a. The graph of a linear equation slopes upward unless the slope is 0.
b. The value of they-intercept has no effect on the direction that the graph of a linear equation slopes.
14.5 Rental-Car Costs. During one month, theAvis Rent-A- Carrate for renting a Buick LeSabre in Mobile, Alabama, was
$68.22 per day plus 25c/per mile. For a 1-day rental, letx de- note the number of miles driven and letydenote the total cost, in dollars.
a. Find the equation that expressesyin terms ofx.
b. Determineb0andb1.
c. Construct a table similar to Table 14.1 on page 630 for the x-values 50, 100, and 250 miles.
d. Draw the graph of the equation that you determined in part (a) by plotting the points from part (c) and connecting them with a line.
e. Apply the graph from part (d) to estimate visually the cost of driving the car 150 miles. Then calculate that cost exactly by using the equation from part (a).
14.6 Air-Conditioning Repairs. Richard’s Heating and Cool- ingin Prescott, Arizona, charges $55 per hour plus a $30 service charge. Letxdenote the number of hours required for a job, and letydenote the total cost to the customer.
a. Find the equation that expressesyin terms ofx.
b. Determineb0andb1.
c. Construct a table similar to Table 14.1 on page 630 for the x-values 0.5, 1, and 2.25 hours.
d. Draw the graph of the equation that you determined in part (a) by plotting the points from part (c) and connecting them with a line.
e. Apply the graph from part (d) to estimate visually the cost of a job that takes 1.75 hours. Then calculate that cost exactly by using the equation from part (a).
14.7 Measuring Temperature. The two most commonly used scales for measuring temperature are the Fahrenheit and Celsius scales. If you lety denote Fahrenheit temperature andx denote Celsius temperature, you can express the relationship between those two scales with the linear equationy=32+1.8x.
a. Determineb0andb1.
b. Find the Fahrenheit temperatures corresponding to the Celsius temperatures−40◦, 0◦, 20◦, and 100◦.
c. Graph the linear equation y=32+1.8x, using the four points found in part (b).
d. Apply the graph obtained in part (c) to estimate visually the Fahrenheit temperature corresponding to a Celsius tempera- ture of 28◦. Then calculate that temperature exactly by using the linear equationy=32+1.8x.
14.8 A Law of Physics. A ball is thrown straight up in the air with an initial velocity of 64 feet per second (ft/sec). According to the laws of physics, if you letydenote the velocity of the ball afterx seconds,y=64−32x.
a. Determineb0andb1for this linear equation.
b. Determine the velocity of the ball after 1, 2, 3, and 4 sec.
c. Graph the linear equationy=64−32x, using the four points obtained in part (b).
d. Use the graph from part (c) to estimate visually the velocity of the ball after 1.5 sec. Then calculate that velocity exactly by using the linear equationy=64−32x.
In Exercises14.9–14.12,
a. find the y-intercept and slope of the specified linear equation.
b. explain what the y-intercept and slope represent in terms of the graph of the equation.
c. explain what the y-intercept and slope represent in terms relating to the application.
14.9 Rental-Car Costs. y=68.22+0.25x (from Exer- cise 14.5)
14.10 Air-Conditioning Repairs. y=30+55x (from Exer- cise 14.6)
14.11 Measuring Temperature. y=32+1.8x (from Exer- cise 14.7)
14.12 A Law of Physics. y=64−32x(from Exercise 14.8) In Exercises14.13–14.22, we give linear equations. For each equation,
a. find the y-intercept and slope.
b. determine whether the line slopes upward, slopes downward, or is horizontal, without graphing the equation.
c. use two points to graph the equation.
14.13 y=3+4x 14.14 y= −1+2x 14.15 y=6−7x 14.16 y= −8−4x 14.17 y=0.5x−2 14.18 y= −0.75x−5
14.19 y=2 14.20 y= −3x
14.21 y=1.5x 14.22 y= −3
In Exercises14.23–14.30, we identify the y-intercepts and slopes, respectively, of lines. For each line,
a. determine whether it slopes upward, slopes downward, or is horizontal, without graphing the equation.
b. find its equation.
c. use two points to graph the equation.
14.23 5 and 2 14.24 −3 and 4 14.25 −2 and−3 14.26 0.4 and 1 14.27 0 and−0.5 14.28 −1.5 and 0 14.29 3 and 0 14.30 0 and 3
634 CHAPTER 14 Descriptive Methods in Regression and Correlation
Extending the Concepts and Skills
14.31 Hooke’s Law. According toHooke’s lawfor springs, de- veloped by Robert Hooke (1635–1703), the force exerted by a spring that has been compressed to a length x is given by the formulaF= −k(x−x0), wherex0 is the natural length of the spring andkis a constant, called thespring constant. A certain spring exerts a force of 32 lb when compressed to a length of 2 ft and a force of 16 lb when compressed to a length of 3 ft. For this spring, find the following.
a. The linear equation that relates the force exerted to the length compressed
b. The spring constant
c. The natural length of the spring
14.32 Road Grade. Thegradeof a road is defined as the dis- tance it rises (or falls) to the distance it runs horizontally, usually expressed as a percentage. Consider a road with positive grade,g.
Suppose that you begin driving on that road at an altitudea0.
a. Find the linear equation that expresses the altitude,a, when you have driven a distance,d, along the road. (Hint:Draw a graph and apply the Pythagorean Theorem.)
b. Identify and interpret the y-intercept and slope of the linear equation in part (a).
c. Apply your results in parts (a) and (b) to a road with a 5% grade and an initial altitude of 1 mile. Express your an- swer for the slope to four decimal places.
d. For the road in part (c), what altitude will you reach after driv- ing 10 miles along the road?
e. For the road in part (c), how far along the road must you drive to reach an altitude of 3 miles?
14.33 In this section, we stated that any nonvertical line can be described by an equation of the formy=b0+b1x.
a. Explain in detail why a vertical line can’t be expressed in this form.
b. What is the form of the equation of a vertical line?
c. Does a vertical line have a slope? Explain your answer.