Learning objectives
Linear regression for two variables is based on a linear equation with one independent variable. The equation has the form
where a and b are constant numbers.
The variable x is the independent variable; y is the dependent variable. Typically, you choose a value to substitute for the independent variable and then solve for the dependent variable.
The following examples are linear equations.
The graph of a linear equation of the form y = a + bx is a straight line. Any line that is not vertical can be described by this equation.
Graph the equation y = –1 + 2x.
View original figure (external; image not reproduced in this text edition)
Original figure description: Graph of the equation y = -1 + 2x. This is a straight line that crosses the y-axis at -1 and is sloped up and to the right, rising 2 units for every one unit of run.
Aaron’s Word Processing Service does word processing. The rate for services is $32 per hour plus a $31.50 one-time charge. The total cost to a customer depends on the number of hours it takes to complete the job.
Find the equation that expresses the total cost in terms of the number of hours required to complete the job.
Let x = the number of hours it takes to get the job done.
Let y = the total cost to the customer.
The $31.50 is a fixed cost. If it takes x hours to complete the job, then (32)(x) is the cost of the word processing only. The total cost is y = 31.50 + 32x.
Slope and y-interceptof a Linear Equation
For the linear equation y = a + bx, b = slope and a = y-inttercept. From algebra, recall that the slope is a number that describes the steepness of a line; the y-intercept is the y-coordinate of the point (0, a), where the line crosses the y-axis.
Please note that in previous courses you learned was the slope-intercept form of the equation, where m represented the slope and b represented the y-intercept. In this text, the form is used, where a is the y-intercept and b is the slope. The key is remembering the coefficient of x is the slope, and the constant number is the y-intercept.
View original figure (external; image not reproduced in this text edition)
Original figure description: Three possible graphs of the equation y = a + bx. For the first graph, (a), b > 0 and so the line slopes upward to the right. For the second, b = 0 and the graph of the equation is a horizontal line. In the third graph, (c), b < 0 and the line slopes downward to the right.
Svetlana tutors to make extra money for college. For each tutoring session, she charges a one-time fee of $25 plus $15 per hour of tutoring. A linear equation that expresses the total amount of money Svetlana earns for each session she tutors is y = 25 + 15x.
What are the independent and dependent variables? What is the y-intercept, and what is the slope? Interpret them using complete sentences.
The independent variable (x) is the number of hours Svetlana tutors each session. The dependent variable (y) is the amount, in dollars, Svetlana earns for each session.
The y-intercept is 25 (a = 25). At the start of the tutoring session, Svetlana charges a one-time fee of $25 (this is when x = 0). The slope is 15 (b = 15). For each session, Svetlana earns $15 for each hour she tutors.
References
Centers for Disease Control and Prevention. (n.d.). Retrieved from https://www.cdc.gov/
National Center for HIV/AIDS, Viral Hepatitis, STD, and TB Prevention. (n.d.). Centers for Disease Control and Prevention. Retrieved from https://www.cdc.gov/nchhstp/default.htm
Chapter Review
The most basic type of association is a linear association. This type of relationship can be defined algebraically by the equations used (numerically with actual or predicted data values) or graphically from a plotted curve. Lines are classified as straight curves. Algebraically, a linear equation typically takes the form y = mx + b, where m and b are constants, x is the independent variable, and y is the dependent variable. In a statistical context, a linear equation is written in the form y = a + bx, where a and b are the constants. This form is used to help you distinguish the statistical context from the algebraic context. In the equation y = a + bx, the constant b that multiplies the x variable (b is called a coefficient) is called the slope. The slope describes the rate of change between the independent and dependent variables; in other words, the rate of change describes the change that occurs in the dependent variable as the independent variable is changed. In the equation y = a + bx, the constant a is called the y-intercept. Graphically, the y-intercept is the y-coordinate of the point where the graph of the line crosses the y-axis. At this point, x = 0.
The slope of a line is a value that describes the rate of change between the independent and dependent variables. The slope tells us how the dependent variable (y) changes for every one-unit increase in the independent (x) variable, on average. The y-intercept is used to describe the dependent variable when the independent variable equals zero. Graphically, the slope is represented by three line types in elementary statistics.
Formula Review
y = a + bx, where a is the y-intercept and b is the slope. The variable x is the independent variable and y is the dependent variable.
Use the following information to answer the next three exercises. A vacation resort rents scuba equipment to certified divers. The resort charges an up-front fee of $25 and another fee of $12.50 an hour.
What are the dependent and independent variables?
dependent variable: fee amount
independent variable: time
Find the equation that expresses the total fee in terms of the number of hours the equipment is rented.
Graph the equation from eip-683.
View original figure (external; image not reproduced in this text edition)
Original figure description: This is a graph of the equation y = 25 + 12.50x. The x-axis is labeled in intervals of 1 from 0 - 7; the y-axis is labeled in intervals of 25 from 0 - 100. The equation's graph is a line that crosses the y-axis at 25 and is sloped up and to the right, rising 12.50 units for every one unit of run.
Use the following information to answer the next two exercises. A credit card company charges $10 when a payment is late and $5 a day each day the payment remains unpaid.
Find the equation that expresses the total fee in terms of the number of days the payment is late.
Graph the equation from eip-23.
View original figure (external; image not reproduced in this text edition)
Original figure description: This is a graph of the equation y = 10 + 5x. The x-axis is labeled in intervals of 1 from 0 - 7; the y-axis is labeled in intervals of 10 from 0 - 50. The equation's graph is a line that crosses the y-axis at 10 and is sloped up and to the right, rising 5 units for every one unit of run.
Is the equation y = 10 + 5x – 3x2 linear? Why or why not?
Which of the following equations are linear?
a. y = 6x + 8
b. y + 7 = 3x
c. y – x = 8x2
d. 4y = 8
y = 6x + 8, 4y = 8, and y + 7 = 3x are all linear equations.
Does the graph in fs-idm48879424 show a linear equation? Why or why not?
View original figure (external; image not reproduced in this text edition)
Original figure description: This is a graph of an equation. The x-axis is labeled in intervals of 1 from -5 to 5; the y-axis is labeled in intervals of 1 from 0 - 8. The equation's graph is a parabola, a u-shaped curve that has a minimum value at (0, 0).
Use the following information to answer the next exercise. element-806 contains real data for the first two decades of flu reporting.
| Year | Number of Flu Cases Diagnosed | Number of Flu Deaths |
| Pre-1981 | 91 | 29 |
| 1981 | 319 | 121 |
| 1982 | 1,170 | 453 |
| 1983 | 3,076 | 1,482 |
| 1984 | 6,240 | 3,466 |
| 1985 | 11,776 | 6,878 |
| 1986 | 19,032 | 11,987 |
| 1987 | 28,564 | 16,162 |
| 1988 | 35,447 | 20,868 |
| 1989 | 42,674 | 27,591 |
| 1990 | 48,634 | 31,335 |
| 1991 | 59,660 | 36,560 |
| 1992 | 78,530 | 41,055 |
| 1993 | 78,834 | 44,730 |
| 1994 | 71,874 | 49,095 |
| 1995 | 68,505 | 49,456 |
| 1996 | 59,347 | 38,510 |
| 1997 | 47,149 | 20,736 |
| 1998 | 38,393 | 19,005 |
| 1999 | 25,174 | 18,454 |
| 2000 | 25,522 | 17,347 |
| 2001 | 25,643 | 17,402 |
| 2002 | 26,464 | 16,371 |
| Total | 802,118 | 489,093 |
Use the columns Year and Number of Flu Cases Diagnosed. Why is year the independent variable and number of flu cases diagnosed the dependent variable (instead of the reverse)?
The number of flu cases depends on the year. Therefore, year becomes the independent variable and the number of flu cases is the dependent variable.
Use the following information to answer the next two exercises. A specialty cleaning company charges an equipment fee and an hourly labor fee. A linear equation that expresses the total amount of the fee the company charges for each session is y = 50 + 100x.
What are the independent and dependent variables?
What is the y-intercept, and what is the slope? Interpret them using complete sentences.
The y-intercept is 50 (a = 50). At the start of the cleaning, the company charges a one-time fee of $50 (this is when x = 0). The slope is 100 (b = 100). For each session, the company charges $100 for each hour they clean.
Use the following information to answer the next three questions. As a result of erosion, a river shoreline is losing several thousand pounds of soil each year. A linear equation that expresses the total amount of soil lost per year is y = 12,000x.
What are the independent and dependent variables?
How many pounds of soil does the shoreline lose in a year?
12,000 lb of soil
What is the y-intercept? Interpret its meaning.
Use the following information to answer the next two exercises. The price of a single issue of stock can fluctuate throughout the day. A linear equation that represents the price of stock for Shipment Express is y = 15 – 1.5x, where x is the number of hours passed in an eight-hour day of trading.
What are the slope and y-intercept? Interpret their meaning.
The slope is –1.5 (b = –1.5). This means the stock is losing value at a rate of $1.50 per hour. The y-intercept is $15 (a = 15). This means the price of stock before the trading day was $15.
If you owned this stock, would you want a positive or negative slope? Why?
Homework
For each of the following situations, state the independent variable and the dependent variable.
- A study is done to determine whether elderly drivers are involved in more motor vehicle fatalities than other drivers. The number of fatalities per 100,000 drivers is compared with the age of drivers.
- A study is done to determine whether the weekly grocery bill changes based on the number of family members.
- Insurance companies base life insurance premiums partially on the age of the applicant.
- Utility bills vary according to power consumption.
- A study is done to determine whether a higher education reduces the crime rate in a population.
- independent variable: age; dependent variable: fatalities
- independent variable: number of family members; dependent variable: grocery bill
- independent variable: age of applicant; dependent variable: insurance premium
- independent variable: power consumption; dependent variable: utility
- independent variable: higher education (years); dependent variable: crime rates
Piece-rate systems are widely debated incentive payment plans. In a recent study of loan officer effectiveness, the following piece-rate system was examined:
| % of goal reached | < 80 | 80 | 100 | 120 |
| Incentive | n/a | $4,000, with an additional $125 added per percentage point from 81% to 99% | $6,500, with an additional $125 added per percentage point from 101% to 119% | $9,500, with an additional $125 added per percentage point starting at 121% |
If a loan officer makes 95 percent of his or her goal, write the linear function that applies based on the incentive plan table. In context, explain the y-intercept and slope.