2.1 Using Lines to Model Data (Page 5 of 20)
2.1 Using Lines to Model Data
Scattergram and Linear Models
The graph of plotted data pairs is called a scattergram. A linear model is a straight line or an equation that describes the relationship between two quantities for a true-to-life situation.
Year / Yearssince 1960
t / Number of Visitors
(millions)
v
1960
1970
1980
1990
2000 / 1.2
2.3
2.6
3.8
4.8
Example 1
Let v represent the number of visitors (in millions) to the Grand Canyon in the year that is t years since 1960.
a. Fill-in the table of values for t. Identify the dependent and independent variables.
b. Make a scattergram of the data.
c. Use a ruler to draw (“eyeball”) a line (linear model) that fits the data well. As always, label and scale both axes.
d. Use the linear model to estimate the number of visitors in 2010 (extrapolation).
e. Use the linear model to estimate in what year there will be 4 million visitors to the Grand Canyon (interpolation).
Interpolation, Extrapolation & Model Breakdown
Interpolation is making prediction within the data given. Extrapolation is making a prediction outside the data given. When a model yields a prediction that does not make sense or an estimate that is not a good approximation, we say that model breakdown has occurred. Model breakdown mostly occurs when trying to make an estimate outside the range given in the data (called extrapolation).
Example 2
Prozac Prescriptions
(millions)
1989
1991
1993
1995
1996 / 6.1
10.0
12.2
18.8
20.7
Prozac is an antidepressant that was approved by the FDA in 1987. Let p be the number of prescriptions of Prozac (in millions) dispensed at t years since 1980.
a. Fill-in the table of values for t.
b. Identify the dependent and independent variables.
c. Make a scattergram of the data.
d. Use a ruler to draw (“eyeball”) a line (linear model) that fits the data well. As always, label and scale both axes.
e. Use the linear model to estimate the number of Prozac prescriptions in 1994.
f. Use the linear model to predict when the number of Prozac prescriptions will reach 30 million.
Year / Salmon Population(millions)
P
19601965
1970
1975
1980
1985
1990 / 10.02
10.00
7.61
3.15
4.59
3.11
2.22
Example 3
The Pacific salmon populations for various years are listed in the table. Let P represent the salmon population (in millions) at t years since 1950.
a. Identify the dependent and independent variables.
b. Make a scattergram of the data.
c. Use a ruler to draw (“eyeball”) a line (linear model) that fits the data well. As always, label and scale both axes.
d. Find the P-intercept of the model and explain its meaning.
e. Find the t-intercept and explain its meaning.
Year
/ Carbon Emissions(billions of tons)
1950
1960
1970
1980
1990 / 1.6
2.6
3.8
4.9
5.9
Example 4 (exercise 2.1 #8)
Over the past 120 years, most scientists agree that Earth’s average temperature has increased by about F. What is under debate is why this happened and whether we should be concerned. Some scientists believe that the warming is due to carbon emissions from burning fossil fuels. Let c represent the carbon emissions at t years since 1950.
a. Identify the dependent and independent variables.
b. Make a scattergram and draw (“eyeball”) a linear model that fits the data well. As always, label and scale the axes.
c. Use your linear model to estimate the carbon emissions in 1998.
d. The actual amount of carbon emissions in 1998 was 6.3 billion tons. Explain the difference between the actual and estimated amounts.
2.2 Finding Regression Equations for Linear Models (Page 7 of 7)
2.2 Finding Regression Equations for Linear Models
In section 2.1 we used scattergrams and an “eyeballed best-fit” line (linear model) to model data and make estimates. In this section we will find the regression equations for linear models.
Linear Regression Function
The linear regression line is the line that mathematically best fits the data. The linear regression equation, or linear regression function, is the equation of the regression line.
Finding the Linear Regression Equation on the TI-83
1. Enter the independent variable data values into list 1 (L1) and the corresponding dependent variable values into list 2 (L2). To access your lists press STAT followed by ENTER.
STAT / 1:Edit
2. Press STAT PLOT followed by ENTER. Then set the Plot1 settings as shown. Press
ZOOM / 9:ZoomStat
to view the scattergram.
3. Run the linear regression program:
STAT / CALC / 4:LinReg (ax+b),
On the TI-83 a is the slope and (0, b) is the “y”-intercept (i.e. the vertical axis intercept). Write a and b to three decimal places.
3. Rewrite the equation using the variables in the application.
Birth Year / Life Expectancy1970
1975
1977
1980
1982
1985
1987
1990
1993
1996
2000 / 70.8
72.6
73.3
73.7
74.5
74.7
74.9
75.4
75.5
76.1
76.9
Example 1
The American life span has been increasing over the last century. Let L represent the life expectancy at birth for an American born t years after 1970. Create a scattergram of the data on your calculator and determine if a linear model is appropriate. If it is, then find the linear regression model for the data.
Number of Years since 1950t / Salmon Population
(millions)
P
1015
20
25
30
35
40 / 10.02
10.00
7.61
3.15
4.59
3.11
2.22
Example 2
The Pacific salmon population for various years are listed in the table. Let P represent the salmon population (in millions) at t years since 1950. Create a scattergram of the data on your calculator and determine if a linear model is appropriate. If it is, then find the linear regression model for the data.
2.3 Function Notation and Making Predictions (Page 9 of 20)
2.3 Function Notation and Making Predictions
In section 2.2 we found the linear regression equations for models. In this section we will use the equations to make estimates and predictions. We will also learn notation for functions.
Example 1
The table shows the average salaries for professors at four-year colleges and universities. Let s represent the average salary (in thousands of dollars) at t years since 1970.
Year / Average Salary(thousands of dollars)
1975
1980
1985
1990
1995
2000 / 16.6
22.1
31.2
41.9
49.1
57.7
a. Verify the linear regression equation for s is .
b. Predict the average salary in 2008.
c. Predict when the average salary will be $75,000.
d. Explain the meaning of the slope in this situation.
Example 2
In example 2 of section 2.2 we found to be a model for the salmon population (in millions) in the year that is t years since 1950.
1a. Predict the salmon population in year 1955.
1b. Use your graphing calculator’s TABLE and TBLSET functions to verify your prediction.
1c. Use the GRAPH and TRACE functions to verify your prediction.
2. Predict the year when the salmon population was 6.4 million.
a. Algebraically
b. Graphically
Example 3
t / Average hours worked per week
W
19751980
1984
1989
1993
1995 / 43.1
46.9
47.3
48.7
50.0
50.6
The average number of hours Americans work in a week gradually increased from 1975 to 1995. Let W be the average number of hours worked per week at t years since 1970.
1. Verify the linear regression model of the data is .
2. Predict the number of hours that the average American will work per week in the year 2003.
3. Predict when the average American will work 60 hours per week.
4. Explain the meaning of the slope in this situation.
Function Notation
When an equation is a function (as all non-vertical lines are) it is often more convenient to use function notation. In example 2 the regression equation is , where the hours worked per week, W, depended on the years elapsed since 1970, t. The expression “W depends on t” is equivalently stated in function terminology as “W is a function of t.” That is,
is read “W equals f of t”
or “W is a function of t”
or “W depends on t”
where f is the name of the function and t is the independent variable. That is,
dependent variable = f(independent variable), i.e.
The only difference is that in equation notation W is the dependent variable, and in function notation f(t) is the dependent variable. There is nothing special about naming the function f, we could have just as easily named it r to remind us it is the regression equation we are talking about. That is, .
Example 4
Let and .
Equation Notation Function Notation
a. Find y when . b. Find .
c. Find x when . d. Find x when .
Example 5
Let . Find
a.
b.
c.
d.
e.
f.
g.
h.
f. x when .
g. x when .
h. x when .
Example 6
The graph of function g is shown. Estimate the following.
1.
2.
3. x when
4. x when
Four-Step Modeling Process
1. Create a scattergram of the data and determine if a linear model is suited for this data.
2. Draw (“eyeball”) a line through the data to represent the linear model. Alternatively, if asked, find the regression line for the data.
3. Verify that the line models the data well.
4. Use the equation or graph for your model to make estimates, make predictions, and draw conclusions.
Example 7
Year / Number of Years since 1950 / Percent Who Smoke1965
1974
1979
1983
1987
1992
1995 / 42.4
37.1
33.5
32.1
28.8
26.5
24.7
Smoking has been on the decline in the United States for decades (see table). Let be the percent of Americans who smoke at t years since 1950.
1. Verify the linear model for p is .
2. Write the regression equation using function notation .
3. Find g(104) and explain its meaning.
4. Find the value for t when g(t) = 30. Explain its meaning.
5. Find the intercepts and explain their meaning.
Example 8
Year
/ Number of Bald Eagle Pairs(thousands)
1993
1994
1995
1996
1997
1998 / 4.2
4.5
4.9
5.1
5.3
5.7
In 1963 there were only 417 male-female pairs of bald eagles in the United States. However, the bald eagle has made a comeback (see table). Let f(t) represent the number of male-female pairs of bald eagles (in thousands) at t years since 1990.
1. Verify the linear model is suited for this data is .
2. In 1999 the bald eagle was taken off the threatened species list. Estimate the number of bald eagle pairs that year.
3. Find . Explain its meaning in this application.
4. Find t when . Explain its meaning in this application.
5. Explain the meaning of the slope in this situation.
2.4 Slope as a Rate of Change (Page 20 of 20)
2.4 Slope as a Rate of Change
Slope as a Rate of ChangeFor a linear function , the slope m is rate of change of y with respect to x.
Example 1
For each of the following, find the rate of change and explain its meaning in the application.
a. Suppose sea level fell steadily by 12 inches in the last four hours as the tide came in.
b. The number of fires in U.S. Hotels declined form 7100 fires in 1990 to 4200 fires in 2002.
c. In San Bruno, CA, the average value of a 2-bedroom home is $543 thousand and the average value of a 5-bedroom home is $793 thousand.
Example 2
(hours)
t / Distance
(miles)
d
0
1
2
3
4
5 / 0
60
120
180
240
300
Suppose a student drives at a constant rate. Let d be the distance (in miles) that the student can drive in t hours. Some values of t and d are shown in the table.
1. Create a scattergram and draw a linear model.
2. Find the slope of the linear model.
3. Find the rate change of distance per hour from t = 2 to t = 3.
4. Find the rate change of distance per hour from t = 0 to t = 4.
5. Find the equation of the linear model.
Example 3
To rent a standard pickup truck, Budget Truck Rental charges a daily fee of $37 plus $0.25 for each mile traveled. Let C represent the cost (in dollars) of renting a pickup truck driven x miles.
x / C0
1
2
3
4
1. Find an equation that models the cost.
2. Identify the dependent and independent variables.
3. What is the slope of the line? Explain its meaning in this application.
Constant Rate of Change PropertyIf the rate of change of y with respect to x is constant, then there is a linear relationship between the variables.
Example 4
A driver fills her 12-gallon gasoline tank and drives at a constant speed. The car consumes 0.04 gallon per mile. Let G be the number of gallons of gasoline remaining in the tank after she has driven d miles since filling up.
a. Is there a linear relationship between d and G? Explain.
b. Find the G-intercept of the linear model and explain its meaning in this application.
c. Find the slope of the linear model.