2.1 Using Lines to Model Data

Transcription

1 2.1 Using Lines to Model Data 2.1 Using Lines to Model Data (Page 1 of 20) Scattergram and Linear Models The graph of plotted data pairs is called a scattergram. A linear model is a straight line or a linear equation that describes the relationship between two quantities for a true-to-life situation. Example 1 Let v represent the number of visitors (in millions) to the Grand Canyon in the year that is t years since a. Fill-in the table of values for t. Identify the dependent and independent variables. b. Make a scattergram of the data. Year Years since 1960 t Number of Visitors (millions) v part d? 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).

2 2.1 Using Lines to Model Data (Page 2 of 20) 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). The following illustration is specific to example 3.

3 Example 2 Prozac is an antidepressant that was approved by the FDA in Let p be the number of prescriptions of Prozac (in millions) dispensed at t years since a. Fill-in the table of values for t. b. Identify the dependent and independent variables. 2.1 Using Lines to Model Data (Page 3 of 20) Year t Number of Prozac Prescriptions (millions) 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 f. Use the linear model to predict when the number of Prozac prescriptions will reach 30 million. g. Identify the p-intercept and interpret its meaning in this application. h. Identify the t-intercept and interpret its meaning in this application.

4 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 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. 2.1 Using Lines to Model Data (Page 4 of 20) Year Salmon Population (millions) P e. Find the t-intercept and explain its meaning.

5 2.2 Finding Regression Equations for Linear Models (Page 5 of 20) 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 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. 3. Press ZOOM / 9:ZoomStat to view the scattergram. 4. Run the linear regression program: STAT / CALC / 4:LinReg (ax+b) L 1, L 2, Y 1 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. 5. Rewrite the equation using the variables in the application.

6 2.2 Finding Regression Equations for Linear Models (Page 6 of 20) Homework Note: The examples and exercises in section 2.2 ask you to find the equation of the linear model using two well- chosen points from the scattergram. I want you to find the linear regression equation and round the constants to two decimal places. Example 1 The American life span has been increasing over the last century. Let L(t) represent the life expectancy at birth for an American born t years after Create a scatter plot of the data on your calculator and determine if a linear model is appropriate. If it is, then (a) Find the linear regression model for the data. Round the constants to two decimal places. Birth Year Life Expectancy at Birth (b) Find a linear model by choosing two representative points from the data set. Round the constants to two decimal places. (c) Use the linear regression model from part (a) to predict the life expectancy for someone born in (d) Use the linear regression model from part (a) to predict the birth year of a person that will have a life expectancy of 80 years.

7 2.2 Finding Regression Equations for Linear Models (Page 7 of 20) 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 Create a scattergram of the data on your calculator and determine if a linear model is appropriate. If it is, then (a) Find the linear regression model for the data. Round the constants to two decimal places. Number of Years since 1950 t Salmon Population (millions) P (b) Find a linear model by choosing two representative points from the data set. Round the constants to two decimal places. (c) Use the linear regression model from part (a) to predict the salmon population in year (d) Use the linear regression model from part (a) to find the t-intercept and explain its meaning in this application.

8 2.3 Function Notation and Making Predictions (Page 8 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 the 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 a. Verify the linear regression equation for s is s = 1.70t b. Predict the average salary in Year Average Salary (thousands of dollars) c. Predict when the average salary will be $75,000. d. Explain the meaning of the slope in this situation.

9 2.3 Function Notation and Making Predictions (Page 9 of 20) Example 2 In example 2 of section 2.2 we found P = 0.29t to be a model for the salmon population (in millions) in the year that is t years since a. Predict the salmon population in year b. 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

10 Example 3 The average number of hours Americans work in a week gradually increased from 1975 to Let W be the average number of hours worked per week at t years since Verify the linear regression model of the data is W = 0.34t Function Notation and Making Predictions (Page 10 of 20) 2. Predict the number of hours that the average American will work per week in the year Predict when the average American will work 60 hours per week. 4. Explain the meaning of the slope in this situation.

11 2.3 Function Notation and Making Predictions (Page 11 of 20) 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 W = 0.34t , 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, W = f (t) = 0.34t 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. W = f (t) 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, r(t) = 0.34t Example 4 Let y = 3x 5 and f (x) = 3x 2. Equation Notation Function Notation a. Find y when x = 2. b. Find f ( 2). c. Find x when y = 10. d. Find x when f (x) = 10.

12 Example 5 Let g(x) = 1 2 a. g( 4) x + 3. Find 2.3 Function Notation and Making Predictions (Page 12 of 20) b. g(10) c. g(0) d. g 2 3 e. g(2.7) f. g(a) g. g(2a) h. g(a + 2) f. x when g(x) = 0. g. x when g(x) = 7. h. x when g(x) = 11.

13 2.3 Function Notation and Making Predictions (Page 13 of 20) Example 6 The graph of function g is shown. Estimate the following. 1. g(4) y 2. g(0) 3. x when g(x) = 3 x 4. x when g(x) = 0 5. Find the equation for g.

14 2.3 Function Notation and Making Predictions (Page 14 of 20) 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 Smoking has been on the decline in the United States for decades (see table). Let p = g(t) be the percent of Americans who smoke at t years since Verify the linear model for p is p = 0.59t Write the regression equation using function notation p = g(t). 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. 6. Explain the meaning of the slope.

15 Example 8 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 Verify the linear model is suited for this data is f (t) = 0.29t In 1999 the bald eagle was taken off the threatened species list. Estimate the number of bald eagle pairs that year. 2.3 Function Notation and Making Predictions (Page 15 of 20) Year Number of Bald Eagle Pairs (thousands) Find f (15). Explain its meaning in this application. 4. Find t when f (t) = 15. Explain its meaning in this application. 5. Explain the meaning of the slope in this situation.

16 2.4 Slope is a Rate of Change 2.4 Slope is a Rate of Change (Page 16 of 20) Slope is a Rate of Change Suppose a quantity y changes steadily from y 1 to y 2 as a quantity x changes steadily from x 1 to x 2. Then the slope m is the rate of change of y with respect to x: m = change in y change in x = change in dep. variable change in indep. variable = y 2 y 1 x 2 x 1 If either quantity changes, but not steadily, then this formula is called the average rate of change. 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 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.

17 Example 2 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. 2.4 Slope is a Rate of Change (Page 17 of 20) Time (hours) t Distance (miles) d Find the slope of the linear model. 3. Find the rate change of distance per hour from t = 2 to t = Find the rate change of distance per hour from t = 0 to t = Find the equation of the linear model.

18 2.4 Slope is a Rate of Change (Page 18 of 20) 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. 1. Find an equation that models the cost. x C 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 Property If the rate of change of y with respect to x is constant, then there is a linear relationship between the variables.

19 2.4 Slope is a Rate of Change (Page 19 of 20) 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. d. Find the equation of the linear model. Unit Analysis A unit analysis of a model s equation is done by determining that the units on both sides of the equation are the same. e. Perform a unit analysis of the equation found in part d.

20 Example 5 Yogurt sales (in billions of dollars) in the United States are shown in the table. Let s by yogurt sales (billions of dollars) in the year that is t years since a. Use a graphing calculator to draw a scattergram and find the linear model for the data. 2.4 Slope is a Rate of Change (Page 20 of 20) Year Sales (billions of dollars) b. Explain the meaning of the slope in this situation. c. Predict the sales in d. Predict when the sales will be $3.5 billion. e. Find the s-intercept and explain it meaning in this situation. f. Find the t-intercept and explain its meaning in this situation.