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Introduction to Regression Section 1.3 1 Mathematical Modeling Mathematical modeling is the process of using mathematics to solve real- world problems. This process can be broken down into three steps: 1. Construct the mathematical model, a problem whose solution will provide information about the real-world problem. (We will do this on our calculator using regression analysis.) 2. Solve the mathematical model. 3. Interpret the solution to the mathematical model in terms of the original real-world problem. 2 Rates of Change Slope can be thought of as a rate of change. Examples: miles per hour, feet per second, price per pound, houses per square mile Note: You will be asked to interpret the slope of your equation in the context of the problem. You will need to be able to describe the rate of change in terms of the data. For example, let’s say the relationship between the weight of a round-shaped diamond and the price can be described as p 6,140c 480, where p is the price in $ and c is the weight in carats. Then the slope is 6,140. To interpret this in terms of p and c, the slope in this problem suggests that a round diamond costs approximately $6,140 per carat. 3 Constructing Models Example Ideal Weight Dr. J.D. Robinson published the following estimate of the ideal body weight of a man: 52 kg + 1.9 kg for each inch over 5 feet a) Find a linear model for Robinson’s estimate of the ideal weight of a man using w for ideal body weight (in kilograms) and h for height over 5 feet (in inches). b) Identify and interpret the slope of the model. 4 Constructing Models-Continued Example Ideal Weight Dr. J.D. Robinson published the following estimate of the ideal body weight of a man: 52 kg + 1.9 kg for each inch over 5 feet c) If a man is 5’8” tall, what does the model predict his weight to be? Show work. d) If a man weighs 70 kilograms, what does the model predict his height to be? Show work. 5 Regression Vocabulary Regression: a process used to relate two quantitative variables Independent Variable: the x variable (or explanatory variable) Dependent Variable: the y variable 6 Interpreting the Scatterplot To interpret the scatterplot, identify these following 4 things: Form: the function that best describes the relationship between the 2 variables. (Some possible forms would be linear, quadratic, cubic, exponential, or logarithmic.) Outlier(s): any values that do not follow the general pattern of the data; stray points.) 7 Interpreting the Scatterplot Direction: a positive or negative direction can be found when looking at linear regression lines only. The direction is found by looking at the sign of the slope. Strength: how closely the points in the data are gathered around the form. 8 Entering in the Data Hit then the Edit menu appears. Select 1:Edit. Then type in the data values for the independent variable in column L1 and the data values for the dependent variable in column L2. 9 Making the Scatterplot Hit then highlight Plot 1 by cursoring up and hitting . Hit . Most likely the scatterplot will not be visible using the standard viewing window. To make the scatterplot viewable, appropriate window settings need to be selected. This can be done by hitting and then entering appropriate values for Xmin, Xmax, Ymin, and Ymax. (OR, a personal favorite, 9:ZoomStat, which adjusts the window settings according to your values.) 10 Some Types of Regression Linear Regression (straight line form)- menu option 4:LinReg(ax+b) Quadratic Regression (parabolic form)- menu option 5:QuadReg Cubic Regression (cubic form)- menu option 6:CubicReg 11 Getting The Regression Equation To obtain the regression and to store it to graph on the scatterplot, Hit then cursor over to CALC to display the CALC menu. Select 4:LinReg(ax+b) then hit then L1 (in yellow above 1) then then then L2 (in yellow above 2) then then then cursor over to Y-VARS then select 1:Function then 1:Y1 then hit . *Note: The directions above refer to linear regression. If a different type of regression is more appropriate, replace 4:LinReg(ax+b) with the more appropriate regression type. 12 Making Predictions Predictions should only be made for values of x within the span of the x- values in the data set. Predictions made outside the data set are called extrapolations and can be dangerous and/or ridiculous, thus they should be done with caution. To make a prediction within the span if the x-values, hit then . Next, arrow up or down until the regression equation appears in the upper-left hand corner then type in the x-value and hit . 13 Clearing Out Old Data To clear out the data in L1 and L2, hit then the Edit menu appears. Select 1:Edit. Use the arrows to place cursor over L1. Hit , then arrow once. Now, use the arrows to place cursor over L2. Hit , then arrow once. (Be sure NOT to hit DELETE…this will remove the list completely!!!) 14 Returning to the Standard Window Hit 6:Standard. (If you are done with regression, don’t forget to un-highlight Plot 1 in the menu.) 15 Example 1: Consumer Debt The table shows the total outstanding consumer debt (excluding home mortgages) in billions of dollars in selected years. (Data is from the Federal Reserve Bulletin.) Year 1985 1990 1995 2000 2003 Consumer Debt 585 789 1096 1693 1987 Let x = 0 correspond to 1985. a) Using a graphing calculator, draw the scatterplot. Interpret the plot: 1. Form 2. Strength 3. Direction 16 Example 1: Consumer Debt (cont) b) Find the regression equation appropriate for this data set. Round values to two decimal places. c) Find and interpret the slope of the regression equation in the context of the scenario. 17 Example 1: Consumer Debt (cont) d) Find the approximate consumer debt in 1998. e) Find the approximate consumer debt in 2008. 18 Example 1: Consumer Debt (cont) f) Using the regression equation, predict the year when consumer debt will reach 2,500 billion dollars. 19 Example 2: Health The table below shows the number of deaths per 100,000 people from heart disease in selected years. (Data is from the U.S. National Center for Health Statistics.) Year 1960 1970 1980 1990 2000 2002 Deaths 559 483 412 322 258 240 Let x = 0 correspond to 1960. a) Using a graphing calculator, draw the scatterplot. Interpret the plot: 1. Form 2. Strength 3. Direction 20 Example 2: Health (cont) b) Find the regression equation appropriate for this data set. Round values to two decimal places. c) Find and interpret the slope of the regression equation in the context of the scenario. 21 Example 2: Health (cont) d) Find an approximation for the number of deaths due to heart disease in 1995. e) Predict the number of deaths from heart disease in 2008. 22

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regression analysis, linear regression, regression model, multiple regression, dependent variable, independent variables, Analysis of Variance, Hypothesis Testing, Symptom Checker, regression models

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posted: | 3/5/2010 |

language: | English |

pages: | 22 |

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