Introduction to Regression

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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.




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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.

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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.




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