Economics 173 Business Statistics

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Economics 173Business Statistics

• Lecture 10
• Fall, 2001
• Professor J. Petry
• http//www.cba.uiuc.edu/jpetry/Econ_173_fa01/

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Simple Linear Regression and Correlation

• Chapter 17

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17.1 Introduction

• In this chapter we employ Regression Analysisto
  examine the relationship among quantitative
  variables.
• The technique is used to predict the value of one
  variable (the dependent variable - y)based on the
  value of other variables (independent variables
  x1, x2,xk.)

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17.2 The Model

• The first order linear model
• y dependent variable
• x independent variable
• b0 y-intercept
• b1 slope of the line
• error variable

b0 and b1 are unknown, therefore, are estimated
from the data.
y
Rise
b1 Rise/Run
Run
b0
x
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17.3 Estimating the Coefficients

• The estimates are determined by
• drawing a sample from the population of interest,
• calculating sample statistics.
• producing a straight line that cuts into the data.

y
w
The question is Which straight line fits best?
w
w
w
w
w w w w
w
w w
w w
w
x
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The best line is the one that minimizes the sum
of squared vertical differences between the
points and the line.
Sum of squared differences
(2 - 1)2
(4 - 2)2
(1.5 - 3)2
(3.2 - 4)2 6.89
Let us compare two lines
(2,4)
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The second line is horizontal
w
(4,3.2)
w
3
2.5
2
w
(1,2)
The smaller the sum of squared differences the
better the fit of the line to the data.
(3,1.5)
w
3
4
2
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To calculate the estimates of the
coefficients that minimize the differences
between the data points and the line, use the
formulas
The regression equation that estimates the
equation of the first order linear model is
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Example 17.1 Relationship between odometer
reading and a used cars selling price.

• A car dealer wants to find the relationship
  between the odometer reading and the selling
  price of used cars.
• A random sample of 100 cars is selected, and the
  data recorded.
• Find the regression line.

Independent variable x
Dependent variable y
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• Solution using first formula
• Solving by hand
• To calculate b0 and b1 we need to calculate
  several statistics first

where n 100.
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• Solution
• Solving by hand using the second formula
• First we obtain the necessary numbers to enter

n 100.
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• Using the computer (see file Xm17-01.xls)

Tools gt Data analysis gt Regression gt Shade the y
range and the x range gt OK
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6533
0
No data
This is the slope of the line. For each
additional mile on the odometer, the price
decreases by an average of 0.0312
The intercept is b0 6533.
Do not interpret the intercept as the Price of
cars that have not been driven
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Example Armanis Pizza

• Armanis Pizza is considering locating at the U
  of I campus. To do their financial analysis, they
  first need to estimate sales for their product.
• They have data from their existing 10 locations
  on other college campuses.
• Estimate sales for the University of Illinois,
  with a college population of 35,000.

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Example Armanis Pizza

• Begin by plotting the data

• Followed by calculating the statistics of your
  regression

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Example Armanis Pizza

• Solve for b1, then b0 using the second formula

• After obtaining values for your regression line,
  plug in the value for your independent variable
  (35, not 35,000) into the formula, and predict
  sales!

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Example Armanis Pizza Excel Output
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17.4 Error Variable Required Conditions

• The error e is a critical part of the regression
  model.
• Four requirements involving the distribution of e
  must be satisfied.
• The probability distribution of e is normal.
• The mean of e is zero E(e) 0.
• The standard deviation of e is se for all values
  of x.
• The set of errors associated with different
  values of y are all independent.

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From the first three assumptions we have y is
normally distributed with mean E(y) b0 b1x,
and a constant standard deviation se
The standard deviation remains constant,
m3
m2
but the mean value changes with x
m1
x1
x2
x3