BA 275 Quantitative Business Methods

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BA 275 Quantitative Business Methods
Agenda

• Simple Linear Regression
• Inference for Regression
• Inference for Prediction

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Regression Analysis

• A technique to examine the relationship between
  an outcome variable (dependent variable, Y) and a
  group of explanatory variables (independent
  variables, X1, X2, Xk).
• The model allows us to understand (quantify) the
  effect of each X on Y.
• It also allows us to predict Y based on X1, X2,
  . Xk.

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Types of Relationship

• Linear Relationship
• Simple Linear Relationship
• Y b0 b1 X e
• Multiple Linear Relationship
• Y b0 b1 X1 b2 X2 bk Xk e
• Nonlinear Relationship
• Y a0 exp(b1Xe)
• Y b0 b1 X1 b2 X12 e
• etc.
• Will focus only on linear relationship.

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Simple Linear Regression Model
population
True effect of X on Y
Estimated effect of X on Y
sample
Key questions 1. Does X have any effect on Y? 2.
If yes, how large is the effect? 3. Given X, what
is the estimated Y?
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Least Squares Method

• Least squares line
• It is a statistical procedure for finding the
  best-fitting straight line.
• It minimizes the sum of squares of the deviations
  of the observed values of Y from those predicted

Sum of Squares is minimized.
Bad fit.
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Initial Analysis

• Summary statistics Plots (e.g., histograms
  scatter plots) Correlations
• Things to look for
• Features of Data (e.g., data range, outliers)
• do not want to extrapolate outside data range
  because the relationship is unknown (or
  un-established).
• Summary statistics and graphs.
• Is the assumption of linearity appropriate?

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Correlation

• r (rho) Population correlation (its value most
  likely is unknown.)
• r Sample correlation (its value can be
  calculated from the sample.)
• Correlation is a measure of the strength of
  linear relationship.
• Correlation falls between 1 and 1.
• No linear relationship if correlation is close to
  0.

r 1 1 lt r lt 0 r 0
0 lt r lt 1 r 1
r 1 1 lt r lt 0 r 0
0 lt r lt 1 r 1
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Correlation (r vs. r)
Sample size
P-value for H0 r 0 Ha r ? 0
r 0.9584
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Fitted Model Least Squares Line
b0
b1
Least squares line estimated_Price 15.1245
76.1745 Area.
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Hypothesis TestingKey Q1 Does X have any effect
on Y?
b0
H0 b1 0 Ha b1 ? 0
SEb1
b1
SEb0
Degrees of freedom n p 1 p of
independent variables used.
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Interval EstimationKey Q2 If so, how large is
the effect?
b0
SEb1
b1
SEb0
Degrees of freedom n p 1 p of
independent variables used.
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Prediction and Confidence IntervalsKey Q3 Given
X, what is the estimated Y?

• What is your estimated price of that 2000-sf
  house on the 9th street?
• Quick answer estimated price -15.1245
  76.1745 (2) 137.2245
• What is the average price of a house that
  occupies 2000 sf?
• Quick answer estimated price -15.1245
  76.1745 (2) 137.2245
• What is the difference?

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Prediction and Confidence Intervals
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Prediction and Confidence Intervals
Prediction interval
Confidence interval
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Model Comparison A Good Fit?
s
SS Sum of Squares ???