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Chapter 16

OMIS 2010 Chapter Notes - Chapter 16: Regression Analysis, Interval Estimation, Royal Guelphic Order

Operations Management and Information System
Course Code
OMIS 2010
Alan Marshall

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Chapter 16: Sample Linear Regression and Correlation
-Regression analysis: used to predict value of one variable on basis of other variables
-Dependent variable: variable to be forecast
-Independent variable: variable that practitioner believes are related to dependent variables (x1, x2, xk)
-Correlation analysis: determines whether relationship exists
-Deterministic models: equation that allow us to determine value of the dependent variable from values of
independent variables
-Probabilistic Model: method to represent randomness
-e is the error variable: accounts for all variables, measurable and immeasurable, that are not part of the
oIts value varies from one ‘sale’ to the next even if ‘x’ remains constant
-First-Order Linear Model (simple linear regression model): y (dependent variable) = B0 (y-int) – (y-int)
B1x(independent variable) + e (error variable)
-X and y must be interval
Estimating the Coefficients
-Draw random samples from population of interest
-Calculate sample statistics to estimate B0 and B1
-Estimators based on drawing straight line though sample data; least squares line: comes closest to sample
data points
oY-hat (predicted/fitted value of y) = b0 +b1x
oB0 and b1 calculated so that sum of squared deviations is minimized
oY-hat on average comes closest to observed values of y
oLeast squares method: produces straight line that minimizes the sum of the squared difference
between the points and the line
o(b0) and (b1) are unbiased estimators of B0 and B1
oResiduals: deviations between the actual data pints and the line, ei
oEi = yi – y-hat
Residuals are observations of the error variable

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Minimized sum of squared deviation called sum of squares for error (SSE)
Residuals are differences between observed values of y1 and y hat
-Note: we can’t determine value of y-hat for value of x that is far outside the range of sample values of x
Error Variable: Required Conditions
-Required conditions for the Error Variable
1. Probability distribution of e is normal
2. Mean of the distribution is 0: E€ = 0
3. Standard deviation of e is sigma e, which is constant regardless of value of x
4. Value of e associate with any particular value of y is independent of e associated with any
other value of y
-For 1, 2, 3: for each value of x, y is a normally distributed random variable with mean: E(y) = B0 + B1x, with
standard deviation of sigma-e
oMean depends on x, std deviation constant for all values of x
oFor each x, y is normally distributed with same standard deviation
Observational and Experimental Data
-Objective is to see how independent variable is related to dependent variable
-When data is observational, both variables are random variable (don’t need to specify which is
dependent and which is not)
-Two variables must be bivariate normally distributed
Assessing the Model
-Least squares method produces best straight line
oStill may not be any relationship or nonlinear relationship between the two variable’s
-Standard error of the estimate: t-test of the slope and coefficient of determination
Sum of Squares for Error
-Least squares method determines coefficient that minimize sum of squared deviations between the points
and the line defined by the coefficients
Standard Error of Estimate
-If sigma e is large, some of the errors will be large
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