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Regression Analysis Simple Regression

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y = mx + b y = a + bx

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where: y dependent variable (value depends on x) a y-intercept (value of y when x = 0) b slope (rate of change in ratio of delta y divided by delta x) x independent variable

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Assumptions Linearity Independence of Error Homoscedasticity Normality

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Linearity The most fundamental assumption is that the model fits the situation [i.e.: the Y variable is linearly related to the value of the X variable].

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Independence of Error The error (residual) is independent for each value of X. [Residual = observed - predicted]

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Homoscedasticity The variation around the line of regression constant for all values of X.

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Normality T he values of Y be normally distributed at each value of X.

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Diagnostic Checking u Linearity u Independence u Examine scatter plot of residuals versus fitted [Y hat ] for evidence of nonlinearity u Plot residuals in time order and look for patterns

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Diagnostic Checking u Homoscedasticity u Normality u Examine scatter plots of residuals versus fitted [Y hat ] and residuals vs time order and look for changing scatter. u Examine histogram of residuals. Look for departures from normal curve.

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Goal Develop a statistical model that can predict the values of a dependent (response) variable based upon the values of the independent (explanatory) variable(s).

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Goal

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Simple Regression quantitative quantitative A statistical model that utilizes one quantitative independent variable “X” to predict the quantitative dependent variable “Y.”

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Mini-Case Since a new housing complex is being developed in Carmichael, management is under pressure to open a new pie restaurant. Assuming that population and annual sales are related, a study was conducted to predict expected sales.

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Mini-Case (Descartes Pie Restaurants)

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Mini-Case u What preliminary conclusions can management draw from the data? u What could management expect sales to be if population of the new complex is approximately 18,000 people?

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Scatter Diagrams u The values are plotted on a two- dimensional graph called a “scatter diagram.” u Each value is plotted at its X and Y coordinates.

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Scatter Plot of Pieshop

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Types of Models No relationship between X and Y Positive linear relationship Negative linear relationship

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Method of Least Squares u The straight line that best fits the data. u Determine the straight line for which the differences between the actual values (Y) and the values that would be predicted from the fitted line of regression (Y-hat) are as small as possible.

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Measures of Variation Explained Unexplained Total

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Explained Variation Sum of Squares (Y hat - Y bar ) 2 due to Regression [SSR]

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Unexplained Variation Sum of Squares (Y obs - Y hat ) 2 Error [SSE]

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Total Variation Sum of Squares (Y obs - Y bar ) 2 Total [SST]

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H0:H0: There is no linear relationship between the dependent variable and the explanatory variable

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Hypotheses H 0 : = 0 H 1 : 0 or H 0 : No relationship exists H 1 : A relationship exists

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Analysis of Variance for Regression

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Standard Error of the Estimate s y.x - the measure of variability around the line of regression

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Relationship When null hypothesis is rejected, a relationship between Y and X variables exists.

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Coefficient of Determination R 2 measures the proportion of variation that is explained by the independent variable in the regression model. R 2 = SSR / SST

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Confidence interval estimates »True mean YX »Individual Y-hat

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Pieshop Forecasting

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Coefficient of Sanity

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Diagnostic Checking u H 0 retain or reject {Reject if p-value 0.05} u R 2 (larger is “better”) u s y.x (smaller is “better”)

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Analysis of Variance for Regression for Pieshop

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Coefficient of Determination R 2 = SSR / SST = 90.27 % thus, 90.27 percent of the variation in annual sales is explained by the population.

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Standard Error of the Estimate s y.x = 13.8293 with SSE = 1,530.0

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Regression Analysis [Simple Regression] *** End of Presentation *** Questions?

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