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Regression Analysis Simple Regression. y = mx + b. y = a + bx. y = a + bx. 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. Assumptions. Linearity

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## Regression Analysis Simple Regression

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**y = mx + b**y = a + bx**y = a + bx**where: ydependent variable(value depends on x) ay-intercept(value of y when x = 0) bslope (rate of change in ratio of delta y divided by delta x) x independentvariable**Assumptions**Linearity Independence of Error Homoscedasticity Normality**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].**Independence of Error**The error (residual) is independent for each value of X. [Residual = observed - predicted]**Homoscedasticity**The variation around the line of regression constant for all values of X.**Normality**The values of Y be normally distributed at each value of X.**Linearity**Independence Examine scatter plot of residuals versus fitted [Yhat] for evidence of nonlinearity Plot residuals in time order and look for patterns Diagnostic Checking**Homoscedasticity**Normality Examine scatter plots of residuals versus fitted [Yhat] and residuals vs time order and look for changing scatter. Examine histogram of residuals. Look for departures from normal curve. Diagnostic Checking**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).**Simple Regression**A statistical model that utilizes onequantitativeindependent variable “X” to predict the quantitativedependent variable “Y.”**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.**Mini-Case**• What preliminary conclusions can management draw from the data? • What could management expect sales to be if population of the new complex is approximately 18,000 people?**Scatter Diagrams**• The values are plotted on a two-dimensional graph called a “scatter diagram.” • Each value is plotted at its X and Y coordinates.**Types of Models**No relationship between X and Y Positive linear relationship Negative linear relationship**Method of Least Squares**• The straight line that best fits the data. • 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.**Measures of Variation**Explained Unexplained Total**Explained Variation**Sum of Squares (Yhat - Ybar)2 due to Regression [SSR]**Unexplained Variation**Sum of Squares (Yobs - Yhat)2 Error [SSE]**Total Variation**Sum of Squares (Yobs - Ybar)2 Total [SST]**H0:**There is no linear relationship between the dependent variable and the explanatory variable**Hypotheses**H0: = 0 H1: 0 or H0: No relationship exists H1: A relationship exists**Standard Error of the Estimate**sy.x -the measure of variability around the line of regression**Relationship**When null hypothesis is rejected, a relationship between Y and X variables exists.**Coefficient of Determination**R2 measures the proportion of variation that is explained by the independent variable in the regression model. R2 = SSR / SST**Confidence interval estimates**• True mean YX • Individual Y-hat**Diagnostic Checking**• H0retain or reject {Reject if p-value 0.05} • R2 (larger is “better”) • sy.x (smaller is “better”)**Coefficient of Determination**R2 = SSR / SST = 90.27 % thus, 90.27 percent of the variation in annual sales is explained by the population.**Standard Error of the Estimate**sy.x = 13.8293 with SSE = 1,530.0**Regression Analysis[Simple Regression]***** End of Presentation *** Questions?

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