Presentation on theme: "Chapter 12 Simple Linear Regression"— Presentation transcript:
1 Chapter 12 Simple Linear Regression Simple Linear Regression ModelLeast Squares MethodCoefficient of DeterminationModel AssumptionsTesting for Significance
2 Simple Linear Regression Model The equation that describes how y is related to x andan error term is called the regression model.The simple linear regression model is:y = b0 + b1x +ewhere:b0 and b1 are called parameters of the model,e is a random variable called the error term.
3 Simple Linear Regression Equation The simple linear regression equation is:E(y) = 0 + 1xGraph of the regression equation is a straight line.b0 is the y intercept of the regression line.b1 is the slope of the regression line.E(y) is the expected value of y for a given x value.
4 Simple Linear Regression Equation Positive Linear RelationshipE(y)xRegression lineInterceptb0Slope b1is positive
5 Simple Linear Regression Equation Negative Linear RelationshipE(y)xInterceptb0Regression lineSlope b1is negative
6 Simple Linear Regression Equation No RelationshipE(y)xRegression lineInterceptb0Slope b1is 0
7 Estimated Simple Linear Regression Equation The estimated simple linear regression equationThe graph is called the estimated regression line.b0 is the y intercept of the line.b1 is the slope of the line.is the estimated value of y for a given x value.
8 Estimation Process Regression Model y = b0 + b1x +e Regression EquationE(y) = b0 + b1xUnknown Parametersb0, b1Sample Data:x yx y1xn ynb0 and b1provide estimates ofEstimatedRegression EquationSample Statisticsb0, b1
9 Least Squares Method Least Squares Criterion where: yi = observed value of the dependent variablefor the ith observation^yi = estimated value of the dependent variablefor the ith observation
10 Least Squares MethodSlope for the Estimated Regression Equation
11 Least Squares Method y-Intercept for the Estimated Regression Equation where:xi = value of independent variable for ithobservationyi = value of dependent variable for ithobservation_x = mean value for independent variable_y = mean value for dependent variablen = total number of observations
12 Simple Linear Regression Example: Reed Auto SalesReed Auto periodically hasa special week-long sale.As part of the advertisingcampaign Reed runs one ormore television commercialsduring the weekend preceding the sale. Data from asample of 5 previous sales are shown on the next slide.
13 Simple Linear Regression Example: Reed Auto SalesNumber ofTV AdsNumber ofCars Sold1321424181727
14 Estimated Regression Equation Slope for the Estimated Regression Equationy-Intercept for the Estimated Regression EquationEstimated Regression Equation
16 Coefficient of Determination Relationship Among SST, SSR, SSESST = SSR SSEwhere:SST = total sum of squaresSSR = sum of squares due to regressionSSE = sum of squares due to error
17 Coefficient of Determination The coefficient of determination is:r2 = SSR/SSTwhere:SSR = sum of squares due to regressionSST = total sum of squares
18 Coefficient of Determination r2 = SSR/SST = 100/114 =The regression relationship is very strong; 88%of the variability in the number of cars sold can beexplained by the linear relationship between thenumber of TV ads and the number of cars sold.
19 Sample Correlation Coefficient where:b1 = the slope of the estimated regressionequation
20 Sample Correlation Coefficient The sign of b1 in the equation is “+”.rxy =
21 Assumptions About the Error Term e 1. The error is a random variable with mean of zero.2. The variance of , denoted by 2, is the same forall values of the independent variable.3. The values of are independent.4. The error is a normally distributed randomvariable.
22 Testing for Significance To test for a significant regression relationship, wemust conduct a hypothesis test to determine whetherthe value of b1 is zero.Two tests are commonly used:t TestF TestandBoth the t test and F test require an estimate of s 2,the variance of e in the regression model.
23 Testing for Significance An Estimate of sThe mean square error (MSE) provides the estimateof s 2, and the notation s2 is also used.s 2 = MSE = SSE/(n - 2)where:
24 Testing for Significance An Estimate of sTo estimate s we take the square root of s 2.The resulting s is called the standard error ofthe estimate.
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