Chapter-12-Simple-Linear-Regression-商務(wù)統(tǒng)計(jì)-教學(xué)課件

1、Chapter-12-Simple-Linear-Regression-商務(wù)統(tǒng)計(jì)-教學(xué)課件Chapter 12 Simple Linear RegressionSimple Linear Regression ModelLeast Squares MethodCoefficient of DeterminationModel AssumptionsTesting for SignificanceUsing the Estimated Regression Equation for Estimation and PredictionComputer SolutionResidual Analys

2、is: Validating Model AssumptionsChapter 12 Simple Linear RegrSimple Linear Regression Modely = b0 + b1x +ewhere: b0 and b1 are called parameters of the model, e is a random variable called the error term. The simple linear regression model is: The equation that describes how y is related to x and an

3、 error term is called the regression model.Simple Linear Regression ModelSimple Linear Regression EquationThe simple linear regression equation is: E(y) is the expected value of y for a given x value. b1 is the slope of the regression line. b0 is the y intercept of the regression line. Graph of the

4、regression equation is a straight line.E(y) = 0 + 1xSimple Linear Regression EquatSimple Linear Regression EquationPositive Linear RelationshipE(y)xSlope b1is positiveRegression lineIntercept b0Simple Linear Regression EquatSimple Linear Regression EquationNegative Linear RelationshipE(y)xSlope b1is

5、 negativeRegression lineIntercept b0Simple Linear Regression EquatSimple Linear Regression EquationNo RelationshipE(y)xSlope b1is 0Regression lineIntercept b0Simple Linear Regression EquatEstimated Simple Linear Regression EquationThe estimated simple linear regression equation is the estimated valu

6、e of y for a given x value. b1 is the slope of the line. b0 is the y intercept of the line. The graph is called the estimated regression line.Estimated Simple Linear RegresEstimation ProcessRegression Modely = b0 + b1x +eRegression EquationE(y) = b0 + b1xUnknown Parametersb0, b1Sample Data:x yx1 y1.

7、 . . . xn ynb0 and b1provide estimates ofb0 and b1EstimatedRegression Equation Sample Statisticsb0, b1Estimation ProcessSample Data:Least Squares MethodLeast Squares Criterionwhere:yi = observed value of the dependent variable for the ith observationyi = estimated value of the dependent variable for

8、 the ith observationLeast Squares MethodLeast SquaSlope for the Estimated Regression EquationLeast Squares MethodSlope for the Estimated Regresy-Intercept for the Estimated Regression EquationLeast Squares Methodwhere:xi = value of independent variable for ith observationn = total number of observat

9、ions_y = mean value for dependent variable_x = mean value for independent variableyi = value of dependent variable for ith observationy-Intercept for the Estimated Reed Auto periodically hasa special week-long sale. As part of the advertisingcampaign Reed runs one ormore television commercialsduring

10、 the weekend preceding the sale. Data from asample of 5 previous sales are shown on the next slide.Simple Linear RegressionExample: Reed Auto SalesReed Auto periodically hasSimSimple Linear RegressionExample: Reed Auto SalesNumber of TV AdsNumber ofCars Sold132131424181727Simple Linear RegressionExa

11、mplEstimated Regression EquationSlope for the Estimated Regression Equationy-Intercept for the Estimated Regression EquationEstimated Regression EquationEstimated Regression EquationSScatter Diagram and Trend LineScatter Diagram and Trend LineCoefficient of DeterminationRelationship Among SST, SSR,

12、SSEwhere: SST = total sum of squares SSR = sum of squares due to regression SSE = sum of squares due to errorSST = SSR + SSECoefficient of DeterminationReThe coefficient of determination is:Coefficient of Determinationwhere:SSR = sum of squares due to regressionSST = total sum of squaresr2 = SSR/SST

13、The coefficient of determinatiCoefficient of Determinationr2 = SSR/SST = 100/114 = .8772 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.Coefficient of Dete

14、rminationr2Sample Correlation Coefficientwhere: b1 = the slope of the estimated regression equationSample Correlation CoefficientThe sign of b1 in the equation is “+”.Sample Correlation Coefficientrxy = +.9366The sign of b1 in the equationAssumptions About the Error Term e1. The error is a random va

15、riable with mean of zero.2. The variance of , denoted by 2, is the same for all values of the independent variable.3. The values of are independent.4. The error is a normally distributed random variable.Assumptions About the Error TeTesting for Significance To test for a significant regression relat

16、ionship, we must conduct a hypothesis test to determine whether the value of b1 is zero. Two tests are commonly used:t TestandF Test Both the t test and F test require an estimate of s 2, the variance of e in the regression model.Testing for Significance To teAn Estimate of s Testing for Significanc

17、ewhere:s 2 = MSE = SSE/(n - 2)The mean square error (MSE) provides the estimateof s 2, and the notation s2 is also used.An Estimate of s Testing for STesting for SignificanceAn Estimate of s To estimate s we take the square root of s 2. The resulting s is called the standard error of the estimate.Te

18、sting for SignificanceAn EstHypotheses Test StatisticTesting for Significance: t TestHypothesesTesting for SignificRejection RuleTesting for Significance: t Testwhere: t is based on a t distributionwith n - 2 degrees of freedomReject H0 if p-value a or t tRejection RuleTesting for Sign1. Determine t

19、he hypotheses.2. Specify the level of significance.3. Select the test statistic.a = .054. State the rejection rule.Reject H0 if p-value 3.182 (with3 degrees of freedom)Testing for Significance: t Test1. Determine the hypotheses.2Testing for Significance: t Test5. Compute the value of the test statis

20、tic.6. Determine whether to reject H0.t = 4.541 provides an area of .01 in the uppertail. Hence, the p-value is less than .02. (Also,t = 4.63 3.182.) We can reject H0.Testing for Significance: t Confidence Interval for 1 H0 is rejected if the hypothesized value of 1 is not included in the confidence

21、 interval for 1. We can use a 95% confidence interval for 1 to test the hypotheses just used in the t test.Confidence Interval for 1 HThe form of a confidence interval for 1 is:Confidence Interval for 1where is the t value providing an areaof a/2 in the upper tail of a t distributionwith n - 2 degre

22、es of freedomb1 is thepointestimatoris themarginof errorThe form of a confidence interConfidence Interval for 1Reject H0 if 0 is not included inthe confidence interval for 1.0 is not included in the confidence interval. Reject H0= 5 +/- 3.182(1.08) = 5 +/- 3.44or 1.56 to 8.44Rejection Rule95% Confid

23、ence Interval for 1ConclusionConfidence Interval for 1RejeHypotheses Test StatisticTesting for Significance: F TestF = MSR/MSEHypothesesTesting for SignificRejection RuleTesting for Significance: F Testwhere:F is based on an F distribution with1 degree of freedom in the numerator andn - 2 degrees of

24、 freedom in the denominatorReject H0 if p-value FRejection RuleTesting for Sign1. Determine the hypotheses.2. Specify the level of significance.3. Select the test statistic.a = .054. State the rejection rule.Reject H0 if p-value 10.13 (with 1 d.f.in numerator and 3 d.f. in denominator)Testing for Si

25、gnificance: F TestF = MSR/MSE1. Determine the hypotheses.2Testing for Significance: F Test5. Compute the value of the test statistic.6. Determine whether to reject H0. F = 17.44 provides an area of .025 in the upper tail. Thus, the p-value corresponding to F = 21.43 is less than 2(.025) = .05. Hence

26、, we reject H0.F = MSR/MSE = 100/4.667 = 21.43 The statistical evidence is sufficient to concludethat we have a significant relationship between thenumber of TV ads aired and the number of cars sold. Testing for Significance: F TSome Cautions about theInterpretation of Significance Tests Just becaus

27、e we are able to reject H0: b1 = 0 and demonstrate statistical significance does not enableus to conclude that there is a linear relationshipbetween x and y. Rejecting H0: b1 = 0 and concluding that therelationship between x and y is significant does not enable us to conclude that a cause-and-effect

28、relationship is present between x and y.Some Cautions about theInterpUsing the Estimated Regression Equationfor Estimation and Predictionwhere:confidence coefficient is 1 - andt/2 is based on a t distributionwith n - 2 degrees of freedomConfidence Interval Estimate of E(yp)Prediction Interval Estima

29、te of ypUsing the Estimated RegressionIf 3 TV ads are run prior to a sale, we expectthe mean number of cars sold to be:Point Estimationy = 10 + 5(3) = 25 carsIf 3 TV ads are run prior toExcels Confidence Interval OutputConfidence Interval for E(yp)Excels Confidence Interval OuThe 95% confidence inte

30、rval estimate of the mean number of cars sold when 3 TV ads are run is:Confidence Interval for E(yp)25 + 4.61 = 20.39 to 29.61 carsThe 95% confidence interval Excels Prediction Interval OutputPrediction Interval for ypExcels Prediction Interval OuThe 95% prediction interval estimate of the number of cars sold in one particular week when 3 TV ads are run is:Prediction Interval for yp25 + 8.28 = 16.72 to 33.28 carsPrediction Interval for yp25 +Residual Analysis Much of the residual analysis is based on an examination of graphi