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 SignificanceUsing the Estimated Regression Equationfor Estimation and PredictionComputer SolutionResidual Analysis: Validating Model Assumptions
2 Simple Linear Regression Model The equation that describes how y is related to x and an error term is called the regression model.The simple linear regression model is:y = b0 + b1x +eb0 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)Regression lineInterceptb0Slope b1is positivex
5 Simple Linear Regression Equation Negative Linear RelationshipE(y)Interceptb0Regression lineSlope b1is negativex
6 Simple Linear Regression Equation No RelationshipE(y)Regression lineInterceptb0Slope b1is 0x
7 Estimated Simple Linear Regression Equation The estimated simple linear regression equation is:The 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 observationyi = estimated value of the dependent variable^
10 The Least Squares Method Slope for the Estimated Regression Equation
11 The Least Squares Method y-Intercept for the Estimated Regression Equationwhere:xi = value of independent variable for ith observationyi = value of dependent variable for ith observationx = mean value for independent variabley = mean value for dependent variablen = total number of observations__
12 Example: Reed Auto Sales Simple Linear RegressionReed Auto periodically has a special week-long sale. As part of the advertising campaign Reed runs one or more television commercials during the weekend preceding the sale. Data from a sample of 5 previous sales are shown on the next slide.
13 Example: Reed Auto Sales Simple Linear RegressionNumber of TV Ads Number of Cars Sold1 143 242 181 173 27
14 Example: Reed Auto Sales Slope for the Estimated Regression Equationb1 = (10)(100)/5 = _____24 - (10)2/5y-Intercept for the Estimated Regression Equationb0 = (2) = _____Estimated Regression Equationy = x^
16 The 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 The Coefficient of Determination The coefficient of determination is:r2 = SSR/SSTwhere:SST = total sum of squaresSSR = sum of squares due to regression
18 Example: Reed Auto Sales Coefficient of Determinationr2 = SSR/SST = 100/114 =The regression relationship is very strong because 88% of the variation in number of cars sold can be explained by the linear relationship between the number of TV ads and the number of cars sold.
19 The Correlation Coefficient Sample Correlation Coefficientwhere:b1 = the slope of the estimated regressionequation
20 Example: Reed Auto Sales Sample Correlation CoefficientThe sign of b1 in the equation is “+”.rxy =
21 Model Assumptions Assumptions About the Error Term The error is a random variable with mean of zero.The variance of , denoted by 2, is the same for all values of the independent variable.The values of are independent.The error is a normally distributed random variable.
22 Testing for Significance To test for a significant regression relationship, we must conduct a hypothesis test to determine whether the value of b1 is zero.Two tests are commonly usedt TestF TestBoth tests require an estimate of s 2, the variance of e in the regression model.
23 Testing for Significance An Estimate of s 2The mean square error (MSE) provides the estimateof s 2, and the notation s2 is also used.s2 = 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 of the estimate.
25 Testing for Significance: t Test HypothesesH0: 1 = 0Ha: 1 = 0Test Statisticwhere
26 Testing for Significance: t Test Rejection RuleReject H0 if t < -tor t > twhere: t is based on a t distributionwith n - 2 degrees of freedom
27 Example: Reed Auto Sales t TestHypothesesH0: 1 = 0Ha: 1 = 0Rejection RuleFor = .05 and d.f. = 3, t.025 = _____Reject H0 if t > t.025 = _____
28 Example: Reed Auto Sales t TestTest Statisticst = _____/_____ = 4.63Conclusionst = 4.63 > 3.182, so reject H0
29 Confidence Interval for 1 We can use a 95% confidence interval for 1 to test the hypotheses just used in the t test.H0 is rejected if the hypothesized value of 1 is not included in the confidence interval for 1.
30 Confidence Interval for 1 The form of a confidence interval for 1 is:where b1 is the point estimateis the margin of erroris the t value providing an areaof a/2 in the upper tail of at distribution with n - 2 degreesof freedom
31 Example: Reed Auto Sales Rejection RuleReject H0 if 0 is not included inthe confidence interval for 1.95% Confidence Interval for 1= 5 +/ (1.08) = 5 +/- 3.44or ____ to ____Conclusion0 is not included in the confidence interval.Reject H0
32 Testing for Significance: F Test HypothesesH0: 1 = 0Ha: 1 = 0Test StatisticF = MSR/MSE
33 Testing for Significance: F Test Rejection RuleReject H0 if F > Fwhere: F is based on an F distributionwith 1 d.f. in the numerator andn - 2 d.f. in the denominator
34 Example: Reed Auto Sales F TestHypothesesH0: 1 = 0Ha: 1 = 0Rejection RuleFor = .05 and d.f. = 1, 3: F.05 = ______Reject H0 if F > F.05 = ______.
35 Example: Reed Auto Sales F TestTest StatisticF = MSR/MSE = ____ / ______ = 21.43ConclusionF = > 10.13, so we reject H0.
36 Some Cautions about the Interpretation of Significance Tests Rejecting H0: b1 = 0 and concluding that the relationship between x and y is significant does not enable us to conclude that a cause-and-effect relationship is present between x and y.Just because we are able to reject H0: b1 = 0 and demonstrate statistical significance does not enable us to conclude that there is a linear relationship between x and y.
37 Using the Estimated Regression Equation for Estimation and Prediction Confidence Interval Estimate of E(yp)Prediction Interval Estimate of ypyp + t/2 sindwhere: confidence coefficient is 1 - andt/2 is based on a t distributionwith n - 2 degrees of freedom
38 Example: Reed Auto Sales Point EstimationIf 3 TV ads are run prior to a sale, we expect the mean number of cars sold to be:y = (3) = ______ cars^
39 Example: Reed Auto Sales Confidence Interval for E(yp)95% confidence interval estimate of the mean number of cars sold when 3 TV ads are run is:= ______ to _______ cars
40 Example: Reed Auto Sales Prediction Interval for yp95% prediction interval estimate of the number of cars sold in one particular week when 3 TV ads are run is:= _____ to ______ cars
41 Residual Analysis Residual for Observation i yi – yi Standardized Residual for Observation iwhere:and^^^^