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1 1 Slide © 2008 Thomson South-Western. All Rights Reserved Slides by JOHN LOUCKS & Updated by SPIROS VELIANITIS

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2 2 Slide © 2008 Thomson South-Western. All Rights Reserved Chapter 14 Simple Linear Regression n Simple Linear Regression Model n Least Squares Method n Coefficient of Determination n Model Assumptions n Testing for Significance n Using the Estimated Regression Equation for Estimation and Prediction for Estimation and Prediction n Residual Analysis: Validating Model Assumptions n Outliers and Influential Observations

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3 3 Slide © 2008 Thomson South-Western. All Rights Reserved Simple Linear Regression Regression analysis can be used to develop an Regression analysis can be used to develop an equation showing how the variables are related. equation showing how the variables are related. Managerial decisions often are based on the Managerial decisions often are based on the relationship between two or more variables. relationship between two or more variables. The variables being used to predict the value of the The variables being used to predict the value of the dependent variable are called the independent dependent variable are called the independent variables and are denoted by x. variables and are denoted by x. Variation in a variable is explained by another variable. The variable being predicted is called the dependent The variable being predicted is called the dependent variable and is denoted by y. variable and is denoted by y.

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4 4 Slide © 2008 Thomson South-Western. All Rights Reserved Simple Linear Regression The relationship between the two variables is The relationship between the two variables is approximated by a straight line. approximated by a straight line. Simple linear regression involves one independent Simple linear regression involves one independent variable and one dependent variable. variable and one dependent variable. Regression analysis involving two or more Regression analysis involving two or more independent variables is called multiple regression. independent variables is called multiple regression.

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5 5 Slide © 2008 Thomson South-Western. All Rights Reserved Simple Linear Regression Model y = 0 + 1 x + where: 0 and 1 are called parameters of the model, is a random variable called the error term. is a random variable called the error term. The simple linear regression model is: The simple linear regression model is: The equation that describes how y is related to x and The equation that describes how y is related to x and an error term is called the regression model. an error term is called the regression model.

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6 6 Slide © 2008 Thomson South-Western. All Rights Reserved Simple Linear Regression Equation n The simple linear regression equation is: E ( y ) is the expected value of y for a given x value. E ( y ) is the expected value of y for a given x value. 1 is the slope of the regression line. 1 is the slope of the regression line. 0 is the y intercept of the regression line. 0 is the y intercept of the regression line. Graph of the regression equation is a straight line. Graph of the regression equation is a straight line. E ( y ) = 0 + 1 x

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7 7 Slide © 2008 Thomson South-Western. All Rights Reserved Simple Linear Regression Equation n Positive Linear Relationship E(y)E(y)E(y)E(y) x Slope 1 is positive Regression line Intercept 0

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8 8 Slide © 2008 Thomson South-Western. All Rights Reserved Simple Linear Regression Equation n Negative Linear Relationship E(y)E(y)E(y)E(y) x Slope 1 is negative Regression line Intercept 0

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9 9 Slide © 2008 Thomson South-Western. All Rights Reserved Simple Linear Regression Equation n No Relationship E(y)E(y)E(y)E(y) x Slope 1 is 0 Regression line Intercept 0

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10 Slide © 2008 Thomson South-Western. All Rights Reserved Estimated Simple Linear Regression Equation n The estimated simple linear regression equation is the estimated value of y for a given x value. is the estimated value of y for a given x value. b 1 is the slope of the line. b 1 is the slope of the line. b 0 is the y intercept of the line. b 0 is the y intercept of the line. The graph is called the estimated regression line. The graph is called the estimated regression line.

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11 Slide © 2008 Thomson South-Western. All Rights Reserved Estimation Process Regression Model y = 0 + 1 x + Regression Equation E ( y ) = 0 + 1 x Unknown Parameters 0, 1 Sample Data: x y x 1 y 1...... x n y n b 0 and b 1 provide estimates of 0 and 1 Estimated Regression Equation Sample Statistics b 0, b 1

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12 Slide © 2008 Thomson South-Western. All Rights Reserved Least Squares Method n The least squares method is a procedure for using sample data to find the estimated regression equation n Least Squares Criterion where: y i = observed value of the dependent variable for the i th observation for the i th observation^ y i = estimated value of the dependent variable for the i th observation for the i th observation

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13 Slide © 2008 Thomson South-Western. All Rights Reserved n Slope for the Estimated Regression Equation is calculated using Differential Calculus aid is: Least Squares Method where: x i = value of independent variable for i th observation observation_ y = mean value for dependent variable _ x = mean value for independent variable y i = value of dependent variable for i th observation observation

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14 Slide © 2008 Thomson South-Western. All Rights Reserved Reed 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 below. Simple Linear Regression n Example: Reed Auto Sales Number of TV Ads ( x ) TV Ads ( x ) Number of Cars Sold ( y ) 132131424181727 x = 10 y = 100

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15 Slide © 2008 Thomson South-Western. All Rights Reserved Scatter Diagram and Trend Line

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16 Slide © 2008 Thomson South-Western. All Rights Reserved Coefficient of Determination n How well does the estimated regression equation fit the data? The coefficient of determination provides a measure of goodness of fit for the estimated regression equation. SSE is the sum of squares due to error sums the residuals or errors. n Relationship Among SST, SSR, SSE where: SST = total sum of squares SST = total sum of squares SSR = sum of squares due to regression SSR = sum of squares due to regression SSE = sum of squares due to error SSE = sum of squares due to error SST = SSR + SSE n The coefficient of determination is: r 2 = SSR/SST

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17 Slide © 2008 Thomson South-Western. All Rights Reserved Coefficient of Determination r 2 = SSR/SST = 100/114 =.8772 The regression relationship is very strong; 87.7% The regression relationship is very strong; 87.7% of the variability in the number of cars sold can be explained by the linear relationship between the number of TV ads and the number of cars sold.

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18 Slide © 2008 Thomson South-Western. All Rights Reserved Sample Correlation Coefficient where: b 1 = the slope of the estimated regression b 1 = the slope of the estimated regression equation equation n The correlation coefficient is a descriptive measure of the strength of a linear equation between two variables x and y. Values of the correlation coefficient are always between -1 (negative or inverse relation) and +1 (positive relation). Zero (0), or close to zero, indicates no relationship.

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19 Slide © 2008 Thomson South-Western. All Rights Reserved The sign of b 1 in the equation is “+”. Sample Correlation Coefficient r xy = +.9366

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20 Slide © 2008 Thomson South-Western. All Rights Reserved Testing for Significance To test for a significant regression relationship, we must conduct a To test for a significant regression relationship, we must conduct a hypothesis test to determine whether the value of 1 is zero hypothesis test to determine whether the value of 1 is zero because if 1 is zero, we would conclude that the two variables are not related. Also, if 1 is not zero the two variables are related. To test for a significant regression relationship, we must conduct a To test for a significant regression relationship, we must conduct a hypothesis test to determine whether the value of 1 is zero hypothesis test to determine whether the value of 1 is zero because if 1 is zero, we would conclude that the two variables are not related. Also, if 1 is not zero the two variables are related. Two tests are commonly used: Two tests are commonly used: t Test and F Test Both the t test and F test require an estimate of 2, the variance Both the t test and F test require an estimate of 2, the variance of in the regression model. Both the t test and F test require an estimate of 2, the variance Both the t test and F test require an estimate of 2, the variance of in the regression model.

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21 Slide © 2008 Thomson South-Western. All Rights Reserved An Estimate of 2 An Estimate of 2 Testing for Significance s 2 = MSE = SSE/( n 2) The mean square error (MSE) provides the estimate of 2, and the notation s 2 is also used. where: SSE = sum of squares due to error SSE = sum of squares due to error

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22 Slide © 2008 Thomson South-Western. All Rights Reserved n Hypotheses Testing for Significance: t Test n Rejection Rule where: t is based on a t distribution with n - 2 degrees of freedom Reject H 0 if p -value < or t t

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23 Slide © 2008 Thomson South-Western. All Rights Reserved 1. Determine the hypotheses. 2. Specify the level of significance. 3. Select the test statistic. =.05 4. State the rejection rule. Reject H 0 if p -value <.05 or | t| > 3.182 (with 3 degrees of freedom) Testing for Significance: t Test

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24 Slide © 2008 Thomson South-Western. All Rights Reserved Testing for Significance: t Test 5. Compute the value of the test statistic. 6. Determine whether to reject H 0. t = 4.541 provides an area of.01 in the upper tail. Hence, the p -value is less than.02. (Also, t = 4.63 > 3.182.) We can reject H 0.

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25 Slide © 2008 Thomson South-Western. All Rights Reserved Confidence Interval for 1 H 0 is rejected if the hypothesized value of 1 is not H 0 is rejected if the hypothesized value of 1 is not included in the confidence interval for 1. included in the confidence interval for 1. We can use a 95% confidence interval for 1 to test We can use a 95% confidence interval for 1 to test the hypotheses just used in the t test. the hypotheses just used in the t test.

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26 Slide © 2008 Thomson South-Western. All Rights Reserved The form of a confidence interval for 1 is: The form of a confidence interval for 1 is: Confidence Interval for 1 where is the t value providing an area of /2 in the upper tail of a t distribution with n - 2 degrees of freedom b 1 is the pointestimator is the margin of error

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27 Slide © 2008 Thomson South-Western. All Rights Reserved Confidence Interval for 1 Reject H 0 if 0 is not included in the confidence interval for 1. 0 is not included in the confidence interval. Reject H 0 = 5 +/- 3.182(1.08) = 5 +/- 3.44 or 1.56 to 8.44 n Rejection Rule 95% Confidence Interval for 1 95% Confidence Interval for 1 n Conclusion

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28 Slide © 2008 Thomson South-Western. All Rights Reserved n Hypotheses n Test Statistic Testing for Significance: F Test F = MSR/MSE

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29 Slide © 2008 Thomson South-Western. All Rights Reserved n Rejection Rule Testing for Significance: F Test where: F is based on an F distribution with 1 degree of freedom in the numerator and n - 2 degrees of freedom in the denominator Reject H 0 if p -value < p -value < or F > F

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30 Slide © 2008 Thomson South-Western. All Rights Reserved 1. Determine the hypotheses. 2. Specify the level of significance. 3. Select the test statistic. =.05 4. State the rejection rule. Reject H 0 if p -value <.05 or F > 10.13 (with 1 d.f. in numerator and 3 d.f. in denominator) 3 d.f. in denominator) Testing for Significance: F Test F = MSR/MSE

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31 Slide © 2008 Thomson South-Western. All Rights Reserved Testing for Significance: F Test 5. Compute the value of the test statistic. 6. Determine whether to reject H 0. 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, we reject H 0. 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, we reject H 0. F = MSR/MSE = 100/4.667 = 21.43 The statistical evidence is sufficient to conclude The statistical evidence is sufficient to conclude that we have a significant relationship between the number of TV ads aired and the number of cars sold.

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32 Slide © 2008 Thomson South-Western. All Rights Reserved Some Cautions about the Interpretation of Significance Tests Just because we are able to reject H 0 : 1 = 0 and Just because we are able to reject H 0 : 1 = 0 and demonstrate statistical significance does not enable demonstrate statistical significance does not enable us to conclude that there is a linear relationship between x and y. Rejecting H 0 : 1 = 0 and concluding that the Rejecting H 0 : 1 = 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.

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33 Slide © 2008 Thomson South-Western. All Rights Reserved If 3 TV ads are run prior to a sale, we expect the mean number of cars sold to be: Point Estimation ^ y = 10 + 5(3) = 25 cars

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34 Slide © 2008 Thomson South-Western. All Rights Reserved The 95% confidence interval estimate of the mean number of cars sold when 3 TV ads are run is: Confidence Interval for E ( y p ) 25 + 4.61 25 + 3.1824(1.4491) 20.39 to 29.61 cars

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35 Slide © 2008 Thomson South-Western. All Rights Reserved The 95% prediction interval estimate of the number of cars sold in one particular week when 3 TV ads are run is: Prediction Interval for y p 25 + 8.28 25 + 3.1824(2.6013) 16.72 to 33.28 cars

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36 Slide © 2008 Thomson South-Western. All Rights Reserved Residual Analysis Much of the residual analysis is based on an Much of the residual analysis is based on an examination of graphical plots. examination of graphical plots. Residual for Observation i Residual for Observation i The residuals provide the best information about . The residuals provide the best information about . If the assumptions about the error term appear If the assumptions about the error term appear questionable, the hypothesis tests about the questionable, the hypothesis tests about the significance of the regression relationship and the significance of the regression relationship and the interval estimation results may not be valid. interval estimation results may not be valid.

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37 Slide © 2008 Thomson South-Western. All Rights Reserved Residual Plot Against x If the assumption that the variance of is the same for all values of x is valid, and the assumed regression model is an adequate representation of the relationship between the variables, then If the assumption that the variance of is the same for all values of x is valid, and the assumed regression model is an adequate representation of the relationship between the variables, then The residual plot should give an overall The residual plot should give an overall impression of a horizontal band of points impression of a horizontal band of points

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38 Slide © 2008 Thomson South-Western. All Rights Reserved x 0 Good Pattern Residual Residual Plot Against x

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39 Slide © 2008 Thomson South-Western. All Rights Reserved Residual Plot Against x x 0 Residual Nonconstant Variance

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40 Slide © 2008 Thomson South-Western. All Rights Reserved Residual Plot Against x x 0 Residual Model Form Not Adequate

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41 Slide © 2008 Thomson South-Western. All Rights Reserved n Residuals Residual Plot Against x

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42 Slide © 2008 Thomson South-Western. All Rights Reserved Residual Plot Against x

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43 Slide © 2008 Thomson South-Western. All Rights Reserved Standardized Residual Plot The standardized residual plot can provide insight about the assumption that the error term has a normal distribution. The standardized residual plot can provide insight about the assumption that the error term has a normal distribution. n If this assumption is satisfied, the distribution of the standardized residuals should appear to come from a standard normal probability distribution.

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44 Slide © 2008 Thomson South-Western. All Rights Reserved n Standardized Residuals Standardized Residual Plot ObservationPredicted YResidualsStandard Residuals 115 -0.535 225 -0.535 320-2 -1.069 415 2 1.069 525 2 1.069

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45 Slide © 2008 Thomson South-Western. All Rights Reserved n Standardized Residual Plot Standardized Residual Plot

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46 Slide © 2008 Thomson South-Western. All Rights Reserved Standardized Residual Plot All of the standardized residuals are between –1.5 and +1.5 indicating that there is no reason to question the assumption that has a normal distribution. All of the standardized residuals are between –1.5 and +1.5 indicating that there is no reason to question the assumption that has a normal distribution.

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47 Slide © 2008 Thomson South-Western. All Rights Reserved Outliers and Influential Observations n Detecting Outliers An outlier is an observation that is unusual in comparison with the other data. An outlier is an observation that is unusual in comparison with the other data. Minitab classifies an observation as an outlier if its standardized residual value is +2. Minitab classifies an observation as an outlier if its standardized residual value is +2. This standardized residual rule sometimes fails to identify an unusually large observation as being an outlier. This standardized residual rule sometimes fails to identify an unusually large observation as being an outlier. This rule’s shortcoming can be circumvented by using studentized deleted residuals. This rule’s shortcoming can be circumvented by using studentized deleted residuals. The | i th studentized deleted residual| will be larger than the | i th standardized residual|. The | i th studentized deleted residual| will be larger than the | i th standardized residual|.

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