Chapter 14 (Part A) Simple Linear Regression PDF

Statistics for Business and Economics (13e) Statistics for Slides by Johnand Economics (13e) Business Loucks Anderson, Sweeney, Williams, Camm, Cochran St. Edward’s © 2017 Cengage Learning University Slides by John Loucks St. Edwards Universit...

Statistics for Business and Economics (13e) Statistics for Slides by Johnand Economics (13e) Business Loucks Anderson, Sweeney, Williams, Camm, Cochran St. Edward’s © 2017 Cengage Learning University Slides by John Loucks St. Edwards University © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 1 otherwise on a password-protected website or school-approved learning management system for classroom use..  Statistics for Business and Economics (13e) Chapter 14, Part A: Simple Linear Regression Simple Linear Regression Model Least Squares Method Coefficient of Determination Model Assumptions Testing for Significance © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 2 otherwise on a password-protected website or school-approved learning management system for classroom use..  Statistics for Business and Economics (13e) Simple Linear Regression Managerial decisions often are based on the relationship between two or more variables. Regression analysis can be used to develop an equation showing how the variables are related. The variable being predicted is called the dependent variable and is denoted by y. The variables being used to predict the value of the dependent variable are called the independent variables and are denoted by x. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 3 otherwise on a password-protected website or school-approved learning management system for classroom use..  Statistics for Business and Economics (13e) Simple Linear Regression Simple linear regression involves one independent variable and one dependent variable. The relationship between the two variables is approximated by a straight line. Regression analysis involving two or more independent variables is called multiple regression. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 4 otherwise on a password-protected website or school-approved learning management system for classroom use..  Statistics for Business and Economics (13e) 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 + e where: b0 and b1 are called parameters of the model. e is a random variable called the error term. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 5 otherwise on a password-protected website or school-approved learning management system for classroom use..  Statistics for Business and Economics (13e) Simple Linear Regression Equation The simple linear regression equation is: E(y) = b0 + b1x Graph 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. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 6 otherwise on a password-protected website or school-approved learning management system for classroom use..  Statistics for Business and Economics (13e) Simple Linear Regression Equation Positive Linear Relationship E(y) Regression line Intercept Slope b1 b0 is positive x © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 7 otherwise on a password-protected website or school-approved learning management system for classroom use..  Statistics for Business and Economics (13e) Simple Linear Regression Equation Negative Linear Relationship E(y) Intercept b0 Regression line Slope b1 is negative x © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 8 otherwise on a password-protected website or school-approved learning management system for classroom use..  Statistics for Business and Economics (13e) Simple Linear Regression Equation No Relationship E(y) Intercept Regression line b0 Slope b1 is 0 x © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 9 otherwise on a password-protected website or school-approved learning management system for classroom use..  Statistics for Business and Economics (13e) Estimated Simple Linear Regression Equation The estimated simple linear regression equation 𝑦 = 𝑏0 + 𝑏1 𝑥 The graph is called the estimated regression line. b0 is the y intercept of the line. b1 is the slope of the line. 𝑦 is the estimated value of y for a given x value. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 10 otherwise on a password-protected website or school-approved learning management system for classroom use..  Statistics for Business and Economics (13e) Estimation Process Regression Model Sample Data: y = b0 + b1x +e x y Regression Equation x1 y1 E(y) = b0 + b1x.. Unknown Parameters.. b0, b1 xn yn Estimated b0 and b1 Regression Equation provide estimates of 𝑦 = 𝑏0 + 𝑏1 𝑥 b0 and b1 Sample Statistics b 0, b 1 © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 11 otherwise on a password-protected website or school-approved learning management system for classroom use..  Statistics for Business and Economics (13e) Least Squares Method Least Squares Criterion min (𝑦𝑖 − 𝑦𝑖 )2 where: yi = observed value of the dependent variable for the ith observation 𝑦𝑖 = estimated value of the dependent variable for the ith observation © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 12 otherwise on a password-protected website or school-approved learning management system for classroom use..  Statistics for Business and Economics (13e) Least Squares Method Slope for the Estimated Regression Equation 𝑥𝑖 − 𝑥 𝑦𝑖 − 𝑦 𝑏1 = 𝑥𝑖 − 𝑥 2 where: xi = value of independent variable for ith observation yi = value of dependent variable for ith observation 𝑥 = mean value for independent variable 𝑦 = mean value for dependent variable © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 13 otherwise on a password-protected website or school-approved learning management system for classroom use..  Statistics for Business and Economics (13e) Least Squares Method y-Intercept for the Estimated Regression Equation 𝑏0 = 𝑦 − 𝑏1 𝑥 © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 14 otherwise on a password-protected website or school-approved learning management system for classroom use..  Statistics for Business and Economics (13e) Simple Linear Regression Example: Reed Auto Sales 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 on the next slide. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 15 otherwise on a password-protected website or school-approved learning management system for classroom use..  Statistics for Business and Economics (13e) Simple Linear Regression Example: Reed Auto Sales Number of Number of TV Ads (x) Cars Sold (y) 1 14 3 24 2 18 1 17 3 27 Sx = 10 Sy = 100 𝑥=2 𝑦 = 20 © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 16 otherwise on a password-protected website or school-approved learning management system for classroom use..  Statistics for Business and Economics (13e) Estimated Regression Equation Slope for the Estimated Regression Equation 𝑥𝑖 − 𝑥 𝑦𝑖 − 𝑦 20 𝑏1 = 2 = =5 𝑥𝑖 − 𝑥 4 y-Intercept for the Estimated Regression Equation 𝑏0 = 𝑦 − 𝑏1 𝑥 = 20 − 5 2 = 10 Estimated Regression Equation 𝑦 = 10 + 5𝑥 © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 17 otherwise on a password-protected website or school-approved learning management system for classroom use..  Statistics for Business and Economics (13e) Using Excel’s Chart Tools for Scatter Diagram & Estimated Regression Equation Reed Auto Sales Estimated Regression Line 30 25 20 Cars Sold y = 5x + 10 15 10 5 0 0 1 2 3 4 TV Ads © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 18 otherwise on a password-protected website or school-approved learning management system for classroom use..  Statistics for Business and Economics (13e) Coefficient of Determination Relationship Among SST, SSR, SSE SST = SSR + SSE 𝑦𝑖 − 𝑦 2 = 𝑦𝑖 − 𝑦 2 + 𝑦𝑖 − 𝑦𝑖 2 where: SST = total sum of squares SSR = sum of squares due to regression SSE = sum of squares due to error © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 19 otherwise on a password-protected website or school-approved learning management system for classroom use..  Statistics for Business and Economics (13e) Coefficient of Determination The coefficient of determination is: r2 = SSR/SST where: SSR = sum of squares due to regression SST = total sum of squares © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 20 otherwise on a password-protected website or school-approved learning management system for classroom use..  Statistics for Business and Economics (13e) Coefficient of Determination r2 = SSR/SST = 100/114 =.8772 The regression relationship is very strong; 87.72% 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. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 21 otherwise on a password-protected website or school-approved learning management system for classroom use..  Statistics for Business and Economics (13e) Using Excel to Compute the Coefficient of Determination Adding r 2 Value to Scatter Diagram Reed Auto Sales Estimated Regression Line 30 25 Cars Sold 20 y = 5x + 10 15 2 R = 0.8772 10 5 0 0 1 2 3 4 TV Ads © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 22 otherwise on a password-protected website or school-approved learning management system for classroom use..  Statistics for Business and Economics (13e) Sample Correlation Coefficient 𝑟𝑥𝑦 = (sign of 𝑏1 ) Coefficient of Determination 𝑟𝑥𝑦 = (sign of 𝑏1 ) 𝑟 2 where: b1 = the slope of the estimated regression equation 𝑦 = 𝑏0 + 𝑏1 𝑥 © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 23 otherwise on a password-protected website or school-approved learning management system for classroom use..  Statistics for Business and Economics (13e) Sample Correlation Coefficient 𝑟𝑥𝑦 = (sign of 𝑏1 ) 𝑟 2 The sign of b1 in the equation 𝑦 = 10 + 5x is "+". 𝑟𝑥𝑦 = +.8772 rxy = +.9366 © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 24 otherwise on a password-protected website or school-approved learning management system for classroom use..  Statistics for Business and Economics (13e) Assumptions About the Error Term e 1. The error e is a random variable with mean of zero. 2. The variance of e , denoted by  2, is the same for all values of the independent variable. 3. The values of e are independent. 4. The error e is a normally distributed random variable. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 25 otherwise on a password-protected website or school-approved learning management system for classroom use..  Statistics for Business and Economics (13e) 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 used: t Test and F Test Both the t test and F test require an estimate of  2, the variance of e in the regression model. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 26 otherwise on a password-protected website or school-approved learning management system for classroom use..  Statistics for Business and Economics (13e) Testing for Significance An Estimate of  2 The mean square error (MSE) provides the estimate of  2, and the notation s2 is also used. s 2 = MSE = SSE/(n - 2) where: SSE= 𝑦𝑖 − 𝑦𝑖 2 = 𝑦𝑖 − 𝑏0 − 𝑏1 𝑥𝑖 2 © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 27 otherwise on a password-protected website or school-approved learning management system for classroom use..  Statistics for Business and Economics (13e) Testing for Significance An Estimate of  To estimate , we take the square root of s2. The resulting s is called the standard error of the estimate. SSE s = MSE = 𝑛−2 © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 28 otherwise on a password-protected website or school-approved learning management system for classroom use..  Statistics for Business and Economics (13e) Testing for Significance: t Test Hypotheses H0: b1 = 0 H a : b1 ≠ 0 Test Statistic 𝑏1 𝑠 𝑡= where 𝑠𝑏1 = 𝑠𝑏1 𝑥𝑖 − 𝑥 2 © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 29 otherwise on a password-protected website or school-approved learning management system for classroom use..  Statistics for Business and Economics (13e) Testing for Significance: t Test Rejection Rule Reject H0 if p-value <  or t < -tor t > t where: t is based on a t distribution with n - 2 degrees of freedom © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 30 otherwise on a password-protected website or school-approved learning management system for classroom use..  Statistics for Business and Economics (13e) Testing for Significance: t Test 1. Determine the hypotheses. H0: b1 = 0 H a : b1 ≠ 0 2. Specify the level of significance.  =.05 𝑏1 3. Select the test statistic. 𝑡= 𝑠𝑏1 4. State the rejection rule. Reject H0 if p-value <.05 or |t| > 3.182 (with 3 degrees of freedom) © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 31 otherwise on a password-protected website or school-approved learning management system for classroom use..  Statistics for Business and Economics (13e) Testing for Significance: t Test 𝑏1 5 5. Compute the value of the test statistic. 𝑡= = = 4.63 𝑠𝑏1 1.08 6. Determine whether to reject H0. 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 H0. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 32 otherwise on a password-protected website or school-approved learning management system for classroom use..  Statistics for Business and Economics (13e) Confidence Interval for b1 We can use a 95% confidence interval for b1 to test the hypotheses just used in the t test. H0 is rejected if the hypothesized value of b1 is not included in the confidence interval for b1. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 33 otherwise on a password-protected website or school-approved learning management system for classroom use..  Statistics for Business and Economics (13e) Confidence Interval for b1 The form of a confidence interval for b1 is: 𝑏1 ± 𝑡𝞪/2 𝑠𝑏1 where b1 is the point estimator, 𝑡𝞪/2 𝑠𝑏1 is the margin of error, and ta/2 is the t value providing an area of /2 in the upper tail of a t distribution with n - 2 degrees of freedom © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 34 otherwise on a password-protected website or school-approved learning management system for classroom use..  Statistics for Business and Economics (13e) Confidence Interval for b1 Rejection Rule Reject H0 if 0 is not included in the confidence interval for b1. 95% Confidence Interval for b1 𝑏1 ± 𝑡𝞪/2 𝑠𝑏1 = 5 +/- 3.182(1.08) = 5 +/- 3.44 or 1.56 to 8.44 Conclusion 0 is not included in the confidence interval. Reject H0 © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 35 otherwise on a password-protected website or school-approved learning management system for classroom use..  Statistics for Business and Economics (13e) Testing for Significance: F Test Hypotheses H0: b1 = 0 H a : b1 ≠ 0 Test Statistic F = MSR/MSE © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 36 otherwise on a password-protected website or school-approved learning management system for classroom use..  Statistics for Business and Economics (13e) Testing for Significance: F Test Rejection Rule Reject H0 if p-value <  or F > F 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 © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 37 otherwise on a password-protected website or school-approved learning management system for classroom use..  Statistics for Business and Economics (13e) Testing for Significance: F Test 1. Determine the hypotheses. H0: b1 = 0 H a : b1 ≠ 0 2. Specify the level of significance.  =.05 3. Select the test statistic. F = MSR/MSE 4. State the rejection rule. Reject H0 if p-value <.05 or F > 10.13 (with 1 d.f. in numerator and 3 d.f. in denominator) © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 38 otherwise on a password-protected website or school-approved learning management system for classroom use..  Statistics for Business and Economics (13e) Testing for Significance: F Test 5. Compute the value of the test statistic. F = MSR/MSE = 100/4.667 = 21.43 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.025. Hence, we reject H0. 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. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 39 otherwise on a password-protected website or school-approved learning management system for classroom use..  Statistics for Business and Economics (13e) 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. © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 40 otherwise on a password-protected website or school-approved learning management system for classroom use..  Statistics for Business and Economics (13e) End of Chapter 14, Part A © 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or 41 otherwise on a password-protected website or school-approved learning management system for classroom use..