Examp ole: District Sales Use both target population and per capita discretionary income to forecast district sales District Sales(gross of jars; Target populationPer capita discretionary ns)(000 persons income($) Y 274 2450 3254 223 375 3802 131 2838 67 86 2347 265 3782 330 2450 l16 2137 11 430 12 232 372 4427 144 2660 14 157 15 212 370 2605 Slide 11
Slide 11 Example: District Sales • Use both target population and per capita discretionary income to forecast district sales. District i Sales (gross of jars; 1 gross = 12 dozens) Yi Target population (‘000 persons) X1i Per capita discretionary income ($) X2i 1 162 274 2450 2 120 180 3254 3 223 375 3802 4 131 205 2838 5 67 86 2347 6 169 265 3782 7 81 98 3008 8 192 330 2450 9 116 195 2137 10 55 53 2560 11 252 430 4020 12 232 372 4427 13 144 236 2660 14 103 157 2088 15 212 370 2605
Examp ole District Sales Excel output SUMMARY OUTPUT Regression Statistics ple R 0.999722 R Square 0998944673 Adjusted R Square.998768791 Standard Error 2.177222343 Observations anova MS 2538447164326922365679466138137E18 Residual 1256883565564740297 Total 539016 Coefficients Standard Error t StatP-value ntercept 345261279243065049314204480.180935 04960049760.00604411819241573E18 000919981000068114950206562E.7 Slide 12
Slide 12 Example: District Sales • Excel output
Exampl ole district sales Multiple regression model Y=Bo+BX1+P2X2+8 where y= district sales XI=target population X=per capita discretionary income Multiple Regression equation Using the assumption E(a)=o, we obtain E()=B0+B1+B2X2 Slide 13
Slide 13 Example: District Sales • Multiple regression model where Y = district sales X1 = target population X2 = per capita discretionary income • Multiple Regression Equation Using the assumption E( ) = 0, we obtain = + + + Y 0 1 X1 2 X2 0 1 1 2 2 E(Y) = + X + X
Exampl ole District Sales Estimated Regression equation bo, b1 b2 are the least squares estimates of Bo Bl B2. Th nus Y=b+6X1+b2X2 For this example Y=34526+0.4960X1+0.0092X2 Predicted sales are expected to increase by 0.496 gross when the target population increases by one thousand, holding per capita discretionary income constant Predicted sales are expected to increase by 0.0092 gross when per capita discretionary income increase by one dollar, holding population constant Slide 14
Slide 14 Example: District Sales • Estimated Regression Equation b0 , b1 , b2 are the least squares estimates of 0 , 1 , 2 . Thus • For this example, – Predicted sales are expected to increase by 0.496 gross when the target population increases by one thousand, holding per capita discretionary income constant. – Predicted sales are expected to increase by 0.0092 gross when per capita discretionary income increase by one dollar, holding population constant. 0 1 1 2 2 Y ˆ = b +b X +b X 1 0092 2 3.4526 0.4960 0. Y ˆ = + X + X
Examp ole District Sales t Test for Significance of Individual Parameters ypothesIs H:B2=0 Ha:B≠0 Decision rule Fora=.05andd.f.=15-2-1=12,to25=2.179 Reject Ho if t>2.179 Test statistic 0.49600 0.00920 81.92 950 0.00605 S40.000968 Conclusions Reject Ho: B=0 Reject Ho: B2=0
Slide 15 Example: District Sales • t Test for Significance of Individual Parameters – Hypothesis – Decision rule For = .05 and d.f. = 15 – 2 – 1 = 12, t.025 = 2.179 Reject H0 if |t| > 2.179 – Test statistic – Conclusions Reject H0 : 1 = 0 Reject H0 : 2 = 0 : 0 0 : 0 = a i i H H 81.92 0.00605 0.49600 1 1 = = = b s b t 9.50 0.000968 0.00920 2 2 = = = b s b t