1. Part III - Forecast Confidence Intervals Chapter 4 Statistics for Managers Using Microsoft Excel, 7e © 2014 Pearson Prentice-Hall, Inc. Philip A. Vaccaro, PhD MGMT E-5070

2. Confidence Interval Estimates  A confidence interval estimate can be developed to make inferences about the predicted average value of ‘ Y ‘.  Increased variation around the line of regression, as measured by the standard error of the estimate, results in a wider interval.  Increased sample size reduces the width of the interval.  The width of the estimate varies at different values of ‘ X ‘.

3. Confidence Interval Estimates  When predicting Y for values of X that are close to the mean of X, the confidence interval is much narrower than for predictions for X values more distant from the mean.  In other words, the interval estimate of the true mean of Y varies as a function of the closeness of the given X to the mean of X.  When predictions are to be made for X values that are distant from the mean of X, a much wider interval will occur.

4. The Regression Line average value of X average value of Y

5. Where:

7. Requirements 7. Compute the coefficient of correlation ( r ). 8. Set up a 95% and 99% confidence interval estimate of the average annual sales volume in a city in which eight ( 8 ) ads are broadcast daily. 9. At the a =.01 and.05 level of significance, is there a relationship between sales volume and the number of radio ads broadcast? 10. Set up the 99% confidence interval estimate of the true slope. 11. Discuss why you should not predict annual sales volume in a city which has fewer than 7 broadcasts daily or more than 14 daily.

8. Prediction Yi = b + b Xi o1 ^ = - 2,107.14 + ( 910.714 ) ( 8 ) = - 2,107.14 + 7,285.712 = 5,178.572 The predicted average number of annual sales is 5,178.572 when 8 radio ads are broadcast daily. SIMPLE LINEAR REGRESSION MODEL ( sample based ) This is a point estimate only ! We need to construct a confidence interval around it!

9. 95% Confidence Interval Estimate Y i +/- t Syx √ hihi n - 2 ^ where: hi = ( Xi – X ) - 2 1 n + ΣXi 2 i = 1 n - ΣXi 2 i = 1 n n Yi = 5,179 sales when Xi = 8 radio ads are broadcast ^ The critical value of the ‘ t ’ statistic, n - 2 degrees of freedom

10. Critical Values of ‘ t ‘ statistic ‘ t ‘, 95% confidence interval estimate with 8 degrees of freedom, 5% level of significance, 2 - tailed test ≈ 2.31

11. Regression Calculations CityXi ( Ads )Yi ( Sales )XiXi YiYi XiYiXiYi A118,00012164,000,00088,000 B75,0004925,000,00035,000 C129,00014481,000,000108,000 D84,0006416,000,00032,000 E108,00010064,000,00080,000 F1310,000169100,000,000130,000 G85,0006425,000,00040,000 H107,00010049,000,00070,000 I1410,000196100,000,000140,000 J74,0004916,000,00028,000 Σ = 100Σ = 70,000Σ = 1,056 Σ = 540,000,000 Σ = 751,000 22 RED THE CALCULATIONS IN RED ARE USED IN THE FORMULAS _ _ X = 10 Y = 7,000 n = 10

12. 95% Confidence Interval Estimate Y i +/- t Syx √ hihi n - 2 ^ where: hi = ( Xi – X ) - 2 1 n + ΣXi 2 i = 1 n - ΣXi 2 i = 1 n n Yi = 5,179 sales when Xi = 8 radio ads are broadcast ^ The critical value of the ‘ t ’ statistic, n - 2 degrees of freedom 8 ads daily average number of ads run 10 cities in sample standard error of the estimate

13. 95% Confidence Interval Estimate 5,179 + / - 2.3060 ( 666.48 ) 1 ( 8 – 10 ) 1,056 -( 100 ) 10 √ + 2 2 5,179 + / - 1,537 √.10 + 4 56 5,179 + / - 1,537 √.171429 5,179 + / - ( 1,537 ) (.4140 ) 5,179 + / - 636.32 4,542.68 ≤ μ yx ≤ 5,815.32 We are 95% confident that the true number of sales, when 8 radio ads are broadcast daily, is between 4,543 and 5,815 units ( t ≈ 2.31 or precisely )

14. 95% Confidence Interval Estimate Y i +/- t Syx √ 1 + h i n - 2 ^ where: hi = ( Xi – X ) - 2 1 n + ΣXi 2 i = 1 n - ΣXi 2 i = 1 n n Yi = 5,179 sales when Xi = 8 radio ads broadcast ^ For an individual city when eight (8) radio ads are broadcast daily

15. FOR AN INDIVIDUAL CITY WHEN 8 RADIO ADS ARE BROADCAST DAILY 5,179 + / - 2.3060 ( 666.48 ) 1 ( 8 -10 ) 1,056 -( 100 ) 10 √ + 2 2 5,179 + / - 1,537 √ 1 +.10 + 4 56 5,179 + / - 1,537 √ 1.1 +.07142 5,179 + / - ( 1,537 ) ( 1.17142 ) 5,179 + / - 1,800.47 3,378.53 ≤ Yi ≤ 6,979.47 1 + ^ We are 95% confident that the average sales in an individual city, when 8 radio ads are broadcast daily is between 3,379 and 6,979 units ( t ≈ 2.31 or precisely ) 95% Confidence Interval Estimate

16. Y i +/- t Syx √ hihi n - 2 ^ where: hi = ( Xi – X ) - 2 1 n + ΣXi 2 i = 1 n - ΣXi 2 i = 1 n n Recall that 5,179 are the average sales when eight (8) radio ads are broadcast daily 99% Confidence Interval Estimate

17. Critical Values of ‘ t ‘ Statistic ‘ t ‘, 99% confidence interval estimate with 8 degrees of freedom, 1% level of significance, 2 - tailed test ≈ 3.36

18. 5,179 + / - 3.3554 ( 666.48 ) 1( 8 - 10 ) 1,056 - ( 100 ) 10 √ + 2 2 5,179 + / - 2,236.3069 √.10 + 4 56 5,179 + / - 2,236.3069 √.171429 5,179 + / - 2,236.3069 (.4140 ) 5,179 + / - 926 4,253 ≤ μ yx ≤ 6,106 We are 99% sure that the true number of sales when 8 radio ads are broadcast daily is between 4,253 and 6,106 units ( t ≈ 3.36 or precisely ) 99% Confidence Interval Estimate

19. Requirements 7. Compute the coefficient of correlation ( r ). 8. Set up a 95% and 99% confidence interval estimate of the average annual sales volume in a city in which eight ( 8 ) ads are broadcast daily. 9. At the a =.01 and.05 level of significance, is there a relationship between sales volume and the number of radio ads broadcast? 10. Set up the 99% confidence interval estimate of the true slope. 11. Discuss why you should not predict annual sales volume in a city which has fewer than 7 broadcasts daily or more than 14 daily.

20. Given : = 910.714 = 89.0623 t = 3.3554 NOTE NOTE: TWO (2) DEGREES OF FREEDOM ARE LOST IN THE PROCESS OF INFERENCE BECAUSE TWO PARAMETER ESTIMATES, b o and b 1 ARE INCLUDED IN THE REGRESSION EQUATION. The estimated value of the regression slope based on sample data The standard deviation of the regression slope based on sample data The critical value of t df = 8 (n-2) a =.01 two-tailed test b 1 S b 1 99% Confidence Interval Estimate of the True Slope ( β )

21. b 1 sb 1 + / -t n - 2 = 910.714 + / - 3.3554 ( 89.0623 ) = 910.714 + / - 298.840 611.874 ≤ β ≤ 1,209.554 1

22. Part III - Forecast Confidence Intervals Chapter 4 Statistics for Managers Using Microsoft Excel, 7e © 2014 Pearson Prentice-Hall, Inc. Philip A. Vaccaro, PhD MGMT E-5070