1. Statistics for Business and Economics 8 th Edition Chapter 11 Simple Regression Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-1

2. Statistical Inference: Hypothesis Tests and Confidence Intervals The variance of the regression slope coefficient (b 1 ) is estimated by where: = Estimate of the standard error of the least squares slope = Standard error of the estimate Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-2 11.5

3. Excel Output Regression Statistics Multiple R0.76211 R Square0.58082 Adjusted R Square0.52842 Standard Error41.33032 Observations10 ANOVA dfSSMSFSignificance F Regression118934.9348 11.08480.01039 Residual813665.56521708.1957 Total932600.5000 CoefficientsStandard Errort StatP-valueLower 95%Upper 95% Intercept98.2483358.033481.692960.12892-35.57720232.07386 Square Feet0.109770.032973.329380.010390.033740.18580 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-3

4. Comparing Standard Errors of the Slope Y X Y X is a measure of the variation in the slope of regression lines from different possible samples Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-4

5. Inference about the Slope: t Test t test for a population slope Is there a linear relationship between X and Y? Null and alternative hypotheses H 0 : β 1 = 0(no linear relationship) H 1 : β 1  0(linear relationship does exist) Test statistic where: b 1 = regression slope coefficient β 1 = hypothesized slope s b1 = standard error of the slope Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-5

6. Inference about the Slope: t Test House Price in $1000s (y) Square Feet (x) 2451400 3121600 2791700 3081875 1991100 2191550 4052350 3242450 3191425 2551700 Estimated Regression Equation: The slope of this model is 0.1098 Does square footage of the house significantly affect its sales price? (continued) Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-6

7. Inferences about the Slope: t Test Example H 0 : β 1 = 0 H 1 : β 1  0 From Excel output: CoefficientsStandard Errort StatP-value Intercept98.2483358.033481.692960.12892 Square Feet0.109770.032973.329380.01039 t b1b1 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-7

8. Inferences about the Slope: t Test Example H 0 : β 1 = 0 H 1 : β 1  0 Test Statistic: t = 3.329 There is sufficient evidence that square footage affects house price From Excel output: Reject H 0 CoefficientsStandard Errort StatP-value Intercept98.2483358.033481.692960.12892 Square Feet0.109770.032973.329380.01039 tb1b1 Decision: Conclusion: Reject H 0  /2=.025 -t n-2,α/2 Do not reject H 0 0  /2=.025 -2.30602.3060 3.329 d.f. = 10-2 = 8 t 8,.025 = 2.3060 (continued) t n-2,α/2 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-8

9. Inferences about the Slope: t Test Example H 0 : β 1 = 0 H 1 : β 1  0 P-value = 0.01039 There is sufficient evidence that square footage affects house price From Excel output: Reject H 0 CoefficientsStandard Errort StatP-value Intercept98.2483358.033481.692960.12892 Square Feet0.109770.032973.329380.01039 P-value Decision: P-value < α so Conclusion: (continued) This is a two-tail test, so the p-value is P(t > 3.329)+P(t < -3.329) = 0.01039 (for 8 d.f.) Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-9

10. Confidence Interval Estimate for the Slope Confidence Interval Estimate of the Slope: Excel Printout for House Prices: At 95% level of confidence, the confidence interval for the slope is (0.0337, 0.1858) CoefficientsStandard Errort StatP-valueLower 95%Upper 95% Intercept98.2483358.033481.692960.12892-35.57720232.07386 Square Feet0.109770.032973.329380.010390.033740.18580 d.f. = n - 2 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-10

11. Confidence Interval Estimate for the Slope Since the units of the house price variable is $1000s, we are 95% confident that the average impact on sales price is between $33.70 and $185.80 per square foot of house size CoefficientsStandard Errort StatP-valueLower 95%Upper 95% Intercept98.2483358.033481.692960.12892-35.57720232.07386 Square Feet0.109770.032973.329380.010390.033740.18580 This 95% confidence interval does not include 0. Conclusion: There is a significant relationship between house price and square feet at the.05 level of significance (continued) Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-11

12. Hypothesis Test for Population Slope Using the F Distribution F Test statistic: where where F follows an F distribution with k numerator and (n – k - 1) denominator degrees of freedom (k = the number of independent variables in the regression model) Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-12

13. Hypothesis Test for Population Slope Using the F Distribution An alternate test for the hypothesis that the slope is zero: Use the F statistic The decision rule is reject H 0 if F ≥ F 1,n-2,α Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-13 (continued) H 0 : β 1 = 0 H 1 : β 1  0

14. Excel Output Regression Statistics Multiple R0.76211 R Square0.58082 Adjusted R Square0.52842 Standard Error41.33032 Observations10 ANOVA dfSSMSFSignificance F Regression118934.9348 11.08480.01039 Residual813665.56521708.1957 Total932600.5000 CoefficientsStandard Errort StatP-valueLower 95%Upper 95% Intercept98.2483358.033481.692960.12892-35.57720232.07386 Square Feet0.109770.032973.329380.010390.033740.18580 With 1 and 8 degrees of freedom P-value for the F-Test Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-14

15. F-Test for Significance H 0 : β 1 = 0 H 1 : β 1 ≠ 0  =.05 df 1 = 1 df 2 = 8 Test Statistic: Decision: Conclusion: Reject H 0 at  = 0.05 There is sufficient evidence that house size affects selling price 0  =.05 F.05 = 5.32 Reject H 0 Do not reject H 0 Critical Value: F 1,8,0.05 = 5.32 (continued) F Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-15