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Session 2

Session 2. Outline for Session 2. More Simple Regression Bottom Part of the Output Hypothesis Testing Significance of the slope and intercept parameters Interval Estimation Confidence intervals for the slope and intercept parameters . Computer Repair Example.

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Session 2

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  1. Session 2

  2. Outline for Session 2 • More Simple Regression • Bottom Part of the Output • Hypothesis Testing • Significance of the slope and intercept parameters • Interval Estimation • Confidence intervals for the slope and intercept parameters Applied Regression -- Prof. Juran

  3. Computer Repair Example Applied Regression -- Prof. Juran

  4. Interpreting the Coefficients Applied Regression -- Prof. Juran

  5. Things to remember: • The Y value we might calculate by plugging an X into this equation is only an estimate (we will discuss this more later). • These coefficients are only estimates; they are probably wrong (and we would therefore like to be able to think about how wrong they might be). Applied Regression -- Prof. Juran

  6. Significance of the Coefficients • Are the slope and intercept significantly different from zero? • Can we construct a confidence interval around these coefficients? • We need measures of dispersion for the estimated parameters. Applied Regression -- Prof. Juran

  7. Statistics of the Regression Estimates • If the true model is linear, the regression estimates are unbiased (correct expected value). • If the true model is linear, the errors are uncorrelated, and the residual variance is constant in X, the regression estimates are also efficient (low variance relative to other estimators). Applied Regression -- Prof. Juran

  8. The Statistics of the Regression Estimates • If, in addition, the residuals are normally distributed, the estimates are random variables with distributions related to the t and 2 distributions. This permits a variety of hypothesis tests and confidence and prediction intervals to be computed. • If the sample size is reasonably large and the residuals are not bizarrely non-normal, the hypothesis tests and confidence intervals are good approximations. Applied Regression -- Prof. Juran

  9. Statistics of (estimated slope of the regression line) The true slope of the regression line, 1, is the most critical parameter. Under our full set of assumptions its estimate, , has the following properties: • It is unbiased: • It has variance: RABE 2.21 Applied Regression -- Prof. Juran

  10. RABE 2.24 • It has standard error • The “t” statistic below has a t-distribution with n-2 degrees of freedom. • A 2-sided confidence interval on 1 is RABE 2.28 RABE 2.33 Applied Regression -- Prof. Juran

  11. Statistics of (estimated intercept of the regression line) The true intercept of the regression line, 0, is sometimes of interest. Under our full set of assumptions its estimate, , has the following properties: • It is unbiased: • It has variance Applied Regression -- Prof. Juran

  12. It has standard error • The “t” statistic below has a t-distribution with n-2 degrees of freedom Applied Regression -- Prof. Juran

  13. 4-Step Hypothesis Testing Procedure 1. Formulate Two Hypotheses 2. Select a Test Statistic 3. Derive a Decision Rule 4. Calculate the Value of the Test Statistic; Invoke the Decision Rule in light of the Test Statistic Applied Regression -- Prof. Juran

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  16. Testing the significance of a simple linear regression ( 1 = 0? ) If the slope is zero (or, equivalently, if the correlation is zero) we do not have a relationship. Thus, a fundamental test is: H0:1 = 0 versus HA:1  0 Applied Regression -- Prof. Juran

  17. This can be carried out by a 2-sided t-test as follows: Reject H0 if Equivalently, we can examine whether the confidence interval on 1 contains 0. Note: Parallel tests and confidence intervals exist for 0 . Applied Regression -- Prof. Juran

  18. Hypothesis Testing Approach Is the effect of “number of units repaired” on “minutes” different from zero? In other words, based on our sample data, which of these hypotheses is true? Applied Regression -- Prof. Juran

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  24. The p-value Applied Regression -- Prof. Juran

  25. What about the intercept? It would appear that our estimated intercept here is not significantly different from zero (see the p-value of 0.2385). It is not uncommon in practical situations to ignore a lack of significance in the intercept — the intercept is held to a lower standard of significance than the slope (or slopes). Applied Regression -- Prof. Juran

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  27. Note that: • the zero-intercept line is not much different from the best-fit line • The best- fit line fits our data best (duh!) • We seem to be able to make better predictions in our practical range of data using the best-fit model Applied Regression -- Prof. Juran

  28. Confidence Interval Approach Applied Regression -- Prof. Juran

  29. Excel Formulas Applied Regression -- Prof. Juran

  30. Excel Formulas Applied Regression -- Prof. Juran

  31. Note that both approaches (hypothesis testing and confidence intervals) use the same basic “picture” of sampling error: In the hypothesis testing approach, we center the picture on a hypothesized parameter value and see whether the data are consistent with the hypothesis. In the confidence interval approach, we center the picture on the data, and speculate that the true population parameter is probably nearby. Applied Regression -- Prof. Juran

  32. The Regression Output Applied Regression -- Prof. Juran

  33. Hypothesis Testing: Gardening Analogy Dirt Rocks Applied Regression -- Prof. Juran

  34. Hypothesis Testing: Gardening Analogy Applied Regression -- Prof. Juran

  35. Hypothesis Testing: Gardening Analogy Applied Regression -- Prof. Juran

  36. Hypothesis Testing: Gardening Analogy Applied Regression -- Prof. Juran

  37. Hypothesis Testing: Gardening Analogy Screened out stuff: Correct decision or Type I Error? Stuff that fell through: Correct decision or Type II Error? Applied Regression -- Prof. Juran

  38. Stock Betas Applied Regression -- Prof. Juran

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  46. Summary • More Simple Regression • Bottom Part of the Output • Hypothesis Testing • Significance of the slope and intercept parameters • Interval Estimation • Confidence intervals for the slope and intercept parameters Applied Regression -- Prof. Juran

  47. For Sessions 3 & 4 • Organize a Project Team • Review your regression notes from core • Two cases: • All-Around Movers • Manley’s Insurance Applied Regression -- Prof. Juran

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