Session 2

Applied Regression -- Prof. Juran2 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. Juran3 Computer Repair Example

Applied Regression -- Prof. Juran4 Interpreting the Coefficients

Applied Regression -- Prof. Juran5 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. Juran6 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. Juran7 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. Juran8 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. Juran9 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. Juran10 –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 2.24 RABE 2.28 RABE 2.33

Applied Regression -- Prof. Juran11 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. Juran12 –It has standard error –The “ t ” statistic below has a t - distribution with n -2 degrees of freedom

Applied Regression -- Prof. Juran13 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

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Applied Regression -- Prof. Juran16 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: H 0 :  1 = 0 versus H A :  1  0

Applied Regression -- Prof. Juran17 This can be carried out by a 2-sided t -test as follows: Reject H 0 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. Juran18 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? Hypothesis Testing Approach

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Applied Regression -- Prof. Juran24 The p -value

Applied Regression -- Prof. Juran25 What about the intercept? It would appear that our estimated intercept here is not significantly different from zero (see the p -value of ). 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).

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Applied Regression -- Prof. Juran27 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. Juran28 Confidence Interval Approach

Applied Regression -- Prof. Juran29 Excel Formulas

Applied Regression -- Prof. Juran30 Excel Formulas

Applied Regression -- Prof. Juran31 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. Juran32 The Regression Output

Applied Regression -- Prof. Juran33 Hypothesis Testing: Gardening Analogy Rocks Dirt

Applied Regression -- Prof. Juran34 Hypothesis Testing: Gardening Analogy

Applied Regression -- Prof. Juran35 Hypothesis Testing: Gardening Analogy

Applied Regression -- Prof. Juran36 Hypothesis Testing: Gardening Analogy

Applied Regression -- Prof. Juran37 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. Juran38 Stock Betas

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Applied Regression -- Prof. Juran46 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 Summary

Applied Regression -- Prof. Juran47 For Sessions 3 & 4 Organize a Project Team Review your regression notes from core stats Two cases: –All-Around Movers –Manley’s Insurance