Hypothesis Testing in a Regression ContextReview: What Do These Regression Terms Mean?Topics of the Day…Review: Populations and SamplesSlide 5Review: Populations and SamplesSlide 7Populations and SamplesPopulations and Samples: ExamplePopulations and Samples: Example #2Would Our “b”s Be Uniformly Distributed?Or Would Our “b”s Be Normally Distributed?The Distribution of Sample Regression Coefficients (b)Normality of Sample Regression Coefficients (b)Slide 15Recap to This Point…Slide 17Example: b=85.22, standard error=1.136Reviewing the Steps of Hypothesis TestingSlide 20Example: Education and EarningsSlide 22Slide 23Example: Education and EarningsSlide 25Another Sample Regression…A Graphical OverviewSlide 28Slide 29Slide 30Slide 31Extreme Sample ExampleResearch ExampleSlide 34Research Example: Hypothesis TestingSlide 36Research Example: Hypothesis TestingSlide 38Slide 39Slide 40Slide 41Slide 42Two-Tailed Hypothesis TestsExample: Marriage on EarningsSlide 45Example: Marriage on EarningsSlide 4710% Cut Points in Two-Tailed Tests5% Cut Points in Two-Tailed Tests1% Cut Points in Two-Tailed TestsSummary of Cut PointsSlide 52Slide 53Hypothesis TestingSlide 55Slide 56Slide 57Slide 58Slide 59Slide 60Slide 61Slide 62Hypothesis Testing: Points to RememberMinitab: Does Tests Against Null Automatically in RegressionLimitations of Hypothesis TestingSlide 66Slide 67Slide 68Hypothesis Testing on Small SamplesDegrees of Freedom in Regression ContextsDegrees of Freedom: ExampleHypothesis Testing in a Regression ContextLIR 832Review: What Do These Regression Terms Mean?Regression Analysis: weekearn versus Education, age, female, hoursThe regression equation isweekearn = - 1053 + 65.1 Education + 7.07 age - 230 female + 18.3 hours44839 cases used 10319 cases contain missing valuesPredictor Coef SE Coef T PConstant -1053.01 19.43 -54.20 0.000Educatio 65.089 1.029 63.27 0.000age 7.0741 0.1929 36.68 0.000female -229.786 4.489 -51.19 0.000hours 18.3369 0.2180 84.11 0.000S = 459.0 R-Sq = 31.9% R-Sq(adj) = 31.9%Topics of the Day…1. Populations and samples in the context of regression.2. The distribution of the error term in a regression model.3. Hypothesis testing.One-tailed.Two-tailed.4. Tests of group variables.Review:Populations and SamplesIn earlier lectures, we learned that there is a true parameter in a population () that we try to estimate by a sample (x-bar). In regression, there are similarities.In other words, there is a “true” relationship between the variables in the population that we are trying to estimate: yi = 0 + 1*X1i + iSince we often cannot see the entire population, we estimate this relationship through finding the equation within a sample: ory X ei i i    0 1 1y b b X ei i i  0 1 1Review:Populations and SamplesAs with all sample results, there are lots of different samples which might be drawn from a population. These samples will typically provide somewhat different estimates of the coefficients. This is, once more, a byproduct of sampling variation.Review: Populations and SamplesEstimate a simple regression model of weekly earnings for all of the data on managers and professionals (what we’ll consider as the “population”), then take random 10% sub-samples of the data and compare the estimates. Weekly Earnings = 0 + 1*education + Upon generating five random sub-samples of the data, we find the following:Review: Populations and SamplesEstimate0 (Intercept)1 (Coefficient on Education)POPULATION -484.57 87.49Sample 1 -333.24 79.21Sample 2 -488.51 88.16Sample 3 -460.15 85.93Sample 4 -502.18 88.44Sample 5 -485.19 87.88Populations and SamplesImportant point:Sampling variability is responsible for the fact that the sample regression coefficients do not exactly reproduce the population regression coefficients, which is what we are after.Q: Why does this happen?A: We pull samples out of populations. We typically take a single sample. But, in fact, there are many many samples we could pull from a given population.Populations and Samples: ExampleQ: How many unique samples (no two samples have the exact same individuals) in a population of 15 with sample size 5?A: 1 5 51 55 1 0 !3 0 0 3C !! *,Populations and Samples: Example #2Q: Now suppose we are taking samples of 5,000 from our 50,000-person data set on managers and professionals. How many unique samples can we draw?A: As a result, we have many possible samples upon which to estimate a regression model. This will produce many different sample regression coefficients (b)… which may be different than the population coefficients ().0000000000000000000000000000000000000000000000000000000000003000000000,2500050000C69500050000103.2 CWould Our “b”s Be Uniformly Distributed?00.0050.010.0150.020.0250.030.0350.040.0450.058080.58181.58282.58383.58484.58585.58686.58787.58888.58989.590betaProbabilityOr Would Our “b”s Be Normally Distributed?The Distribution of Sample Regression Coefficients (b)By the Gauss-Markov Theorem, so long that we have 31 degrees of freedom in our regression:A.) The average of b’s is B.) The variance of b declines as sample size becomes large.C.) The b’s are normally distributed.The Gauss-Markov Theorem can be thought of as the Central Limit Theorem for regression; this allows us to engage in inference from samples to populations.Enini()   11Normality of Sample Regression Coefficients (b)Q: What does normality of the sample regression coefficients (b) buy us?A: This allows us to do inference using a normal distribution, as we have already done.Thus, we can use Z transformations and hypothesis testing just as before.Normality of Sample Regression Coefficients (b)Distribution of beta-hatProbability of Beta-hat3210-1-2-343210prob(b-hat)n-=9n=100VariableEffect of Sample Size on Distribution of Estimated Regression CofficientsRecap to This Point…1. We wish to know about population regression (i.e., the “true” model).2. We do not observe the population; instead, we use samples to make inferences about the population and the population regression.3. Samples are characterized by sampling variability.a.) Samples do not exactly reproduce the