Lessons 3 and 4: Statistics• Working with Stata1. Probability Example: CLT in action2. Populations have parameters, Samples have estimators3. Estimators & Estimates4. Parameters have distributions: From Probability to Statistics5. Confidence Intervals for ParametersE.g. Pick a digit6. Hypothesis Testing: terms7. Hypothesis Testing: steps8. P-value9. Properties of Estimators10. Efficiency of Sample Mean11. Monte Carlo Demonstration of CLTCopyright © 2003 by Pearson Education, Inc. 3-21. Probability Example: Confidence Interval for sample mean•Ai - the outcome of some event for individual i E.g. p(lefthanded)=.085• How often should that happen in a class of this size? • frequency of “4”s = N x sample mean of “4”s• Standard error (std. deviation of mean)-V(Ai) = p(Ai=1)[1-p(Ai=1)]• Normal approximation• Confidence interval for sample mean• Application: Fair bet?Copyright © 2003 by Pearson Education, Inc. 3-3Copyright © 2003 by Pearson Education, Inc. 3-4Calculating Confidence interval for leftiesCopyright © 2003 by Pearson Education, Inc. 3-52. Populations have parameters, Samples have estimatorsSample meanSample variancee.g. Pop. meanPop. VarianceEstimatorsParameters(typically Greek)C.P.S.N coin flipse.g. U.S. ResidentsAll coin flipsSamplePopulationCopyright © 2003 by Pearson Education, Inc. 3-63. Estimators and EstimatesCopyright © 2003 by Pearson Education, Inc. 3-74. Parameters have distributions: From Probability to Statistics• Probability: use information from populations to learn about samples (Lefthanded example)• Statistics: use information from samples to learn about populations• How to make the transitionCopyright © 2003 by Pearson Education, Inc. 3-8From Probability to Statistics…Copyright © 2003 by Pearson Education, Inc. 3-95. Confidence Intervals for parameters: Pick a number example• Find the distribution of estimator• Interpret as distribution of parameterCopyright © 2003 by Pearson Education, Inc. 3-10Confidence IntervalsCopyright © 2003 by Pearson Education, Inc. 3-116. Hypothesis testing: termsCopyright © 2003 by Pearson Education, Inc. 3-127. Hypothesis Testing: StepsCopyright © 2003 by Pearson Education, Inc. 3-138. P-value: How likely were we to miss by at least that much, if Ho is true?Copyright © 2003 by Pearson Education, Inc. 3-149. Properties of EstimatorsCopyright © 2003 by Pearson Education, Inc. 3-1510 Efficiency of Sample MeanCopyright © 2003 by Pearson Education, Inc. 3-1611. Monte Carlo Demonstration of CLT• Imagine estimating the mean hourly wage :, by drawing samples of size N from the distribution of hourly wages, say in 1984.• Stata will do this for us, maybe 10,000 times.Copyright © 2003 by Pearson Education, Inc. 3-17N=40, 10,000 drawsFractionm68 10 120.0705Copyright © 2003 by Pearson Education, Inc. 3-18N=60, 10,000 drawsFractionm68 10 120.0786Copyright © 2003 by Pearson Education, Inc. 3-19Copyright © 2003 by Pearson Education, Inc. 3-20The next two slides each present one half of Figure 3.3.Copyright © 2003 by Pearson Education, Inc. 3-21Summary - Lessons 3 and 4: Statistics1. Probability Example: CLT in action2. Populations have parameters, Samples have estimators3. Estimators & Estimates4. Parameters have distributions: From Probability to Statistics5. Confidence Intervals for ParametersE.g. Pick a digit6. Hypothesis Testing: terms7. Hypothesis Testing: steps8. P-value9. Properties of Estimators10. Efficiency of Sample Mean11. Monte Carlo Demonstration of CLTCopyright © 2003 by Pearson Education, Inc. 3-23Appendix