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## Appendix C

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**Review of Statistical Inference**Appendix C Prepared by Vera Tabakova, East Carolina University**Appendix C: Review of Statistical Inference**• C.1 A Sample of Data • C.2 An Econometric Model • C.3 Estimating the Mean of a Population • C.4 Estimating the Population Variance and Other Moments • C.5 Interval Estimation Principles of Econometrics, 3rd Edition**Appendix C: Review of Statistical Inference**• C.6 Hypothesis Tests About a Population Mean • C.7 Some Other Useful Tests • C.8 Introduction to Maximum Likelihood Estimation • C.9 Algebraic Supplements Principles of Econometrics, 3rd Edition**C.1 A Sample of Data**Principles of Econometrics, 3rd Edition**C.1 A Sample of Data**Figure C.1 Histogram of Hip Sizes Principles of Econometrics, 3rd Edition**C.2 An Econometric Model**Principles of Econometrics, 3rd Edition**C.3 Estimating the Mean of a Population**Principles of Econometrics, 3rd Edition**C.3 Estimating the Mean of a Population**Principles of Econometrics, 3rd Edition**C.3 Estimating the Mean of a Population**Principles of Econometrics, 3rd Edition**C.3.1 The Expected Value of**Principles of Econometrics, 3rd Edition**C.3.2 The Variance of**Principles of Econometrics, 3rd Edition**C.3.3 The Sampling Distribution of**Figure C.2 Increasing Sample Size and Sampling Distribution of Principles of Econometrics, 3rd Edition**C.3.4 The Central Limit Theorem**Principles of Econometrics, 3rd Edition**C.3.4 The Central Limit Theorem**Figure C.3 Central Limit Theorem Principles of Econometrics, 3rd Edition**C.3.5 Best Linear Unbiased Estimation**• A powerful finding about the estimator of the population mean is that it is the best of all possible estimators that are both linear and unbiased(線性不偏). • A linear estimator is simply one that is a weighted average of the Yi’s, such as , where the ai are constants. • “Best” means that it is the linear unbiased estimator with the smallest possible variance. Principles of Econometrics, 3rd Edition**C.4 Estimating the Population Variance and Other Moments**Principles of Econometrics, 3rd Edition**C.4.1 Estimating the population variance**Principles of Econometrics, 3rd Edition**C.4.1 Estimating the population variance**Principles of Econometrics, 3rd Edition**C.4.2 Estimating higher moments**In statistics the Law of Large Numbers(大數法則)says that sample means converge to population averages (expected values) as the sample size N → ∞. Principles of Econometrics, 3rd Edition**C.4.2 Estimating higher moments**Principles of Econometrics, 3rd Edition**C.5 Interval Estimation**• C.5.1 Interval Estimation: σ2 Known Principles of Econometrics, 3rd Edition**C.5.1 Interval Estimation: σ2 Known**Figure C.4 Critical Values for the N(0,1) Distribution Principles of Econometrics, 3rd Edition**C.5.1 Interval Estimation: σ2 Known**Principles of Econometrics, 3rd Edition**C.5.1 Interval Estimation: σ2 Known**Principles of Econometrics, 3rd Edition**C.5.3 Interval Estimation: σ2 Unknown**• When σ2 is unknown it is natural to replace it with its estimator Principles of Econometrics, 3rd Edition**C.5.3 Interval Estimation: σ2 Unknown**Principles of Econometrics, 3rd Edition**C.5.3 Interval Estimation: σ2 Unknown**Principles of Econometrics, 3rd Edition**C.5.5 Interval estimation using the hip data**Given a random sample of size N = 50 we estimated the mean U.S. hip width to be = 17.158 inches. Principles of Econometrics, 3rd Edition**C.6 Hypothesis Tests About A Population Mean**Principles of Econometrics, 3rd Edition**C.6.1 Components of Hypothesis Tests**• The Null Hypothesis （虛無假設） The “null” hypothesis, which is denoted H0 (H-naught), specifies a value c for a parameter. We write the null hypothesis as A null hypothesis is the belief we will maintain until we are convinced by the sample evidence that it is not true, in which case we reject the null hypothesis. Principles of Econometrics, 3rd Edition**C.6.1 Components of Hypothesis Tests**• The Alternative Hypothesis （對立假設） • H1: μ > c If we reject the null hypothesis that μ = c, we accept the alternative that μ is greater than c. • H1: μ < c If we reject the null hypothesis that μ = c, we accept the alternative that μ is less than c. • H1: μ ≠ c If we reject the null hypothesis that μ = c, we accept the alternative that μ takes a value other than (not equal to) c. Principles of Econometrics, 3rd Edition**C.6.1 Components of Hypothesis Tests**• The Test Statistic （檢定統計量） A test statistic’s probability distribution is completely known when the null hypothesis is true, and it has some other distribution if the null hypothesis is not true. Principles of Econometrics, 3rd Edition**C.6.1 Components of Hypothesis Tests**Principles of Econometrics, 3rd Edition**C.6.1 Components of Hypothesis Tests**• The Rejection Region • If a value of the test statistic is obtained that falls in a region of low probability, then it is unlikely that the test statistic has the assumed distribution, and thus it is unlikely that the null hypothesis is true. • If the alternative hypothesis is true, then values of the test statistic will tend to be unusually “large” or unusually “small”, determined by choosing a probability , called the level of significance of the test. • The level of significance（顯著水準 ）of the test is usually chosen to be .01, .05 or .10. Principles of Econometrics, 3rd Edition**C.6.1 Components of Hypothesis Tests**• A Conclusion • When you have completed a hypothesis test you should state your conclusion, whether you reject, or do not reject, the null hypothesis. • Say what the conclusion means in the economic context of the problem you are working on, i.e., interpret the results in a meaningful way. Principles of Econometrics, 3rd Edition**C.6.2 One-tail Tests with Alternative “Greater Than” (>)**Figure C.5 The rejection region for the one-tail test of H1: μ = c against H1: μ > c Principles of Econometrics, 3rd Edition**C.6.3 One-tail Tests with Alternative “Less Than” (<)**Figure C.6 The rejection region for the one-tail test of H1: μ = c against H1: μ < c Principles of Econometrics, 3rd Edition**C.6.4 Two-tail Tests with Alternative “Not Equal To”**(≠) Figure C.7 The rejection region for a test of H1: μ = c against H1: μ ≠ c Principles of Econometrics, 3rd Edition**C.6.5 Example of a One-tail Test Using the Hip Data**• The null hypothesis is The alternative hypothesis is • The test statistic if the null hypothesis is true. • The level of significance =.05. Principles of Econometrics, 3rd Edition**C.6.5 Example of a One-tail Test Using the Hip Data**• The value of the test statistic is • Conclusion: Since t = 2.5756 > 1.68 we reject the null hypothesis. The sample information we have is incompatible with the hypothesis that μ = 16.5. We accept the alternative that the population mean hip size is greater than 16.5 inches, at the =.05 level of significance. Principles of Econometrics, 3rd Edition**C.6.6 Example of a Two-tail Test Using the Hip Data**• The null hypothesis is The alternative hypothesis is • The test statistic if the null hypothesis is true. • The level of significance =.05, therefore Principles of Econometrics, 3rd Edition**C.6.6 Example of a Two-tail Test Using the Hip Data**• The value of the test statistic is • Conclusion: Since we do not reject the null hypothesis. The sample information we have is compatible with the hypothesis that the population mean hip size μ = 17. Principles of Econometrics, 3rd Edition**C.6.7 The p-value**Principles of Econometrics, 3rd Edition**C.6.7 The p-value**• How the p-value is computed depends on the alternative. If t is the calculated value [not the critical value tc] of the t-statistic with N−1 degrees of freedom, then: • if H1: μ > c , p = probability to the right of t • if H1: μ < c , p = probability to the left of t • if H1: μ ≠ c , p = sum of probabilities to the right of |t| and to the left of –|t| Principles of Econometrics, 3rd Edition**C.6.7 The p-value**Figure C.8 The p-value for a right-tail test Principles of Econometrics, 3rd Edition**C.6.7 The p-value**Figure C.9 The p-value for a two-tailed test Principles of Econometrics, 3rd Edition**C.6.9 Type I and Type II errors**Principles of Econometrics, 3rd Edition**C.6.10 A Relationship Between Hypothesis Testing and**Confidence Intervals • If we fail to reject the null hypothesis at the level of significance, then the value c will fall within a 100(1)% confidence interval estimate of μ. • If we reject the null hypothesis, then c will fall outside the 100(1)% confidence interval estimate of μ. Principles of Econometrics, 3rd Edition**C.6.10 A Relationship Between Hypothesis Testing and**Confidence Intervals • We fail to reject the null hypothesis when or when Principles of Econometrics, 3rd Edition**C.7 Some Useful Tests**• C.7.1 Testing the population variance Principles of Econometrics, 3rd Edition