C.1 A Sample of DataC.2 An Econometric ModelC.3 Estimating the Mean of a PopulationC.4 Estimating the Population Variance and Other MomentsC.5 Interval EstimationC.6 Hypothesis Tests About a Population MeanC.7 Some Other Useful TestsC.8 Introduction to Maximum Likelihood Estimation C.9 Algebraic Supplements Figure C.1 Histogram of Hip SizesFigure C.2 Increasing Sample Size and Sampling Distribution ofCentral Limit Theorem: If Y1,YN are independent and identically distributed（i.i.d.） random variables with mean and variance 2, and , then has a probability distribution that converges to the standard normal N(0,1) as N .Figure C.3 Central Limit TheoremA 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 Yis, such as , where the ai are constants. “Best” means that it is the linear unbiased estimator with the smallest possible variance. 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 . C.5.1 Interval Estimation: 2 KnownFigure C.4 Critical Values for the N(0,1) Distribution When 2 is unknown it is natural to replace it with its estimator Remark: The confidence interval (C.15) is based upon the assumption that the population is normally distributed, so that is normally distributed. If the population is not normal, then we invoke the central limit theorem, and say that is approximately normal in “large” samples, which from Figure C.3 you can see might be as few as 30 observations. In this case we can use (C.15), recognizing that there is an approximation error introduced in smaller samples.Given a random sample of size N = 50 we estimated the mean U.S. hip width to be = 17.158 inches. Components of Hypothesis TestsA null hypothesis, H0 (虛無假設)An alternative hypothesis, H1 （對立假設）A test statistic （檢定統計量）A rejection region （拒絕域）A conclusion （結論）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.The Alternative Hypothesis （對立假設）H1: c If we reject the null hypothesis that = c, we accept the alternative that is greater than c. H1: c Figure C.6 The rejection region for the one-tail test of H1: = c against H1: 1.68 we reject the null hypothesis. The sample information we have is incompatible with the hypothesis that = 16.5. We accept the alternative that the population mean hip size is greater than 16.5 inches, at the =.05 level of significance.The null hypothesis is The alternative hypothesis is The test statistic if the null hypothesis is true.The level of significance =.05, therefore The value of the test statistic isConclusion: Since we do not reject the null hypothesis. The sample information we have is compatible with the hypothesis that the population mean hip size = 17. p-value rule: Reject the null hypothesis when the p- value is less than, or equal to, the level of significance . That is, if p then reject H0. If p then do not reject H0How 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 N1 degrees of freedom, then:if H1: c , p = probability to the right of tif H1: c , p = probability to the left of tif H1: c , p = sum of probabilities to the right of |t| and to the left of |t|Figure C.8 The p-value for a right-tail test Figure C.9 The p-value for a two-tailed test Correct Decisions The null hypothesis is false and we decide to reject it. The null hypothesis is true and we decide not to reject it.Incorrect Decisions The null hypothesis is true and we decide to reject it (a Type I error) The null hypothesis is false and we decide not to reject it (a Type II error)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 . If we reject the null hypothesis, then c will fall outside the 100(1)% confidence interval estimate of .We fail to reject the null hypothesis when or whenC.7.1 Testing the population varianceCase 1: Population variances are equalCase 2: Population variances are unequalThe normal distribution is symmetric, and has a bell-shape with a peakedness and tail-thickness leading to a kurtosis of 3. We can test for departures from normality by checking the skewness（偏態） and kurtosis（峰態） from a sample of data.S