西南交通大學(xué)設(shè)備采購(gòu)招標(biāo)書ppt課件

1、Appendix CReview of Statistical InferencePrepared by Vera Tabakova, East Carolina UniversityAppendix C: Review of Statistical InferenceC.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 Estimation2Ap

2、pendix C: Review of Statistical InferenceC.6 Hypothesis Tests About a Population MeanC.7 Some Other Useful TestsC.8 Introduction to Maximum Likelihood Estimation C.9 Algebraic Supplements 3C.1 A Sample of Data4C.1 A Sample of DataFigure C.1 Histogram of Hip Sizes5C.2 An Econometric Model (C.1)(C.2)6

3、C.3 Estimating the Mean of a Population (C.3)(C.4)7C.3 Estimating the Mean of a Population (C.3)(C.4)8C.3 Estimating the Mean of a Population 9C.3.1 The Expected Value of (C.5)10C.3.2 The Variance of (C.6)11C.3.3 The Sampling Distribution ofFigure C.2 Increasing Sample Size and Sampling Distribution

4、 of12C.3.4 The Central Limit TheoremCentral 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 .13C.3.4 The Central Limit TheoremFigure C.3

5、Central Limit Theorem14C.3.5 Best Linear Unbiased EstimationA 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

6、 the ai are constants. “Best means that it is the linear unbiased estimator with the smallest possible variance. 15C.4 Estimating the Population Variance and Other Moments 16C.4.1 Estimating the population variance (C.7)17C.4.1 Estimating the population variance (C.9)(C.8)18C.4.2 Estimating higher m

7、omentsIn statistics the Law of Large Numbers(大數(shù)法則) says that sample means converge to population averages (expected values) as the sample size N . 19C.4.2 Estimating higher moments20C.5 Interval EstimationC.5.1 Interval Estimation: 2 Known(C.10)21C.5.1 Interval Estimation: 2 Known Figure C.4 Critica

8、l Values for the N(0,1) Distribution 22C.5.1 Interval Estimation: 2 Known (C.11)23C.5.1 Interval Estimation: 2 Known (C.13)(C.12)24C.5.3 Interval Estimation: 2 UnknownWhen 2 is unknown it is natural to replace it with its estimator (C.14)25C.5.3 Interval Estimation: 2 Unknown (C.15)26C.5.3 Interval

9、Estimation: 2 UnknownRemark: 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 “l(fā)arge” samples, w

10、hich 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.27C.5.5 Interval estimation using the hip dataGiven a random sample of size N = 50 we estimated the mean U.S. hip width

11、to be = 17.158 inches. 28C.6 Hypothesis Tests About A Population Mean Components of Hypothesis TestsA null hypothesis, H0 (虛無假設(shè))An alternative hypothesis, H1 （對(duì)立假設(shè)）A test statistic （檢定統(tǒng)計(jì)量）A rejection region （拒絕域）A conclusion （結(jié)論）29C.6.1 Components of Hypothesis TestsThe Null Hypothesis 虛無假設(shè)The “null

12、 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.30C.6.1 Components of Hypo

13、thesis TestsThe Alternative Hypothesis 對(duì)立假設(shè)H1: c If we reject the null hypothesis that = c, we accept the alternative that is greater than c. H1: ) Figure C.5 The rejection region for the one-tail test of H1: = c against H1: c 36C.6.3 One-tail Tests with Alternative “Less Than () Figure C.6 The reje

14、ction 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.40C

15、.6.6 Example of a Two-tail Test Using the Hip DataThe null hypothesis isThe alternative hypothesis is The test statistic if the null hypothesis is true.The level of significance =.05, therefore 41C.6.6 Example of a Two-tail Test Using the Hip DataThe value of the test statistic isConclusion: Since w

16、e do not reject the null hypothesis. The sample information we have is compatible with the hypothesis that the population mean hip size = 17. 42C.6.7 The p-value 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

17、 H0. If p then do not reject H043C.6.7 The p-valueHow 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 t

18、if H1: c , p = sum of probabilities to the right of |t| and to the left of |t|44C.6.7 The p-value Figure C.8 The p-value for a right-tail test 45C.6.7 The p-value Figure C.9 The p-value for a two-tailed test 46C.6.9 Type I and Type II errors Correct DecisionsThe null hypothesis is false and we decid

19、e to reject it.The null hypothesis is true and we decide not to reject it.Incorrect DecisionsThe 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)47C.6.10 A Relationship Between Hypothesis Testing and Conf

20、idence IntervalsIf 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 .48C.6.10 A Relationship Between

21、 Hypothesis Testing and Confidence IntervalsWe fail to reject the null hypothesis when or when49C.7 Some Useful TestsC.7.1 Testing the population variance 50C.7.1 Testing the Population Variance 51C.7.2 Testing the Equality of two Population MeansCase 1: Population variances are equal52C.7.2 Testing

22、 the Equality of two Population MeansCase 2: Population variances are unequal53C.7.3 Testing the ratio of two population variances54C.7.4 Testing the normality of a populationThe 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(tài) and kur