第五章假設(shè)檢驗(yàn)與統(tǒng)計(jì)推斷

1、5-1,Chapter 5: Hypothesis Testing and Statistical Inference,一、假設(shè)檢驗(yàn)的概念與思想,什么是假設(shè)(hypothesis),對總體參數(shù)的的數(shù)值所作的一種陳述 總體參數(shù)包括總體均值、比例、方差等 分析之前必需陳述 其動機(jī)主要是企圖利用人們掌握的反映現(xiàn)實(shí)的數(shù)據(jù)來找出假設(shè)與現(xiàn)實(shí)之間的矛盾，從而否定這個(gè)假設(shè),我認(rèn)為該地區(qū)新生嬰兒的平均體重為3190克,什么是假設(shè)檢驗(yàn)(hypothesis testing),事先對總體參數(shù)或分布形式作出某種假設(shè)，然后利用樣本信息來判斷原假設(shè)是否成立 有參數(shù)假設(shè)檢驗(yàn)和非參數(shù)假設(shè)檢驗(yàn) 采用邏輯上的反證法，依據(jù)統(tǒng)計(jì)上的

2、小概率原理,假設(shè)檢驗(yàn)的基本思想,因此我們拒絕假設(shè) = 50,樣本均值,m,50,抽樣分布,H0,假設(shè)檢驗(yàn)的過程,5-7,Hypothesis Testing,Hypothesis testing involves drawing inferences about two contrasting propositions (hypotheses) relating to the value of a population parameter, one of which is assumed to be true in the absence of contradictory data. We s

3、eek evidence to determine if the hypothesis can be rejected; if not, we can only assume it to be true but have not statistically proven it true,5-8,Hypothesis Testing Procedure,Formulate the hypothesis Select a level of significance, which defines the risk of drawing an incorrect conclusion that a t

4、rue hypothesis is false Determine a decision rule Collect data and calculate a test statistic Apply the decision rule and draw a conclusion,5-9,1.Hypothesis Formulation,Null hypothesis, H0 a statement that is accepted as correct Alternative hypothesis, H1 a proposition that must be true if H0 is fal

5、se Tests involving a single population parameter are called one-sample tests; tests involving two populations are called two-sample tests,5-10,Types of Hypothesis Tests,One Sample Tests H0: population parameter constant vs. H1: population parameter constant H0: population parameter = constant vs. H1

6、: population parameter constant Two Sample Tests H0: population parameter (1) - population parameter (2) 0 vs. H1: population parameter (1) - population parameter (2) 0 H0: population parameter (1) - population parameter (2) = 0 vs. H1: population parameter (1) - population parameter (2) 0,5-11,Form

7、ulating Hypotheses,Formulating the correct set of hypotheses depends on “burden of proof” what you wish to prove statistically should be H1 Example: To seek evidence that technical support calls average less than 30 minutes (Customer Support Survey file), the correct hypotheses are: H0: Mean respons

8、e time 30 minutes H1: Mean response time 30 minutes,5-12,2.顯著性水平Four Outcomes,The null hypothesis is actually true, and the test correctly fails to reject it. The null hypothesis is actually false, and the hypothesis test correctly reaches this conclusion. The null hypothesis is actually true, but t

9、he hypothesis test incorrectly rejects it (Type I error). The null hypothesis is actually false, but the hypothesis test incorrectly fails to reject it (Type II error,5-13,Quantifying Outcomes,Probability of Type I error (rejecting H0 when it is true) = a = level of significance Probability of corre

10、ctly failing to reject H0 = 1 a = confidence coefficient Probability of Type II error (failing to reject H0 when it is false) = b Probability of correctly rejecting H0 when it is false = 1 b = power of the test,假設(shè)檢驗(yàn)中的兩類錯(cuò)誤,1.第一類錯(cuò)誤（棄真錯(cuò)誤） 原假設(shè)為真時(shí)拒絕原假設(shè) 會產(chǎn)生一系列后果 第一類錯(cuò)誤的概率為 被稱為顯著性水平 2.第二類錯(cuò)誤（取偽錯(cuò)誤） 原假設(shè)為假時(shí)接受原假

11、設(shè) 第二類錯(cuò)誤的概率為(Beta,H0: 無罪,假設(shè)檢驗(yàn)中的兩類錯(cuò)誤 （決策結(jié)果,假設(shè)檢驗(yàn)就好像一場審判過程,統(tǒng)計(jì)檢驗(yàn)過程,錯(cuò)誤和 錯(cuò)誤的關(guān)系,5-17,3.Decision Rules,Compute a test statistic from sample data and compare it to the hypothesized sampling distribution of the test statistic Divide the sampling distribution into a rejection region and non-rejection region. If

12、 the test statistic falls in the rejection region, reject H0 (concluding that H1 is true); otherwise, fail to reject H0,5-18,Rejection Regions,5-19,4.Hypothesis Tests and Spreadsheet Support,5-20,Hypothesis Tests and Spreadsheet Support (contd,5-21,二、單樣本假設(shè)檢驗(yàn)1.One Sample Tests for Means Standard Devi

13、ation Unknown,Example hypothesis H0: m m0 versus H1: m m0 Test statistic: Reject H0 if t -tn-1,5-22,Example,For the Customer Support Survey.xls data, test the hypotheses H0: mean response time 30 minutes H1: mean response time 30 minutes Sample mean = 21.91; sample standard deviation = 19.49; n = 44

14、 observations Reject H0 because t = 2.75 -t43,0.05 = -1.6811,5-23,PHStat Tool: t-Test for Mean,PHStat menu One Sample Tests t-Test for the Mean, Sigma Unknown,Enter null hypothesis and alpha Enter sample statistics or data range Choose type of test,5-24,Results,5-25,2.Using p-Values,p-value = probab

15、ility of obtaining a test statistic value equal to or more extreme than that obtained from the sample data when H0 is true, shown as areas under the sampling distributions below,Test Statistic,Lower one-tailed test？ Two-tailed test,m0,m0,Test Statistic,5-26,Example p-Value,p = probability of obtaini

16、ng a test statistic of -2.75 or less = 0.0043,5-27,Two-Tailed Test,Consumer Transportation Survey H0: Mean age = 40 H1: Mean age 40 Sample mean = 37.9; sample standard deviation = 11,5-28,Results,5-29,3.One Sample Tests for Proportions,Example hypothesis H0: p p0 versus H1: p p0 Test statistic: Reje

17、ct if z -za,5-30,Example,For the Customer Support Survey data, test the hypothesis that the proportion of overall quality responses in the top two boxes（3很好，4 非常好） is at least 0.75 H0: p .75 H0: p .75 Sample proportion = 0.682; n = 44 For a level of significance of 0.05, the critical value of z is -

18、1.645; therefore, we cannot reject the null hypothesis,5-31,PHStat Tool: One Sample z-Test for Proportions,PHStat One Sample Tests z-Tests for the Proportion,Enter null hypothesis, significance level, number of successes, and sample size Enter type of test,5-32,Results,5-33,4.Type II Errors and the

19、Power of a Test,The probability of a Type II error, b, and the power of the test (1 b) cannot be chosen by the experimenter. The power of the test depends on the true value of the population mean, the level of confidence used, and the sample size. A power curve shows (1 b) as a function of m1,5-34,F

20、inding the Probability of a Type II Error,5-35,How b Depends on H1,5-36,How b Depends on Sample Size,5-37,Example Power Curve,5-38,三、兩樣本假設(shè)檢驗(yàn)1.Two Sample Tests for Means Standard Deviation Known,Example hypothesis H0: m1 m2 0 versus H1: m1 - m2 0 Test Statistic: Reject if z -za,5-39,Two Sample Tests

21、for Means Sigma Unknown and Equal,Example hypothesis H0: m1 m2 0 versus H1: m1 - m2 0 Test Statistic: Reject if z za,5-40,Two Sample Tests for Means Sigma Unknown and Unequal,Example hypothesis H0: m1 m2 = 0 versus H1: m1 - m2 0 Test Statistic: Reject if z za/2 or z - za/2,t = (x1 - x2 ),with df,5-4

22、1,Spreadsheet Tools: Two Sample t-Tests, Population Variance Known,Excel z-test: Two Sample for Means PHStat Z Test for Differences in Two Means,5-42,Spreadsheet Tools: Two Sample t-Tests, Population Variance Unknown,Population variances assumed unequal Excel t-test: Two Sample Assuming Unequal Vari

23、ances Population variances assumed unequal Excel t-test: Two Sample Assuming Equal Variances PHStat t-test for Differences in Two Means,5-43,Interpreting Excel Output,If t Stat is negative, provides the correct p-value for a lower-tail test; however, for an upper-tail test, you must subtract this nu

24、mber from 1.0 to get the correct p-value. If t Stat is nonnegative then provides the correct p-value for an upper tail test; consequently, for a lower tail test, you must subtract this number from 1.0 to get the correct p-value. Also, for a lower tail test, you must change the sign on t Critical one

25、-tail,5-44,Comparison of Excel and PHStat Results Lower-Tail Test,5-45,2.Two Sample Test for Means With Paired Samples,Example hypothesis H0: average difference = 0 versus H1: average difference 0 Test Statistic: Reject if t tn-1,a/2 or t - tn-1,a/2,5-46,Excel Tool Example: Pile Foundation Data,H0:

26、Average difference = 0 H1: Average difference 0 Sample mean difference = 6.3 Sample standard deviation = 10.31,5-47,3.Two Sample Tests for Proportions,Example hypothesis H0: p1 p2 = 0 versus H1: p1 - p2 0 Test Statistic: Reject if z za/2 or z - za/2,where,5-48,Example: Accounting Professionals Data,

27、H0: Proportion of females with graduate degree proportion of males with graduate degree = 0 H1: Proportion of females with graduate degree proportion of males with graduate degree 0 PHStat tool: z-Test for Differences in Two Proportions,5-49,Results,5-50,4.Hypothesis Tests and Confidence Intervals,If a 100(1 a)% confidence interval for a two-tailed test does not contain the hypothesized value, then we would reject the null hypothesis. If a 100(1 a)% confidence interval for a lower (upper)