经管学会网盘光华方向专业概统lecture

1、1Lecture 20Testing for Mean When Variance is Known and Testing for Probabilityp valueEquivalence of Tests and Confidence IntervalsExample 1Suppose that X=(X1,Xn) is a random sample from a normal distribution with mean m and known variance . We wish to test the hypothesesOur test is to reject H0 if f

2、or some c.Example 1To make the test having a specified significance level a0, we wantAmong all such tests, we want to be as large as possible for any . Therefore, c should be made as large as that makesreject H0of allor valuesIfExample 2Suppose that X1,Xn are i.i.d. with Bernoulli distribution with

3、parameter p. Suppose we wish to test the hypotheses:Let . The larger p is, the larger we expect Y to be.Our test is to reject H0 if for some c.We want the size of the test to be as close to a0 as possible without exceeding a0, so c would be the smallest number such that E.g. n=10, p0=0.3, a0=0.1. Be

4、causeAny value of c in the interval (5,6 would be fine, because Y can only take integer values.p-valueThe smallest level a0 such that we would reject the null hypothesis H0 at level a0 with the observed data.The probability of getting a value of the test statistic as extreme as or more extreme than

5、that observed by chance alone, if the null hypothesis H0, is true.The null hypothesis H0 would be rejected for every larger value of a0 and would not be rejected for any smaller value.Example 1 ContinuedSuppose that X=(X1,Xn) is a random sample from a normal distribution with mean m and known varian

6、ce . We wish to test the hypothesesOur test is to reject H0 if for some c.Example 1 Continuedp/2-|Z|p/2|Z|If p0.0473.The researcher would typically report the observed value of the test statistic and the corresponding p-value.The researcher does not have to select beforehand an arbitrary level of si

7、gnificance at which the test is to be carried out.DiscussionSuppose that a random sample X=(X1,Xn) is to be taken from a distribution that depends on an unknown parameter .If a random set w(X) satisfies for every , we call w(x) a coefficient g Confidence Interval for q.Equivalence of Tests and Confi

8、dence IntervalsConsider testing the hypotheses:Suppose that for every point and every value of g (0g1), we can construct a level 1-g test such thatFor each possible set of values x=(x1,xn), let w(x) denote the set of all points for which the test accepts the hypothesis H0 when the observed data are

9、X=x. Hypothesis Test Confidence IntervalSuppose that w(x) is a Confidence Interval for q with confidence coefficient g. Then for each point , define a test procedure as follows: accepting H0 if and only ifSo is a level 1-g test for every q0. Confidence Interval Hypothesis Test Example 3 A Confidence

10、 Interval for the Mean of a Normal DistributionSuppose that X=(X1,Xn) is a random sample from a normal distribution with mean m and known variance . Based on the observed value x, we need to construct a coefficient g confidence interval for m.Let a0=1-g . We have known that a level a0 test for testing the hypothesesis to reject H0 if The equivalent confidence interval is Example 4 Constructing a Test from a Confidence IntervalSuppose that X=(X1,Xn) is a random sample from a normal distribution with mean m and v