Hand08 - lecuture 8
Statistics (FEB21007X)
Erasmus Universiteit Rotterdam
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Lecture 8 Hypothesis tests 8 Hypothesis tests Please read Bain and Engelhardt Sec 12. 8.1 Two types of error Two types of error A statistical hypothesis is a claim about the value of a parameter. For example: More than of Dutch inhabitants feel very happy. There are only two types of statistical hypotheses: the null hypothesis H0 : the claim that is initially assumed to be true, and the alternative hypothesis Ha : the complement to H0 . Based on sample evidence we will either reject or not reject the null hypothesis. The test may give us good reason to believe that H a is true, but it will never give us good reason to believe that H0 is true. Two types of error Apart from the distinction between null and alternative hypothesis, we may also distinguish between simple corresponds to one single parameter 69 8. Hypothesis tests 70 composite corresponds to more than one parameter value. Denote the set of parameter values belonging to a composite null hypothesis N . Denote the set of parameter values belonging to a composite alternative hypothesis A. The alternative hypothesis is usually composite. For example, 0. Two types of error A statistical test uses sample data to decide whether the null hypothesis should be rejected. The test statistic T is the statistic on which this decision is based. Thus, there should exist a rejection region RR: the set of all possible test statistic values for which H0 will be rejected. Decision rule: the null hypothesis will be rejected if and only if the computed value of the test statistic falls in the rejection region. Two types of error T Test conclusion RR Do not reject H0 RR Reject H0 H0 true H0 false correct Type II Type I correct When doing a statistical hypothesis test, there will always be some probability of making Type I or Type II error: the probability of committing a Type I error is P (Type I error) P (T RR H0 true) , the probability of committing a Type II error is P (Type II error) P (T RR H0 false) Two types of error We will construct the rejection region such that c 2016, Erasmus School of Economics (ESE). This work is made available exclusively for use students enrolled in the ESE course and must not be distributed beyond that limited group. 8. Hypothesis tests 8.1 72 Pivotal quantities Pivotal quantities As we have seen already, the random variable often is a pivotal quantity. Then we may test the null hypothesis H 0 statistic. T : means of the test . Do we know the distribution of T under the null hypothesis H0 : ? Do we know the distribution of T under the alternative hypothesis? Pivotal quantities In general, if Q Y1 , Y2 , . . . , Yn ) is a pivotal quantity, then we may test the null hypothesis H0 : means of the test statistic. T Q Y1 , Y2 , . . . , Yn ) . Do we know the distribution of T under the null hypothesis H0 : ? Do we know the distribution of T under the alternative hypothesis? c 2016, Erasmus School of Economics (ESE). This work is made available exclusively for use students enrolled in the ESE course and must not be distributed beyond that limited group. 8. More about hypothesis tests 8 73 More about hypothesis tests Please read Bain and Engelhardt Sec 12. 8.2 testing testing Let denote a parameter as Usual tests: H0 : vs Ha : H0 : vs Ha : H0 : vs Ha : The form of the rejection region depends on the alternative hypothesis. Choose the one that gives maximal power. That is, choose the form of your rejection region in accordance with the expected value of your test statistic under the alternative hypothesis. Reject if the test statistic is large. Reject if the test statistic is small. Reject if the test statistic is either small or large. 8.2 Three approaches Three approaches Three approaches to testing: classical confidence interval approach. All three equivalent. Classical approach Decision rule in classical approach: If the test statistic T falls in the rejection region RR, then reject the null hypothesis at size c 2016, Erasmus School of Economics (ESE). This work is made available exclusively for use students enrolled in the ESE course and must not be distributed beyond that limited group. 8. More about hypothesis tests 8.2 75 Steps Steps Steps in classical approach: 1. Identify the parameter(s) of interest and describe them in the context of the problem. State the and alternative hypothesis in terms of these parameters. 2. Select a test statistic T , and determine its distribution under the null hypothesis. 3. Determine the form of the rejection region: Reject if the test statistic is large. Reject if the test statistic is small. Reject if the test statistic is either small or large. 4. Set the size Steps Steps in classical approach (continued): 5. Determine the rejection region looking up the critical value of the test. 6. Compute the observed value t of the test statistic T . If t falls inside the rejection region, reject the null hypothesis at size otherwise, do not reject the null hypothesis at size 7. State your conclusion in the context of the problem. That is, communicate your conclusion to a large audience. Steps Steps in P approach: 1. Identify the parameter(s) of interest and describe them in the context of the problem. State the and alternative hypothesis in terms of these parameters. 2. Select a test statistic T , and determine its distribution under the null hypothesis. 3. Determine the form of the rejection region. Reject if the test statistic is large. c 2016, Erasmus School of Economics (ESE). This work is made available exclusively for use students enrolled in the ESE course and must not be distributed beyond that limited group. 8. More about hypothesis tests 76 Reject if the test statistic is small. Reject if the test statistic is either small or large. 4. Set the size Steps Steps in P approach (continued): 5. Compute the observed value t of the test statistic T . 6. Determine the If then reject the null hypothesis at size otherwise, do not reject the null hypothesis at size 7. State your conclusion in the context of the problem. That is, communicate your conclusion to a large audience. Steps Steps in confidence interval approach: 1. Identify the parameter(s) of interest and describe them in the context of the problem. State the and alternative hypothesis in terms of these parameters. 2. Select a test statistic T , and determine its distribution under the null hypothesis. 3. Give the formula of the corresponding confidence interval, take Ha into account. 4. Set the size Steps Steps in confidence interval approach (continued): 5. Compute the confidence interval with confidence level 1 6. If the computed confidence interval contains the value of parameter under the null hypothesis, then do not reject the null hypothesis at size otherwise, reject the null hypothesis at size 7. State your conclusion in the context of the problem. That is, communicate your conclusion to a large audience. c 2016, Erasmus School of Economics (ESE). This work is made available exclusively for use students enrolled in the ESE course and must not be distributed beyond that limited group. Bibliography Lee J. Bain and Max Engelhardt. Introduction to Probability and Mathematical Statistics. Cengage Learning, second edition, ISBN 0534380204. David Blackwell. An Oral History with David Blackwell. Regional Oral History Office, The Bancroft Library, University of California, Berkeley, 2003. URL Conducted Nadine Wilmot in 2002 and 2003. Gary W. Oehlert. A note on the delta method. The American Statistician, 1992. Dennis Wackerly, William Mendenhall and Richard L. Scheaffer. Mathematical Statistics with Applications. Cengage Learning, 7th edition, 2008. ISBN 0495110817. 78
Hand08 - lecuture 8
Vak: Statistics (FEB21007X)
Universiteit: Erasmus Universiteit Rotterdam
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