Lecture 5 Section B Class Web page URL

Lecture 5: Section B Class Web page URL: http: //www. econ. uiuc. edu/ECON 173 Data used in some examples can be found in: http: //www. econ. uiuc. edu/ECON 173/hmodata. xls http: //www. econ. uiuc. edu/ECON 173/hmodata_ans. xls 1

Lecture 5: Today’s Topics l Recap: Confidence Interval and sample size l Hypothesis testing Methodology l Decision Making: Type I and II Errors l Test of Mean with known variance l P-value Approach l Test of Mean with unknown variance 2

Decision Making and Consequences STATES OF NATURE A C T I O Do Not Reject H 0 H 0 True Correct Confidence=1 -a Type I Error P(Type I)=a H 0 False Type II Error P( Type II)=b Correct Power=1 -b N S 3

a & b Have an Inverse Relationship Reduce probability of one error and the other one goes up. b a 4

To buy a mp 3 player A C T I O N STATES OF NATURE Napster around Confidence level Buy mp 3 of the test player =1 -0. 2=0. 8 P(Type I Don’t buy mp 3 player error)=0. 2 Napster dead P( Type II error)=0. 1 Power of the test =1 -0. 1=0. 9 S 5

Hypothesis Testing Process Assume the population mean age is 50. (Null Hypothesis) No, not likely! Population The Sample Mean Is 20 REJECT Null Hypothesis Sample 6

Definitions-I Null Hypothesis (H 0): The hypothesis that depicts the traditional belief or the conventional wisdom and is maintained unless there is sufficient evidence to prove otherwise. l Alternative Hypothesis (H 1): The hypothesis which serves as a plausible alternative to replace the null hypothesis given there is sufficient evidence against the null hypothesis. l 7

Definitions-II Type I Error: The error which occurs when you reject H 0 given that it is indeed true. l Type II Error: The error which occurs when you do not reject H 0 given that it is indeed false. l Level of Significance (a) : The maximum probability of committing a Type I Error. Sometimes (1 -a) is called confidence coefficient. l Power (1 -b) : The probability of correctly rejecting the null hypothesis when it is really false. l 8

Z test of hypothesis for Mean (test for m, s known, critical value approach) Critical values of z Area=0. 1 0. 90 Rejection Region 0 Region of Acceptance Rejection Region At 10% level, reject H 0 if z is in the Rejection region. Do not reject if z is in the Acceptance region at 10% level. 9

Level of Significance, a and the Rejection Region a H 0: m ³ 3 Lower one-tailed H 1: m < 3 Rejection Regions H 0: m £ 3 Upper one-tailed H 1: m > 3 H 0: m = 3 H 1: m ¹ 3 0 0 Critical Value(s) a a/2 Two-tailed 0 10

Z test of hypothesis for Mean (test for m, s known, p-value approach) Area=p-value Rejection Region 0 Rejection Region of Acceptance Test statistic z At 10% level, reject H 0 if p-value<0. 1. Do not reject if p-value 0. 1. 11

Definitions-III l test statistic: The measured value of the statistic which is used to test a hypothesis. l critical value: The tabulated value of the test statistics, beyond which we reject the null hypothesis. l p-value: The smallest level of significance at which the null hypothesis is rejected. 12

Steps for Hypothesis Testing-I l Step 1: Setup the null and alternative hypothesis. e. g. H 0: m=20 vs. H 1: m 20 l Step 2: Collect data and decide on a. e. g. Data on a sample of Doritos and a=0. 05. l Step 3: Calculate summary sample statistics. e. g. Calculate and s. l Step 4: Calculate the test statistic z (or t). e. g. 13

Steps for Hypothesis Testing-II Step 5: Find out the distribution of the test statistics under H 0. e. g. follows a standard normal distribution if H 0 is true. l Step 6: Obtain the Rejection region using the pvalue or otherwise. e. g. reject if p-value<a or teststatistic is z<lower critical value (or z>upper critical value or either) l Step 7: Make your decision of whether to accept or reject H 0. e. g. Reject the null hypothesis that each bag of Doritos contain 20 oz of chips. l Step 8: Draw your conclusion. e. g. On an average the weight of each bag of Doritos is different from 20 oz. 14 l

One-Tail Z Test for Mean (s Known) l Assumptions – – – l Population Is Normally Distributed If Not Normal, use large samples Null Hypothesis Has £ or ³ Sign Only Z Test Statistic: 15

Rejection Region H 0: m ³ 0 H 1: m < 0 H 0: m £ 0 H 1: m > 0 Reject H 0 a a 0 Must Be Significantly Below m = 0 Z Z 0 Small values don’t contradict H 0 Don’t Reject H 0! 16

Example: One Tail Test l. Does an average box of cereal contain more than 368 grams of cereal? A random sample of 25 boxes _ showed X = 372. 5. The company has specified s to be 15 grams. Test at the a=0. 05 level. 368 gm. H 0: m £ 368 H 1: m > 368 17

Example Solution: One Tail H 0: m £ 368 H 1: m > 368 Test Statistic: la = 0. 025 ln = 25 l. Critical Value: 1. 645 Reject. 05 0 1. 645 Z Decision: Do Not Reject at a =. 05 Conclusion: No Evidence True Mean Is More than 368 18

p Value Solution p Value is P(Z ³ 1. 50) = 0. 0668 Use the alternative hypothesis to find the direction of the test. p Value . 0668 1. 0000 -. 9332. 0668 . 9332 0 1. 50 From Z Table: Lookup 1. 50 Z Z Value of Sample Statistic 19

t-Test: s Unknown l Assumptions – Population is normally distributed – If not normal, only slightly skewed & a large sample taken l Parametric lt test procedure test statistic 20

Example: One Tail t-Test Does an average box of cereal contain more than 368 grams of cereal? A random sample of 36 boxes showed X = 372. 5, and s = 15. Test at the a=0. 01 level. s is not given, 368 gm. H 0: m £ 368 H 1: m > 368 21

Example Solution: One Tail H 0: m £ 368 H 1: m > 368 Test Statistic: la = 0. 01 ln = 36, df = 35 l. Critical Value: 2. 4377 Reject. 01 0 2. 4377 Z Decision: Do Not Reject at a =. 01 Conclusion: No Evidence that True Mean Is More than 368 22