1 Lecture 5: Section B Class Web page URL: Data used in some examples can be found in:

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

3 Decision Making and Consequences H 0 TrueH 0 False Do Not Reject H 0 Correct Confidence=1-  Type II Error P( Type II)=  Reject H 0 Type I Error P(Type I)=  Correct Power=1-  STATES OF NATURE ACTIONSACTIONS

4   Reduce probability of one error and the other one goes up.  &  Have an Inverse Relationship

5 To buy a mp3 player Napster aroundNapster dead Buy mp3 player Confidence level of the test =1-  P( Type II error)=  Don’t buy mp3 player P(Type I error)=  Power of the test =1-0.1=0.9 STATES OF NATURE ACTIONSACTIONS

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

7 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. 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.

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

9 Z test of hypothesis for Mean (test for  known, critical value approach) Rejection Region Region of Acceptance 0 Area=0.1 Critical values of z 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. 0.90

10 Level of Significance,  and the Rejection Region H 0 :   3 H 1 :  < H 0 :   3 H 1 :  > 3 H 0 :   3 H 1 :   3    /2 Critical Value(s) Rejection Regions Lower one-tailed Upper one-tailed Two-tailed

11 Z test of hypothesis for Mean (test for  known, p-value approach) Rejection Region Region of Acceptance 0 Area=p-value At 10% level, reject H 0 if p-value<0.1. Do not reject if p-value  Test statistic z

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

13 Steps for Hypothesis Testing-I Step 1: Setup the null and alternative hypothesis. e.g. H 0 :  =20 vs. H 1 :  20 Step 2: Collect data and decide on . e.g. Data on a sample of Doritos and  =0.05. Step 3: Calculate summary sample statistics. e.g. Calculate and s. Step 4: Calculate the test statistic z (or t). e.g.

14 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. Step 6: Obtain the Rejection region using the p- value or otherwise. e.g. reject if p-value upper critical value or either) Step 7: Make your decision of whether to accept or reject H0. e.g. Reject the null hypothesis that each bag of Doritos contain 20oz of chips. Step 8: Draw your conclusion. e.g. On an average the weight of each bag of Doritos is different from 20 oz. Steps for Hypothesis Testing-II

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

16 Z 0  Reject H 0 Z 0 0  H 0 :  H 1 :  < 0 H 0 :  0 H 1 :  > 0 Must Be Significantly Below  = 0 Small values don’t contradict H 0 Don’t Reject H 0 ! Rejection Region

17 Does an average box of cereal contain more than 368 grams of cereal? A random sample of 25 boxes showed X = The company has specified  to be 15 grams. Test at the  0.05 level. 368 gm. Example: One Tail Test H 0 :  368 H 1 :  > 368 _

18  = n = 25 Critical Value: Test Statistic: Decision: Conclusion: Do Not Reject at  =.05 No Evidence True Mean Is More than 368 Z Reject Example Solution: One Tail H 0 :  368 H 1 :  > 368

19 Z p Value Z Value of Sample Statistic From Z Table: Lookup Use the alternative hypothesis to find the direction of the test p Value is P(Z  1.50) = p Value Solution

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

21 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   15. Test at the  0.01 level. 368 gm. H 0 :  368 H 1 :  368  is not given,

22  = 0.01 n = 36, df = 35 Critical Value: Test Statistic: Decision: Conclusion: Do Not Reject at  =.01 No Evidence that True Mean Is More than 368 Z Reject Example Solution: One Tail H 0 :  368 H 1 :  368