Week 8 Fundamentals of Hypothesis Testing: One-Sample Tests

1. Statistics Week 8 Fundamentals of Hypothesis Testing: One-Sample Tests

2. Goals of this note After completing this noe, you should be able to: • Formulate null and alternative hypotheses for applications involving a single population mean or proportion • Formulate a decision rule for testing a hypothesis • Know how to use the p-value approaches to test the null hypothesis for both mean and proportion problems • Know what Type I and Type II errors are

3. What is a Hypothesis? • A hypothesis is a claim (assumption) about a population parameter: • population mean • population proportion • The average number of TV sets in U.S. homes is equal to three ( ) A marketing company claims that it receives 8% responses from its mailing. ( p=.08 )

4. The Null Hypothesis, H0 • States the assumption to be testedExample: The average number of TV sets in U.S. Homes is equal to three ( ) • Is always about a population parameter, not about a sample statistic

5. The Null Hypothesis, H0 (continued) • Begins with the assumption that the null hypothesis is true • Similar to the notion of innocent until proven guilty • Refers to the status quo • Always contains “=” , “≤” or “” sign • May or may not be rejected

6. The Alternative Hypothesis, H1 • Is the opposite of the null hypothesis • e.g.: The average number of TV sets in U.S. homes is not equal to 3 ( H1: μ≠ 3 ) • Challenges the status quo • Never contains the “=” , “≤” or “” sign • Is generally the hypothesis that is believed (or needs to be supported) by the researcher

7. Hypothesis Testing • We assume the null hypothesis is true • If the null hypothesis is rejected we have proven the alternate hypothesis • If the null hypothesis is not rejected we have proven nothing as the sample size may have been to small

8. Hypothesis Testing Process Claim:the population mean age is 50. (Null Hypothesis: Population H0: μ = 50 ) Now select a random sample X = likely if μ = 50? Is 20 Suppose the sample If not likely, REJECT mean age is 20: X = 20 Sample Null Hypothesis

9. Sampling Distribution of H0: μ = 50 H1: μ¹ 50 • There are two cutoff values (critical values), defining the regions of rejection /2 /2 X 50 Reject H0 Do not reject H0 Reject H0 0 20 Likely Sample Results Lower critical value Upper critical value

10. Level of Significance,  • Defines the unlikely values of the sample statistic if the null hypothesis is true • Defines rejection region of the sampling distribution • Is designated by , (level of significance) • Typical values are .01, .05, or .10 • Is the compliment of the confidence coefficient • Is selected by the researcher before sampling • Provides the critical value of the test

11. Level of Significance and the Rejection Region a Level of significance = Represents critical value a a H0: μ = 3 H1: μ≠ 3 /2 /2 Rejection region is shaded Two tailed test 0 H0: μ≤ 3 H1: μ > 3 a 0 Upper tail test H0: μ≥ 3 H1: μ < 3 a Lower tail test 0

12. Errors in Making Decisions • Type I Error • When a true null hypothesis is rejected • The probability of a Type I Error is  • Called level of significance of the test • Set by researcher in advance • Type II Error • Failure to reject a false null hypothesis • The probability of a Type II Error is β

13. Example Possible Jury Trial Outcomes The Truth Verdict Innocent Guilty Innocent No error Type II Error Guilty Type I Error No Error

14. Outcomes and Probabilities Possible Hypothesis Test Outcomes Actual Situation Decision H0 True H0 False Key: Outcome (Probability) Do Not No error (1 - ) Type II Error ( β ) Reject a H 0 Reject Type I Error ( ) No Error ( 1 - β ) H a 0

15. Type I & II Error Relationship • Type I and Type II errors can not happen at the same time • Type I error can only occur if H0 is true • Type II error can only occur if H0 is false If Type I error probability (  ) , then Type II error probability ( β )

16. p-Value Approach to Testing • p-value: Probability of obtaining a test statistic more extreme ( ≤ or  ) than the observed sample value given H0 is true • Also called observed level of significance

17. p-Value Approach to Testing (continued) X • Convert Sample Statistic (e.g. ) to Test Statistic (e.g. t statistic ) • Obtain the p-value from a table or computer • Compare the p-value with  • If p-value < , reject H0 • If p-value  , do not reject H0

18. 8 Steps in Hypothesis Testing 1. State the null hypothesis, H0 State the alternative hypotheses, H1 2. Choose the level of significance, α 3. Choose the sample size, n 4. Determine the appropriate test statistic to use 5. Collect the data 6. Compute the p-value for the test statistic from the sample result 7. Make the statistical decision: Reject H0 if the p-value is less than alpha 8. Express the conclusion in the context of the problem

19. Hypothesis Tests for the Mean Hypothesis Tests for   Known  Unknown

20. Hypothesis Testing Example Test the claim that the true mean # of TV sets in U.S. homes is equal to 3. • 1. State the appropriate null and alternative hypotheses H0: μ = 3 H1: μ≠ 3 (This is a two tailed test) • 2. Specify the desired level of significance Suppose that  = .05 is chosen for this test • 3.Choose a sample size Suppose a sample of size n = 100 is selected

21. Hypothesis Testing Example (continued) • 4. Determine the appropriate Test σ is unknown so this is a t test • 5.Collect the data Suppose the sample results are n = 100, = 2.84 s = 0.8 • 6.So the test statistic is: The p value for n=100, =.05, t=-2 is .048

22. Hypothesis Testing Example (continued) • 7. Is the test statistic in the rejection region? Reject H0 if p is < alpha; otherwise do not reject H0 The p-value .048 is < alpha .05, we reject the null hypothesis

23. Hypothesis Testing Example (continued) • 8. Express the conclusion in the context of the problem Since The p-value .048 is < alpha .05, we have rejected the null hypothesis Thereby proving the alternate hypothesis Conclusion: There is sufficient evidence that the mean number of TVs in U.S. homes is not equal to 3 If we had failed to reject the null hypothesis the conclusion would have been: There is not sufficient evidence to reject the claim that the mean number of TVs in U.S. home is 3

24. One Tail Tests • In many cases, the alternative hypothesis focuses on a particular direction This is a lower tail test since the alternative hypothesis is focused on the lower tail below the mean of 3 H0: μ≥ 3 H1: μ < 3 This is an upper tail test since the alternative hypothesis is focused on the upper tail above the mean of 3 H0: μ≤ 3 H1: μ > 3

25. Lower Tail Tests H0: μ≥ 3 H1: μ < 3 • There is only one critical value, since the rejection area is in only one tail a Reject H0 Do not reject H0 -t 3 Critical value

26. Upper Tail Tests H0: μ≤ 3 H1: μ > 3 • There is only one critical value, since the rejection area is in only one tail a Do not reject H0 Reject H0 t tα 3 Critical value

27. Assumptions of the One-Sample t Test • The data is randomly selected • The population is normally distributed orthe sample size is over 30 and the population is not highly skewed

28. Hypothesis Tests for Proportions • Involves categorical values • Two possible outcomes • “Success” (possesses a certain characteristic) • “Failure” (does not possesses that characteristic) • Fraction or proportion of the population in the “success” category is denoted by p

29. Proportions (continued) • Sample proportion in the success category is denoted by ps • When both np and n(1-p) are at least 5, pscan be approximated by a normal distribution with mean and standard deviation

30. Hypothesis Tests for Proportions • The sampling distribution of ps is approximately normal, so the test statistic is a Z value: Hypothesis Tests for p np  5 and n(1-p)  5 np < 5 or n(1-p) < 5 Not discussed in this chapter

31. Example: Z Test for Proportion A marketing company claims that it receives 8% responses from its mailing. To test this claim, a random sample of 500 were surveyed with 25 responses. Test at the  = .05 significance level. Check: np = (500)(.08) = 40 n(1-p) = (500)(.92) = 460 

32. a= .05 n = 500, ps = .05 Z Test for Proportion: Solution Test Statistic: H0: p = .08 H1: p ¹ .08 Critical Values: ± 1.96 p-value for -2.47 is .0134 Decision: Reject H0 at  = .05 Reject Reject .025 .025 There is sufficient evidence to reject the company’s claim of 8% response rate. Conclusion: z -1.96 0 1.96 -2.47

33. Potential Pitfalls and Ethical Considerations • Use randomly collected data to reduce selection biases • Do not use human subjects without informed consent • Choose the level of significance, α, before data collection • Do not employ “data snooping” to choose between one-tail and two-tail test, or to determine the level of significance • Do not practice “data cleansing” to hide observations that do not support a stated hypothesis • Report all pertinent findings

34. Summary • Addressed hypothesis testing methodology • Discussed critical value and p–value approaches to hypothesis testing • Discussed type 1 and Type2 errors • Performed two tailed t test for the mean (σ unknown) • Performed Z test for the proportion • Discussed one-tail and two-tail tests • Addressed pitfalls and ethical issues