Hypothesis Testing

1. Hypothesis Testing

2. Hypothesis Testing • Hypothesisis a claim or statement about a property of a population. • Hypothesis Testing is to test the claim or statement • Example: A conjecture is made that “the average starting salary for computer science gradate is Rs 45,000 per month”.

3. Hypothesis Testing • Null Hypothesis (H0): is the statement being tested in a test of hypothesis. • Alternative Hypothesis (H1): is what is believe to be true if the null hypothesis is false.

4. Null Hypothesis • Must contain condition of equality =, ³, or £ • Test the Null Hypothesis directly • Reject H0 or fail to reject H0

5. Alternative Hypothesis • Must be true if H0 is false • ¹, <, > • ‘opposite’ of Null Hypothesis Example H0 : µ = 30 versus H1 : µ > 30 or H1 : µ < 30

6. Identify the Problem • State the Null Hypothesis (H0: m³ 3) • State its opposite, the Alternative Hypothesis (H1: m < 3) • Hypotheses are mutually exclusive

7. Hypothesis Testing Process Assumethe population meanage is 50. (NullHypothesis) Population The Sample Mean Is 20 No, not likely! REJECT Null Hypothesis Sample

8. Reason for RejectingH0 Sampling Distribution It is unlikely that we would get a sample mean of this value ... ... Therefore, we reject the null hypothesis that m = 50. ... if in fact this were the population mean. m = 50 Sample Mean 20 H0

9. Level of Significance,a • Defines Unlikely Values of Sample Statistic if Null Hypothesis Is True • Called Rejection Region of Sampling Distribution • Designateda(alpha) • Typical values are 0.01, 0.05, 0.10 • Selectedby the Researcherat the Start • Provides theCritical Value(s)of the Test

10. Level of Significanceand the Rejection Region Critical Value(s) a H0: m³ 3 H1: m < 3 Rejection Regions 0 a H0: m£ 3 H1: m > 3 0 a/2 H0: m= 3 H1: m¹ 3 0

11. Errors in Making Decisions • Type I Error • Reject True Null Hypothesis • Has Serious Consequences • Probability of Type I Error is a • Called Level of Significance • Type II Error • Do Not Reject False Null Hypothesis • Probability of Type II Error Is b (Beta)

12. Result Possibilities H0: Innocent Hypothesis Test Jury Trial Actual Situation Actual Situation Innocent Guilty H True H False Verdict Decision 0 0 Do Not Type II a Correct Error Innocent 1 - Reject b ) Error ( H 0 Type I Reject Error Correct Guilty Error H (1 - ) b 0 ) ( a

13. Type I Error • The mistake of rejecting the null hypothesis when it is true. • The probabilityof doing this is called the significance level, denoted by a(alpha). • Common choices for a: 0.05 and 0.01 • Example: rejecting a perfectly good parachute and refusing to jump

14. Type II Error • The mistake of failing to reject the null hypothesis when it is false. • Denoted by ß (beta) • Example: Failing to reject a defectiveparachute and jumping out of a plane with it.

15. CriticalRegion • Set of all values of the test statistic that would cause a rejection of the null hypothesis. Critical Region

16. Critical Region • Set of all values of the test statistic that would cause a rejection of the null hypothesis. Critical Region

17. Critical Region • Set of all values of the test statistic that would cause a rejection of the null hypothesis. Critical Regions

18. Critical Value Value (s) that separates the critical region from the values that would not lead to a rejection of H0. Reject H0 Fail to reject H0 Critical Value ( z score )

19. Conclusions in Hypothesis Tests Original claim is H0 (This is the only case in which the original claim is rejected). “There is sufficient evidence to warrant rejection of the claim that. . . (original claim).” Yes (Reject H0) Do you reject H0?. No (Fail to reject H0) “There is not sufficient evidence to warrant rejection of the claim that. . . (original claim).” Original claim is H1 (This is the only case in which the original claim is supported). Yes (Reject H0) “The sample data supports the claim that . . . (original claim).” Do you reject H0? No (Fail to reject H0) “There is not sufficient evidence to support the claim that. . . (original claim).”

20. Left-tailed Test H0: µ ³ 200 H1: µ < 200 Points Left Reject H0 Fail to reject H0 Values that differ significantly from 200 200

21. Right-tailed Test H0: µ £ 200 H1: µ > 200 Points Right Reject H0 Fail to reject H0 Values that differ significantly from 200 200

22. Two-tailed Test a is divided equally between the two tails of the critical region H0: µ = 200 H1: µ ¹ 200 Means less than or greater than Reject H0 Reject H0 Fail to reject H0 200 Values that differ significantly from 200

23. Definition Test Statistic: is a sample statistic or value based on sample data Example: x–µx z= s / n

24. Question: How can we justify/test this conjecture? A. What do we need to know to justify this conjecture? B. Based on what we know, how should we justify this conjecture?

25. Answer to A: Randomly select, say 100, computer science graduates and find out their annual salaries ---- We need to have some sample observations, i.e., a sample set!

26. Answer to B: That is what we will learn in this chapter ---- Make conclusions based on the sample observations

27. Statistical Reasoning Analyze the sample set in an attempt to distinguish between results that can easily occur and results that are highly unlikely.

28. Figure 7-1 Central Limit Theorem:

29. Likely sample means µx = 30k Figure 7-1 Central Limit Theorem: Distribution of Sample Means Assume the conjecture is true!

30. Likely sample means µx = 30k z= –1.96 x = 20.2k z = 1.96 x= 39.8k or or Figure 7-1 Central Limit Theorem: Distribution of Sample Means Assume the conjecture is true!

31. Likely sample means µx = 30k z= –1.96 x = 20.2k z = 1.96 x= 39.8k or or Figure 7-1 Central Limit Theorem: Distribution of Sample Means Assume the conjecture is true! Sample data: z= 2.62 x= 43.1k or

32. Components of aFormal Hypothesis Test

33. Two-tailed,Left-tailed,Right-tailedTests

34. Problem 1 Test µ = 0 against µ > 0, assuming normally and using the sample [multiples of 0.01 radians in some revolution of a satellite] 1, -1, 1, 3, -8, 6, 0 (deviations of azimuth) Choose α = 5%.

35. Problem 2 In one of his classical experiments Buffon obtained 2048 heads in tossing a coin 4000 times. Was the coin fair?

36. Problem 3 In one of his classical experiments K Pearson obtained 6019 heads in 12000 trials. Was the coin fair?

37. Problem 5 Assuming normality and known variance б2 = 4, test the hypotheses µ = 30 using a sample of size 4 with mean X = 28.5 and choosing α = 5%.

38. Problem 7 Assuming normality and known variance б2 = 4, test the hypotheses µ = 30 using a sample of size 10 with mean X = 28.5. What is the rejection region in case of a two sided test with α= 5%.

39. Problem 9 A firm sells oil in cans containing 1000 g oil per can and is interested to know whether the mean weight differs significantly from 1000 g at the 5% level, in which case the filling machine has to be adjusted. Set up a hypotheses and an alternative and perform the test, assuming normality and using a sample of 20 fillings with mean 996 g and Standard Deviation 5g.

40. Problem 11 If simultaneous measurements of electric voltage by two different types of voltmeter yield the differences (in volts) 0.8, 0.2, -0.3, 0.1, 0.0, 0.5, 0.7 and 0.2 Can we assert at the 5% level that there is no significant difference in the calibration of the two types of instruments? Assume normality.