Hypothesis Testing: Understanding Null and Alternate Hypotheses

1. Chapter Nine Hypothesis Testing

2. Hypothesis testing is used to make decisions concerning the value of a parameter.

3. Null Hypothesis: H0 • a working hypothesis about the population parameter in question

4. The value specified in the null hypothesis is often: • a historical value • a claim • a production specification

5. Alternate Hypothesis: H1 • any hypothesis that differs from the null hypothesis

6. An alternate hypothesis is constructed in such a way that it is the one to be accepted when the null hypothesis must be rejected.

7. A manufacturer claims that their light bulbs burn for an average of 1000 hours. We have reason to believe that the bulbs do not last that long. Determine the null and alternate hypotheses.

8. A manufacturer claims that their light bulbs burn for an average of 1000 hours. ... • The null hypothesis (the claim) is that the true average life is 1000 hours. • H0: m = 1000

9. … A manufacturer claims that their light bulbs burn for an average of 1000 hours. We have reason to believe that the bulbs do not last that long. ... If we reject the manufacturer’s claim, we must accept the alternate hypothesis that the light bulbs do not last as long as 1000 hours. H1: m < 1000

10. Types of Statistical Tests • Left-tailed: H1 states that the parameter is less than the value claimed in H0. • Right-tailed: H1 states that the parameter is greater than the value claimed in H0. • Two-tailed: H1 states that the parameter is different from (  ) the value claimed in H0.

11. Given the Null Hypothesis H0:  = k If you believe that  is less thank, Use the left-tailed test: H1:  < k

12. Given the Null Hypothesis H0:  = k If you believe that  is more thank, Use the right-tailed test: H1:  > k

13. Given the Null Hypothesis H0:  = k If you believe that  is different fromk, Use the two-tailed test: H1:   k

14. General Procedure for Hypothesis Testing • Formulate the null and alternate hypotheses. • Take a simple random sample. • Compute a test statistic corresponding to the parameter in H0. • Assess the compatibility of the test statistic with H0.

15. Hypothesis Testing about the Mean of a Normal Distribution with a Known Standard Deviation 

16. P-value of a Statistical Test • Assuming H0 is true, the probability that the test statistic (computed from sample data) will take on values as extreme as or more than the observed test statistic is called the P-value of the test • The smaller the P-value computed from sample data, the stronger the evidence against H0.

17. P-values for Testing a Mean Using the Standard Normal Distribution

18. P-value for a Left-tailed Test • P-value = probability of getting a test statistic less than

19. P-value for a Right-tailed Test • P-value = probability of getting a test statistic greater than

20. P-value for a Two-tailed Test • P-value = probability of getting a test statistic lower than or higher than

21. Types of Errors in Hypothesis Testing • Type I • Type II

22. Type I Error • rejecting a null hypothesis which is, in fact, true

23. Type II Error • not rejecting a null hypothesis which is, in fact, false

24. Type I and Type II Errors

25. Level of Significance, Alpha (a) • the probability of rejecting a true hypothesis • Alpha is the probability of a type I error

26. Type II Error • Beta = β =probability of a type II error (failing to reject a false hypothesis) • In hypothesis testing α and β values should be chosen as small as possible. • Usually α is chosen first.

27. Power of the Test = 1 – β • The probability of rejectingH0 when it is in fact false = 1 – b. • The power of the test increases as the level of significance (a) increases. • Using a larger value of alpha increases the power of the test but also increases the probability of rejecting a true hypothesis.

28. Probabilities Associated with a Statistical Test

29. Hypotheses and Types of Errors A fast food restaurant indicated that the average age of its job applicants is fifteen years. We suspect that the true age is lower than 15. We wish to test the claim with a level of significance of a = 0.01. Determine the Null and Alternate hypotheses and describe Type I and Type II errors.

30. … average age of its job applicants is fifteen years. We suspect that the true age is lower than 15. H0: m = 15 H1: m < 15

31. H0: m= 15H1: m < 15 a = 0.01 A type I error would occur if we rejected the claim that the mean age was 15, when in fact the mean age was 15 (or higher). The probability of committing such an error is as much as 1%.

32. H0: m = 15H1: m < 15 a = 0.01 A type II error would occur if we failed to reject the claim that the mean age was 15, when in fact the mean age was lower than 15. The probability of committing such an error is called beta.

33. Concluding a Hypothesis Test Using the P-value and Level of Significance α • If P-value <αreject the null hypothesis and say that the data are statistically significant at the level α. • If P-value > α, do not reject the null hypothesis.

34. Basic Components of a Statistical Test • Null hypothesis, alternate hypothesis and level of significance • Test statistic and sampling distribution • P-value • Test conclusion • Interpretation of the test results

35. Null Hypothesis, Alternate Hypothesis and Level of Significance • If the sample data evidence against H0 is strong enough, we reject H0 and adopt H1. • The level of significance, α, is the probability of rejecting H0 when it is in fact true.

36. Test Statistic and Sampling Distribution • Mathematical tools to measure compatibility of sample data and the null hypothesis

37. P-value The probability of obtaining a test statistic from the sampling distribution that is as extreme as or more extreme than the sample test statistic computed from the data under the assumption that H0 is true

38. Test Conclusion • If P-value <αreject the null hypothesis and say that the data are statistically significant at the level α. • If P-value > α, do not reject the null hypothesis.

39. Interpretation of Test Results • Give a simple explanation of conclusion in the context of the application.

40. Reject or ... • When the sample evidence is not strong enough to justify rejection of the null hypothesis, we fail to reject the null hypothesis. • Use of the term“accept the null hypothesis” should be avoided. • When the null hypothesis cannot be rejected, a confidence interval is frequently used to give a range of possible values for the parameter.

41. Fail to Reject H0 • There is not enough evidence to reject H0. The null hypothesis is retained but not proved.

42. Reject H0 • There is enough evidence to reject H0. Choose the alternate hypothesis with the understanding that it has not been proven.

43. Testing the Mean  When  is Known • Let x be the appropriate random variable. Obtain a simple random sample (of size n) of x values and compute the sample mean x. • State the null and alternate hypotheses and set the level of significance α. • If x has a normal distribution, any sample size will work. If we cannot assume a normal distribution, use n> 30.

44. Testing the Mean  When  is Known • Use the test statistic:

45. Testing the Mean  When  is Known • Use the standard normal distribution and the type of test (one-tailed or two-tailed) to find the P-value corresponding to the test statistic. • If the P-value <α, then reject H0. If the P-value > α, then do not reject H0. • State your conclusion.

46. Testing the Mean When  is Known: Example Your college claims that the mean age of its students is 28 years. You wish to check the validity of this statistic with a level of significance of a = 0.05. Assume the standard deviation is 4.3 years. A random sample of 49 students has a mean age of 26 years.

47. Hypothesis Test Example H0:  = 28 H1: ¹ 28 Perform a ________-tailed test. two Level of significance = α = 0.05

48. Sample Test Statistic

49. Sample Results For a two-tailed test: P-value = 2P(z < 3.26) = 2(0.0006) = 0.0012

50. P-value and Conclusion • P-value = 0.0012 • α = 0.05. Since the P-value < α , we reject the null hypothesis. • We conclude that the true average age of students is not 28.