Hypothesis Testing

1. Hypothesis Testing

2. To define a statistical Test we • Choose a statistic (called the test statistic) • Divide the range of possible values for the test statistic into two parts • The Acceptance Region • The Critical Region

3. To perform a statistical Test we • Collect the data. • Compute the value of the test statistic. • Make the Decision: • If the value of the test statistic is in the Acceptance Region we decide to accept H0 . • If the value of the test statistic is in the Critical Region we decide to reject H0 .

4. The z-test for Proportions Testing the probability of success in a binomial experiment

5. Situation • A success-failure experiment has been repeated n times • The probability of success p is unknown. We want to test • H0: p = p0 (some specified value of p) Against • HA:

6. The Test Statistic • Accept H0 if: • The Acceptance and Critical Region • Reject H0 if: Two-tailed critical region

7. One-tailed critical regions These are used when the alternative hypothesis (HA) is one-sided • Accept H0 if: The Acceptance and Critical Region • Reject H0 if: • Accept H0 if: • Reject H0 if:

8. One-tailed critical regions The Acceptance and Critical Region Accept H0 if: , Reject H0 if:

9. One-tailed critical regions The Acceptance and Critical Region Accept H0 if: , Reject H0 if:

10. Comments • Whether you use a one-tailed or a two-tailed tests is determined by the choice of the alternative hypothesis HA • The alternative hypothesis, HA, is usually the research hypothesis. The hypothesis that the researcher is trying to “prove”.

11. Examples • A person wants to determine if a coin should be accepted as being fair. Let p be the probability that a head is tossed. One is trying to determine if there is a difference (positive or negative) with the fair value of p.

12. A researcher is interested in determining if a new procedure is an improvement over the old procedure. The probability of success for the old procedure is p0(known). The probability of success for the new procedure is p (unknown) . One is trying to determine if the new procedure is better (i.e. p > p0) .

13. A researcher is interested in determining if a new procedure is no longer worth considering. The probability of success for the old procedure is p0(known). The probability of success for the new procedure is p (unknown) . One is trying to determine if the new procedure is definitely worse than the one presently being used (i.e. p < p0) .

14. The z-test for the Mean of a Normal Population We want to test, m, denote the mean of a normal population

15. The Situation • Let x1, x2, x3 , … , xn denote a sample from a normal population with mean m and standard deviation s. • Let • we want to test if the mean, m, is equal to some given value m0. • Obviously if the sample mean is close to m0 the Null Hypothesis should be accepted otherwise the null Hypothesis should be rejected.

16. The Test Statistic

17. The Acceptance and Critical Region • This depends on H0 and HA Two-tailed critical region • Accept H0 if: • Reject H0 if: One-tailed critical regions • Accept H0 if: • Accept H0 if: • Reject H0 if: • Reject H0 if:

18. Example A manufacturer Glucosamine capsules claims that each capsule contains on the average: • 500 mg of glucosamine To test this claim n = 40 capsules were selected and amount of glucosamine (X) measured in each capsule. Summary statistics:

19. Manufacturers claim is correct We want to test: against Manufacturers claim is not correct

20. The Test Statistic

21. The Critical Region and Acceptance Region Using a = 0.05 za/2 = z0.025 = 1.960 We accept H0 if -1.960 ≤ z ≤ 1.960 reject H0 ifz < -1.960 or z > 1.960

22. The Decision Sincez= -2.75 < -1.960 We reject H0 Conclude: the manufacturers’s claim is incorrect:

23. “Students” t-test

24. Recall: The z-test for means The Test Statistic

25. Comments • The sampling distribution of this statistic is the standard Normal distribution • The replacement of s by s leaves this distribution unchanged only the sample size n is large.

26. For small sample sizes: The sampling distribution of Is called “students” t distribution with n –1 degrees of freedom

27. Properties of Student’s t distribution • Similar to Standard normal distribution • Symmetric • unimodal • Centred at zero • Larger spread about zero. • The reason for this is the increased variability introduced by replacing s by s. • As the sample size increases (degrees of freedom increases) the t distribution approaches the standard normal distribution

29. t distribution standard normal distribution

30. The Situation • Let x1, x2, x3 , … , xn denote a sample from a normal population with mean m and standard deviation s. Both m and s are unknown. • Let • we want to test if the mean, m, is equal to some given value m0.

31. The Test Statistic The sampling distribution of the test statistic is the t distribution with n-1 degrees of freedom

32. ta and ta/2 are critical values under the t distribution with n – 1 degrees of freedom

33. a or a/2 Critical values for the t-distribution

34. Critical values for the t-distribution are provided in tables. A link to these tables are given with today’s lecture

35. Look up a Look up df

36. Note: the values tabled for df = ∞ are the same values for the standard normal distribution

37. Example • Let x1, x2, x3 , x4, x5, x6 denote weight loss from a new diet for n = 6 cases. • Assume that x1, x2, x3 , x4, x5, x6 is a sample from a normal population with mean m and standard deviation s. Both m and s are unknown. • we want to test: New diet is not effective versus New diet is effective

38. The Test Statistic The Critical region: Reject if

39. The Data The summary statistics:

40. The Test Statistic The Critical Region (using a= 0.05) Reject if Conclusion: Accept H0:

41. Confidence Intervals

42. Confidence Intervals for the mean of a Normal Population, m, using the Standard Normal distribution Confidence Intervals for the mean of a Normal Population, m, using the t distribution

43. The Data The summary statistics:

44. Example • Let x1, x2, x3 , x4, x5, x6 denote weight loss from a new diet for n = 6 cases. The Data: The summary statistics:

45. Confidence Intervals (use a= 0.05)

46. Comparing Populations Proportions and means

47. Sums, Differences, Combinations of R.V.’s A linear combination of random variables, X, Y, . . . is a combination of the form: L =aX +bY + … where a, b, etc. are numbers – positive or negative. Most common:Sum = X +Y Difference = X –Y Simple Linear combination of X, bX + a

48. Means of Linear Combinations If L =aX +bY + … The mean of Lis: Mean(L) =a Mean(X) +b Mean(Y) + … Most common: Mean( X +Y) = Mean(X) + Mean(Y) Mean(X –Y) = Mean(X) – Mean(Y) Mean(bX + a) = bMean(X) + a

49. Variances of Linear Combinations If X, Y, . . . are independent random variables and L =aX +bY + … then Variance(L) =a2Variance(X) +b2 Variance(Y) + … Most common: Variance( X +Y) = Variance(X) + Variance(Y) Variance(X –Y) = Variance(X) + Variance(Y) Variance(bX + a) = b2Variance(X)

50. Combining Independent Normal Random Variables If X, Y, . . . are independent normal random variables, then L =aX +bY + … is normally distributed. In particular: X +Y is normal with X –Y is normal with