1. Mr. Mark Anthony Garcia, M.S. Mathematics Department De La Salle University

2. Situation: Hypothesis Testing

5. Hypothesis Testing A statistical hypothesis is an assertion or conjecture concerning one or more populations. A test of hypothesis is the method to determine whether the statistical hypothesis is true or not.

6. Types of Statistical Hypothesis We have two types of statistical hypothesis. We have 1. Null Hypothesis 2. Alternative Hypothesis

7. Null Hypothesis

8. Alternative Hypothesis  The alternative hypothesis is the hypothesis that is accepted if the null hypothesis is rejected.  This hypothesis allows for the possibility of several values and it is denoted by H 1.  This hypothesis may be directional (quantifier is ) or non-directional (quantifier is  ).

9. Null and Alternative Hypothesis

10. Example: Null and Alternative Hypothesis In the political analyst situation, the null and alternative hypothesis is given as follows.

11. Example: Null and Alternative Hypothesis No more than 20% of the faculty at the local university contributed to the annual giving fund.

12. Example: Null and Alternative Hypothesis At most 65% of public school children are malnourished.

13. Example: Null and Alternative Hypothesis Fifty-five percent of elected public officials came from the same university.

14. Example: Null and Alternative Hypothesis At least 70% of next year's new cars will be in the compact and subcompact category.

15. Example: Null and Alternative Hypothesis The proportion of voters favoring the incumbent in the upcoming elections is 0.58.

16. Elements of Hypothesis Testing

17. Null hypothesis TRUEFALSE RejectTYPE I ErrorCorrect Decision Do not RejectCorrect DecisionTYPE II Error

18. One-tailed vs. Two-tailed Tests

19. Test Statistic  The test statistic is the value generated from sample data. Its value is then compared with the critical values.  In the PhStat output, it is referred to as the Z Test Statistic.

20. Test Statistic

22. Critical Region The critical region is sometimes called the rejection region. If the critical region is satisfied, then the null hypothesis is rejected.

23. Critical Region: Left-Tailed

24. Critical Region: Right-Tailed

25. Critical Region: Two-tailed

26. Critical Region

28. Conclusion  Since the critical region is NOT satisfied, we do not reject the null hypothesis that the population proportion of voters is at least 70%.  Hence, there is sufficient evidence to conclude that senatorial candidate A will top the senatorial elections in city X.

29. Example 1: Hypothesis Testing A chocolate manufacturer targets an 8 out of 10 public approval of their new chocolate recipe to release in the market. A random sample of 80 people where given a taste test and resulted a 75% approval of the product. Will the company release the product in the market with a 0.05 level of significance?

30. Example 1: Hypothesis Testing

32. Z Test of Hypothesis for the Proportion Data Null Hypothesis p=0.8 Level of Significance0.05 Intermediate Calculations Sample Proportion0.75 Standard Error0.04472136 Z Test Statistic-1.118033989 Two-Tailed Test Lower Critical Value-1.959963985 Upper Critical value1.959963985 p-Value0.263552477

33. Example 1: Hypothesis Testing

34. Example 2: Hypothesis Testing A commonly prescribed drug on the market for relieving nervous tension is believed to be only 60% effective. Experimental results with a new drug administered to a random sample of 100 adults who were suffering from nervous tension showed that 70 received relief. Is this sufficient evidence to conclude that the new drug is superior to the one commonly prescribed? Use a 0.05 level of significance.

35. Example 2: Hypothesis Testing

37. Data Null Hypothesis p=0.6Null Hypothesis p=0.6 Level of Significance0.05Level of Significance0.05 Number of Successes70Number of Successes70 Sample Size100Sample Size100 Intermediate Calculations Sample Proportion0.7Sample Proportion0.7 Standard Error0.048989795Standard Error0.048989795 Z Test Statistic2.041241452Z Test Statistic2.041241452 Upper-Tail TestLower-Tail Test Upper Critical Value1.644853627Lower Critical Value-1.644853627 p-Value0.020613417p-Value0.979386583

38. Example 2: Hypothesis Testing

39. P-value Approach  The p-value or the probability value approaches hypothesis testing in a different manner.  Instead of comparing the test statistic with the critical values, we compare the p-value with the level of significance.

40. P-value Approach  If the test statistic is in the critical region, then the p-value will be less than the level of significance.  It does not matter whether it is a left tail, right tail, or two tail test. This rule always holds.  Reject the null hypothesis if the p- value is less than the level of significance.