1. McGraw-Hill/IrwinCopyright © 2009 by The McGraw-Hill Companies, Inc. All Rights Reserved. Chapter 9 Hypothesis Testing

2. 9-2 Hypothesis Hypothesis is a statement about a certain population parameter. It is usually about a parameter equal to (not equal to), less than or equal to ( greater than), greater than or equal to(or less than) some given number. For example: the supporting rate of a candidate among all voters is greater than 50%. A hypothesis testing is a statistical technique for evaluating if there is enough evidence to support a hypothesis. The strength of evidence is evaluated by the probability of certain statistic, called test statistic.

3. 9-3 Rule of Rare Events If under current assumption, the probability to observe the sample (statistic) we have is extremely small, then we conclude the assumption is not right. How small is small? if the probability to observe the sample is less than the Level of significance .  =0.1,.05,.01 It is determined by the risk requirement of the problem. The risk is the probability to make a mistake.

4. 9-4 Rule of Events, final version If under current assumption (H-null), the probability to observe the test statistic is  , then we conclude the assumption is not right. If under H-null, the p-value is  , then we conclude H-null assumption is not right.

5. 9-5 How to set up the symbolic form of the hypothesis- express relationship with math expressions Find out the hypothesis of interest about the population parameter (mean), and write down its symbolic expression. Write down the symbolic expression of the complement. The expression with equal sign (=, ≤, ≥ ) is called H-null hypothesis (denoted by H 0 ), and the remaining expression not containing equal sign is called the H-A hypothesis (denoted by H a ).

6. 9-6 Types of Hypotheses Right-Sided tailed, “Greater Than” Alternative H 0 :    0 vs.H a :  >  0 Left-Sided tailed, “Less Than” Alternative H 0 :    0 vs.H a :  <  0 Two-Sided tailed, “Not Equal To” Alternative H 0 :  =  0 vs. H a :    0 where  0 is a given constant value (with the appropriate units) that is a comparative value

7. 9-7 Types of Hypotheses -the red part is the conversion followed by some textbooks Right-Sided (Right-Tailed), “Greater Than” Alternative H 0 :    0 vs.H a :  >  0 H 0 :  =  0 vs.H a :  >  0 Left-Sided (Left-Tailed), “Less Than” Alternative H 0 :    0 vs.H a :  <  0 H 0 :  =  0 vs.H a :  <  0 Two-Sided (Two-Tailed), “Not Equal To” Alternative H 0 :  =  0 vs. H a :    0 where  0 is a given constant value (with the appropriate units) that is a comparative value

8. 9-8 Z Tests about a Population Mean: σ known The population standard deviation σ is known. Suppose the population being sampled is normally distributed, or sample size n is at least 30. Under these two conditions, use the Z distribution to calculate the p-value and then use the rule of rare events to perform the hypothesis testing.

9. 9-9 Z Test Statistic-σ is known Use the “test statistic” If the population is normal or n is large *, the test statistic t.s. follows a normal distribution * n ≥ 30, by the Central Limit Theorem

10. 9-10 Z Tests about a Population Mean: σ known AlternativeType of testp-value H a : µ > µ 0 P(Z>t.s.) H a : µ < µ 0 P(Z<t.s.) H a : µ  µ 0 2*P(Z>t.s.) for t.s.>0 2*P(Z<t.s.) for t.s.<0 Reject H 0 if: p-value≤ 

11. 9-11 Right-tailed test: P(Z>t.s.) z 0123 -3-2 Right-tailed Test H 0 : μ ≤ k H a : μ > k Test statistic P is the area to the right of the test statistic.

12. 9-12 Left-tailed Test Left-tailed test: P(Z<t.s.) z 0123- 3 - 2 - 1 Test statistic H 0 : μ  k H a : μ < k P is the area to the left of the test statistic.

13. 9-13 Two-tailed Test Two-tailed test: 2*P(Z>t.s.) for t.s.>0 2*P(Z<t.s.) for t.s.<0 z 0 123-3-2 Test statistic H 0 : μ = k H a : μ  k P is twice the area to the left of the negative test statistic. P is twice the area to the right of the positive test statistic.

14. 9-14 Hypothesis Testing Conclusion If p-value≤ , then we say we reject H- null and accept Ha. If p-value> , we say we fail to reject H- null and do not accept Ha. Never say we accept H-null.

15. 9-15 To use the rule of rare events we only need to know the relationship between p-value and the given significance level. See slide 3. The rejection region method explores the property and sets up rejection regions in which any value corresponds to a p-value less than the given significance level . That means if the t.s. is on the rejection region then we reject H 0. Z Tests about a Population Mean: σ known, rejection region method

16. 9-16 Z a and Right Hand Tail Areas The definition of the critical value Z αThe definition of the critical value Z α Z αZ α The area to the right if 1-αThe area to the right if 1-α Z α is the percentile such thatZ α is the percentile such that P(Z< Z α ) =1-αP(Z< Z α ) =1-α

17. 9-17 Right-tailed test, for any given significance level  z 0123 -3-2 Right-tailed Test, rejection region H 0 : μ ≤ k H a : μ > k Test statistic The area to the left of z  is α. z z

18. 9-18 Left-tailed Test, rejection region Left-tailed test z 0123- 3 - 2 - 1 Test statistic H 0 : μ  k H a : μ < k The area to the left of - z  is α. -z 

19. 9-19 Two-tailed Test, rejection region Two-tailed test z 0 123-3-2 H 0 : μ = k H a : μ  k The area to the left of - z  is α/2. The area to the right of z  is α/2. -z  z 

20. 9-20 Z Tests about a Population Mean: σ known, rejection region method AlternativeReject H 0 if:Rejection region H a : μ > µ 0 t.s. ≥ z  [z  ∞ ) H a : μ < µ 0 t.s. ≤ –z  (-∞  - z  ] H a : μ  µ 0 either t.s. ≥ z  /2 or t.s. ≤ –z  /2 (-∞  - z  ] [z  ∞ ) Where the test statistics is

21. 9-21 t Tests about a Population Mean: σ Unknown The population standard deviation σ is unknown, as is the usual situation, but the sample standard deviation s is given. The population being sampled is normally distributed or sample size is n≥30. Under these two conditions, we can use the t distribution to test hypotheses

22. 9-22 Defining the t Statistic: σ Unknown Let  be the mean of a sample of size n with standard deviation s Also, µ 0 is the claimed value of the population mean Define a new test statistic If the population being sampled is normal or sample size is big enough, and s is given… The sampling distribution of the t.s. is a t distribution with n – 1 degrees of freedom

23. 9-23 t Tests about a Population Mean: σ Unknown Continued AlternativeReject H 0 if:Rejection region H a : µ > µ 0 t.s. ≥ t  [t  ∞ ) H a : µ < µ 0 t.s. ≤ –t  (-∞  - t  ] H a : µ  µ 0 t.s. ≥ t  /2 or t.s. ≤ –t  /2 (-∞  - t  ] [t  ∞ ) t , t  /2, and p-values are based on n – 1 degrees of freedom (for a sample of size n)

24. 9-24 Definition of the critical value t α

25. 9-25 Type I and Type II errors If we reject H 0, then it is possible to make type I error If we fail to reject H 0 and do not accept H a (or equivalently: fail to reject H 0 and reject H a ), then it is possible to make type II error.

26. 9-26 Selecting an Appropriate Test Statistic for a Test about a Population Mean