1. One Sample Inf-1 In statistical testing, we use deductive reasoning to specify what should happen if the conjecture or null hypothesis is true. A study is designed to collect data to challenge this hypothesis. We determine if what we expected to happen is supported by the data. If it is not supported, we reject the conjecture. If it cannot be rejected we (tentatively) fail to reject the conjecture. We have not proven the conjecture, we have simply not been able to disprove it with these data. Statistical Test for Population Mean

2. One Sample Inf-2 Statistical tests are based on the concept of Proof by Contradiction. If P then Q  =  If NOT Q then NOT P Analogy with justice system Logic Behind Statistical Tests Court of Law 1.Start with premise that person is innocent. (Opposite is guilty.) 2.If enough evidence is found to show beyond reasonable doubt that person committed crime, reject premise. (Enough evidence that person is guilty.) 3.If not enough evidence found, we don’t reject the premise. (Not enough evidence to conclude person guilty.) Statistical Hypothesis Test 1.Start with null hypothesis, status quo. (Opposite is alternative hypothesis.) 2.If a significant amount of evidence is found to refute null hypothesis, reject it. (Enough evidence to conclude alternative is true.) 3.If not enough evidence found, we don’t reject the null. (Not enough evidence to disprove null.)

3. One Sample Inf-3 CLAIM: A new variety of turf grass has been developed that is claimed to resist drought better than currently used varieties. CONJECTURE: The new variety resists drought no better than currently used varieties. DEDUCTION:If the new variety is no better than other varieties (P), then areas planted with the new variety should display the same number of surviving individuals (Q) after a fixed period without water than areas planted with other varieties. CONCLUSION:If more surviving individuals are observed for the new varieties than the other varieties (NOT Q), then we conclude the new variety is indeed not the same as the other varieties, but in fact is better (NOT P). Examples in Testing Logic

4. One Sample Inf-4 1.Null Hypothesis (H 0 :) 2.Alternative Hypothesis (H A :) 3.Test Statistic (T.S.) Computed from sample data. Sampling distribution is known if the Null Hypothesis is true. 4.Rejection Region (R.R.) Reject H 0 if the test statistic computed with the sample data is unlikely to come from the sampling distribution under the assumption that the Null Hypothesis is true. 5.Conclusion: Reject H 0 or Do Not Reject H 0 H 0 :  ≤ 530 H A :  > 530 Other H A :  < 530   530 R.R. z * > z  Or z * < -z   r  z * | > z  Five Parts of a Statistical Test

5. One Sample Inf-5 Research Hypotheses: The thing we are primarily interested in “proving”. Average height of the class Null Hypothesis: Things are what they say they are, status quo. Hypotheses (It’s common practice to always write H 0 in this way, even though what is meant is the opposite of H A in each case.)

6. One Sample Inf-6 Some function of the data that uses estimates of the parameters we are interested in and whose sampling distribution is known when we assume the null hypothesis is true. Most good test statistics are constructed using some form of a sample mean. Test Statistic Why? The Central Limit Theorem Of Statistics

7. One Sample Inf-7 Test Statistic: Sample Mean Under H 0 : the sample mean has a sampling distribution that is normal with mean  0. Under H A : the sample mean has a sampling distribution that is also normal but with a mean   that is different from  0. Hypotheses of interest: Developing a Test Statistic for the Population Mean

8. One Sample Inf-8 H 0 : True H A 1 : True 00 11 What would you conclude if the sample mean fell in location (1)? How about location (2)? Location (3)? Which is most likely 1, 2, or 3 when H 0 is true? (1)(2)(3) Graphical View

9. One Sample Inf-9 00 Reject H 0 if the sample mean is in the upper tail of the sampling distribution. Rejection Region C1C1 How do we determine C 1 ? H 0 : Assumed True Testing H A 1 :  0 Rejection Region

10. One Sample Inf-10 Reject H 0 if the sample mean is larger than “expected”. If H 0 were true, we would expect 95% of sample means to be less than the upper limit of a 90% CI for μ. In this case, if we use this critical value, in 5 out of 100 repetitions of the study we would reject H o incorrectly. That is, we would make an error. But, suppose H A 1 is the true situation, then most sample means will be greater than C 1 and we will be making the correct decision to reject more often. From the standard normal table. Determining the Critical Value for the Rejection Region

11. One Sample Inf-11 H 0  =  0 True Testing H A 1 :  0 Rejection Region C1C1 00 C2C2 H 0 :  =  0 True Testing H A 1 :  0 Rejection Region C3LC3L H 0 :  =  0 True Testing H A 1 :  0 C3UC3U 2.5% 5% 2.5% Rejection Regions for Different Alternative Hypotheses

12. One Sample Inf-12 H 0 : True 00 (1) (2) (3) Rejection Region C1C1 If the sample mean is at location (2) and H 0 is actually true, we make the wrong decision. This is called making a TYPE I error, and the probability of making this error is usually denoted by the Greek letter . If  = P(Reject H 0 when H 0 is is the true condition) If then  = 1/20=5/100 or.05 then the Type I error is . If the sample mean is at location (1) or (3) we make the correct decision. Type I Error

13. One Sample Inf-13 H A 1 : True 11 Rejection Region C1C1 00 (1) (2) (3)  = P(Do Not Reject H 0 when H A is the true condition) We will come back to the Type II error later. If the sample mean is at location (1) or (3) we make the wrong decision. If the sample mean is at location (2) we make the correct decision. Type II Error This is called making a TYPE II error, and the probability of making this type error is usually denoted by the Greek letter .

14. One Sample Inf-14 H 0 : True H A 1 : True 00 11 Type I and II Errors C1C1 Rejection Region Type I Error Region Type II Error Region We want to minimize the Type I error, just in case H 0 is true, and we wish to minimize the Type II error just in case H A is true. Problem: A decrease in one causes an increase in the other. Also: We can never have a Type I or II error equal to zero!  

15. One Sample Inf-15 1.Assume the data come from a population with unknown distribution but which has mean (  ) and standard deviation (  ). 2.For any sample of size n from this population we can compute a sample mean,. 3.Sample means from repeated studies having samples of size n will have the sampling distribution of following a normal distribution with mean  and a standard deviation of  /  n (the standard error of the mean). This is the Central Limit Theorem. 4.If the Null Hypothesis is true and  =  0, then we deduce that with probability  we will observe a sample mean greater than  0 + z   /  n. (For example, for  = 0.05, z  =1.645.) The solution to our quandary is to set the Type I Error Rate to be small, and hope for a small Type II error also. The logic is as follows: Setting the Type I Error Rate

16. One Sample Inf-16 Following this same logic, rejection rules for all three possible alternative hypotheses can be derived. Reject H o if: For (1-  )100% of repeated studies, if the true population mean is  0 as conjectured by H 0, then the decision concluded from the statistical test will be correct. On the other hand, in  100% of studies, the wrong decision (a Type I error) will be made. Setting Rejection Regions for Given Type I Error Note: It is just easier to work with a test statistic that is standardized. We only need the standard normal table to find critical values.

17. One Sample Inf-17 The value 0 <  < 1 represents the risk we are willing to take of making the wrong decision. But, what if I don’t wish to take any risk? Why not set  = 0? What if the true situation is expressed by the alternative hypothesis and not the null hypothesis? 00 11 But this! Suppose H A 1 is really the true situation. Then the samples come from the distribution described by the alternative hypothesis and the sampling distribution of the mean will have mean  1, not  0.  = 0.05  = 0.01  = 0.001 Risk

18. One Sample Inf-18 (at an  Type I error probability) Pretty clear cut when  1 much greater than  0 Do not reject H o Reject H o CC 00 11 The Rejection Rule or if

19. One Sample Inf-19 CC 00 11 When H 0 is true When H A is true Error Probabilities If H A is the true situation, then any sample whose mean is larger than will lead to the correct decision (reject H 0, accept H A ).

20. One Sample Inf-20 Then any sample such that its sample mean is less than will lead to the wrong decision (do not reject H 0, reject H A ). Do not reject H A Reject H A C1C1 If H A is the true situation H 0 : TrueH A : True Reject H 0 Type I Error  Correct 1 -  Accept H 0 Correct 1 -  Type II Error  True Situation Decision

21. One Sample Inf-21 Type I Error Probability Type II Error Probability Power of the test. (Reject H 0 when H 0 true) (Reject H A when H A true) (Reject H 0 when H A true) Computing Error Probabilities

22. One Sample Inf-22 Sample Size: n = 50 Sample Mean: Sample Standard deviation: s = 5.6 H 0 :  =  0 = 38 H A :  > 38 What risk are we willing to take that we reject H 0 when in fact H 0 is true? P(Type I Error) =  =.05Critical Value: z.05 = 1.645 Rejection Region Conclusion: Reject H 0 Example

23. One Sample Inf-23 To compute the Type II error for a hypothesis test, we need to have a specific alternative hypothesis stated, rather than a vague alternative Vague H A :  >  0 H A :  <  0 H A :    0 Specific H A :  =  1 >  0 H A :  =  1 <  0 Note: As the difference between  0 and  1 gets larger, the probability of committing a Type II error (  ) decreases. Significant Difference:  =  0 -  1 Type II Error. H A :  = 5 < 10 H A :  = 20 > 10 H A :   10

24. One Sample Inf-24 H 0 :  =  0 H A :  =  1 >  0 H 0 :  =  0 H A :  =  1 <  0 H 0 :  =  0 H a :    0  =  1   0 1 -  = Power of the test 00 11 00 11 Computing the probability of a Type II Error (  )

25. One Sample Inf-25 Assuming s is actually equal to  (usually what is done): Example: Power

26. One Sample Inf-26 Power versus  (fixed n and  )   Type II error 1-  As  gets larger, it is less likely that a Type II error will be made.

27. One Sample Inf-27 Power versus n  (fixed  and  )  Type II error 1-  As n gets larger, it is less likely that a Type II error will be made. n What happens for larger  ?

28. One Sample Inf-28  n  power 150.755.245 250.286.714 450.001.999  n  power 125.857.143 225.565.435 425.054.946 Power vs. Sample Size Power = 1- 

29. One Sample Inf-29 Pr(Type II Error)  1-  0  0large 1-  0  0large Power  n=10 n=50 Power Curve See Table 4, Ott and Longnecker

30. One Sample Inf-30 1) For fixed  and n,  decreases (power increases) as  increases. 2) For fixed  and ,  decreases (power increases) as n increases. 3) For fixed  and n,  decreases (power increases) as  decreases. Summary

31. One Sample Inf-31   1 <  Decreasing  decreases the spread in the sampling dist of Note: z  changes. Same thing happens if you increase n. Same thing happens if  is increased. N(  ,    n) N(  ,    n) Increasing Precision for fixed  increases Power

32. One Sample Inf-32 1) Specify the critical difference,  (assume  is known). 2)Choose P(Type I error) =  and Pr(Type II error) =  based on traditional and/or personal levels of risk. 3)One-sided tests : 4)Two-sided tests: Example: One-sided test,  = 5.6, and we wish to show a difference of  =.5 as significant (i.e. reject H 0 :  =  0 for H A :  =  1 =  0 +  ) with  =.05 and  =.2. Sample Size Determination