Chapter 21 More About Hypothesis Tests Using a Single Sample

2 Even with lots of evidence, we can still make the wrong decision. When we perform a hypothesis test, we can make mistakes in two ways. TYPE I The null hypothesis is true, but we mistakenly reject it. TYPE II The null hypothesis is false, but we fail to reject it. Nobody’s Perfect

3 The probability of a type I error is denoted by  and is called the level of significance of the test. Thus, a test with  = 0.01 is said to have a level of significance of 0.01 or to be a level 0.01 test. You have the option to choose your alpha level. Appropriately for the situation. Ex. Assessing the safety of airbags – low alpha level.01 Ex. Just wondering whether students prefer pizza with or without pepperoni – higher alpha Common use of α is.05 Significance Level

4 The common standard of rejecting the null hypothesis is when the p-value < α When this happens, we conclude “there is statistical significant evidence to support rejecting the null hypothesis, ….. Since p-value < α Significance Level – Type I The First Error

5 When the p value > α, we fail to reject the Null hypothesis. When this happens, we Conclude, “ there is statistical significant evidence to support failing to reject the null hypothesis No Error

6 H o : I will be better off if I take no action. H a : I will be better off if I take action. Type I Error would correspond to taking action When you would have been better off taking action. Type II Error would correspond to taking no action when you would have been better off taking action. Action Or No Action

7 Error

8 Trial by jury example Null HypothesisH o : Innocent DecisionTrueFalse Failed to Reject H o No error Innocent Type II Error β Guilty ; Failed to Go to jail No action taken Reject H o Type I Error α Sent to jail, but innocent Action taken No error Not Guilty

9 Error Analogy Consider a medical test where the hypotheses are equivalent to H 0 : the patient has a specific disease H a : the patient doesn’t have the disease Then, Type I error is equivalent to a false negative (I.e., Saying the patient does not have the disease when in fact, he does.) Type II error is equivalent to a false positive (I.e., Saying the patient has the disease when, in fact, he does not.)

10 Medical example Null HypothesisH o : the patient does have a disease H a : the patient does not have the disease DecisionTrueFalse Failed to Reject H o No error Disease Type II Error β Test shoes Has the disease but in fact, he does not False postive Reject H o Type I Error α Does not have Dissease but if fact does False negative No Error No Disease

11 Medical example Different H o Null HypothesisH o : the patient does not have a disease H a : the patient does have the disease DecisionTrueFalse Failed to Reject H o No error No Disease Type II Error β Sick person diagnosed as Disease Free False negative Reject H o Type I Error α Healthy person is diagnosed with the diseased False positive Disease

12 Examples See Power and Type I and II Errors Worksheet