Justice System Warm-Up: Read “Making Mistakes in the Justice System.” When you are done reading, raise your hand for a questions sheet.
Hypothesis Testing In each test, there are two possible decisions: 1.fail to reject the null hypothesis—any observed difference between the hypothesized value and the sample value is not statistically significant 2.reject the null hypothesis—the difference between the hypothesized value and the sample value is declared to be statistically significant
Fail to Reject the Null Hypothesis You are NOT saying the true population value is the hypothesized value--just that there is no compelling evidence against it (legal analogy) You don’t prove someone innocent, there is just not enough evidence to prove guilt
Reject the Null Hypothesis Declare that the true population value is not equal to the hypothesized value because the sample value is so different from the hypothesized value that the difference is unlikely based on random sampling alone. Rejecting the null hypothesis makes a strong statement that we are quite sure (95% or 99% sure) that the hypothesized value could not be the population value
Hypothesis Testing There are 4 possible outcomes (two for each decision – correct and incorrect). Ho TrueHo False Reject HoType I errorcorrect decision Fail to Reject Ho correct decisionType II error
Hypothesis Testing Each outcome has a certain probability of occurring. We focus on the probability of errors. Ho TrueHo False Reject HoType I errorcorrect decision Fail to Reject Ho correct decisionType II error α β
Hypothesis Testing In which of the four outcomes do we want to be? We want to correctly conclude the alternative. Hence, we want to reject the null in favor of the alternative, when the alternative is in fact true This likelihood is called the “power” of the test. Ho TrueHo False Reject HoType I errorcorrect decision Fail to Reject Ho correct decisionType II error α β power
Hypothesis Testing The “power” is a probability of correctly rejecting the Null Hypothesis. Power = 1 - β Ho TrueHo False Reject HoType I errorcorrect decision Fail to Reject Ho correct decisionType II error α β power
Hypothesis Testing Ho TrueHo False Reject HoType I errorcorrect decision Fail to Reject Ho correct decisionType II error αβ power 1 - β
Type I Error The mistake of rejecting the Ho when it is true. This is not a miscalculation or procedural error but a rare event that happens by chance. (A false positive to an HIV test—null hypothesis = no disease: test shows have the disease when actually do not). The probability of rejecting the null hypothesis when it is true is called the significance level or α (alpha). You can decrease the chance of a type I error by increasing the confidence level.
Hypothesis Testing Ho TrueHo False Reject HoType I errorcorrect decision Fail to Reject Ho correct decisionType II error α β power 1 - β
α, β, and n are all related For any fixed α, an increase in the sample size n will cause a decrease in β, and hence an increase in power! For any fixed sample size n, a decrease in α will cause an increase in β and an increase in α will cause a decrease in β. To decrease both α and β, increase the sample size n.
Consider the Consequences of each type of error. M&M’s have a mean weight of.916 g while Bufferin tablets have a mean weight of 325 mg of aspirin. If the M&Ms’ weights are too large, Mars could lose money but customers will likely not complain. If the weights are too small, unless it was WAY off, customers would probably not notice. However, If Bufferin tablets’ weights are off in either direction, the company could face consumer lawsuits or FDA action. Bristol-Myers, the company that makes Bufferin, is thus likely to use a smaller significance level α and a larger sample size n to do its testing because of the more serious consequences.
Remember We want to make Power as large as possible Making the probability of Type II Error as small as possible will make the Power as large as possible Because we are seeking evidence against the null hypothesis, Type I Errors are more serious
Classwork/Homework Read the Type I and Type II Errors Notes Complete the Type I and Type II Error Worksheet If you complete before the end of the period, you can turn it in. Otherwise, it is due at the beginning of the next period.