More on Type I and Type II Errors - Santa Rosa Junior College



More on Type I and Type II ErrorsAfter you have been doing hypothesis testing for a while, you may start to get a little wary: after call, this is not some sort of bullet-proof strategy for making decision, and there are two types of errors that could occur. At the back of your mind, you probably have the same question that people often ask: Is it better to make a Type II error than Type I error? The short answer is that Type I Error is not necessarily "better" error than Type II. They are just simply two different animals. Let me use an example to illustrate how they are different. Suppose that you buy a 12-oz cup of coffee from Starbucks daily, but after 10 days, you think on average, you are getting less than what you paid for, since the sample mean is oz. So you would have the hypotheses:ozoz?Suppose is true, i.e. you are getting an average of 12 oz or more, but you insist that you are getting less (i.e. rejecting a true ), you would have made a Type I Error. You can think of Type I error as the one made by the "overly critical" person, who sees something wrong within?the smallest evidence.?If it’s helpful, you can use the image of a “conspiracy theorist” to remember that. What about Type II error? (The probability of a Type II error is represented by another greek letter, ). If somehow the baristas at Starbucks were not properly trained, and they got used to putting less than 12 oz in your cup, then is false, and is true. However, if your hypothesis test shows that the P-value > , then you will not be able to reject (i.e. concluding that you are getting less than what you paid for). If someone makes Type II error all the time, then s/he might be a bit too oblivious of things that go wrong. You would probably not want to have this clueless person as your babysitter! Among the two types of errors, Type I error is easier to control, since we can just choose a smaller (significance level) so that we become less "critical" in finding something wrong. The smaller the , the smaller the chance of a Type I error will be. In the coffee example, sufficiently small values would not allow you to complain to the manager if your sample mean is 11 oz, but would prompt you to complain if your sample mean is 10oz. Using smaller might make you look less aggressive, since you are giving them the benefit of doubt that it could be occurring by chance. However, using extremely small has an unintended consequence. If due to some systematic manipulation, the customers are only getting an average of 11.5 oz (i.e. , so is true),? with a very small (say ), it becomes extremely difficult to reject the false .? So the high chance of Type II errors means that coffee shops can get away with serving people less coffee than advertised, since you will have to have a very small sample mean (say oz) in order to reject . Our example shows that using alone to control both type of errors is simply impossible, and it's not exactly useful to make the chance of Type I Error as small as possible either. How can we control the Type II Error () then? The answer to this, not mentioned in the textbook, is that we will have to increase the sample size (guess what? More data never hurts!), which leads to the increase of "power" of the hypothesis test, defined as . You will study more about the power of the test, as well as how you use power to decide an appropriate sample size, in another advanced statistics course. The web applet we used for last week’s discussion can be used to simulate how the sample size affects power of the test, if you are interested.Here is the summary of the decisions again, together with the concept of power: Reality is true is falseDecisionReject Prob of Type I: Power = Do not reject Prob of correct null decision: Prob of Type II: ................
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