Chapter 10 Lesson 10.2 Hypotheses and Test Procedures 10.2: Errors in Hypothesis Testing

When you perform a hypothesis test you make a decision: reject H 0 or fail to reject H 0 Each could possibly be a wrong decision; therefore, there are two types of errors. When you make one of these decisions, there is a possibility that you could be wrong!

Type I error The error of rejecting H 0, when H 0 is true The probability of a Type I error is denoted by .  is called the ___________ of the test

Type II error The error of failing to reject H 0, when H 0 is false The probability of a Type II error is denoted by 

H 0 is true H 0 is false Reject H 0 Fail to reject H 0 Type I error Type II error Suppose H 0 is true and we fail to reject it, what type of decision was made? Suppose H 0 is false and we reject it, what type of decision was made? Suppose H 0 is true and we reject it, what type of decision was made? Suppose H 0 is false and we fail to reject it, what type of decision was made? Here is another way to look at the types of errors: The Truth Your Decision

Type I error – the airline decides to reward the employees when the proportion of on-time flights doesn’t exceeds.72 The U.S. Bureau of Transportation Statistics reports that 72% of all domestic passenger flights arrived on time (meaning within 15 minutes of its scheduled arrival time). Suppose that an airline with a poor on- time record decides to offer its employees a bonus if, in an upcoming month, the airline’s proportion of on- time flights exceeds the overall industry rate of.72. H 0 : p =.72 H a : p >.72 State the hypotheses. State a Type I error in context. Type II error – the airline employees do not receive the bonus when they deserve it. State a Type II error in context. What are the potential consequences of these errors?

In 2004, Vertex Pharmaceuticals, a biotechnology company, issued a press release announcing that it had filed an application with the FDA to begin clinical trials on an experimental drug VX-680 that had been found to reduce the growth rate of pancreatic and colon cancer tumors in animal studies. Data resulting from the planned clinical trials can be used to test: Let  = the true mean growth rate of tumors for patients taking the experimental drug H 0 :  = mean growth rate of tumors for patients not taking the experimental drug H a :  < mean growth rate of tumors for patients not taking the experimental drug State a Type I error in the context of this problem. A Type I error would be to incorrectly conclude that the experimental drug is effective in slowing the growth rate of tumors What is a potential consequence of this error? A potential consequence of making a Type I error would be that the company would continue to devote resources to the development of the drug when it really is not effective.

In 2004, Vertex Pharmaceuticals, a biotechnology company, issued a press release announcing that it had filed an application with the FDA to begin clinical trials on an experimental drug VX-680 that had been found to reduce the growth rate of pancreatic and colon cancer tumors in animal studies. Data resulting from the planned clinical trials can be used to test: H 0 :  = mean growth rate of tumors for patients not taking the experimental drug H a :  < mean growth rate of tumors for patients not taking the experimental drug State a Type II error in the context of this problem. A Type II error would be to conclude that the drug is ineffective when in fact the mean growth rate of tumors is reduced What is a potential consequence of this error? A potential consequence of making a Type II error would be that the company might abandon development of a drug that was effective.

The relationship between  and  The ideal test procedure would result in both  = 0 (probability of a Type I error) and  = 0 (probability of a Type II error). This is impossible to achieve since we must base our decision on sample data.

The relationship between  and  Standard test procedures allow us to select , the significance level of the test, but we have no direct control over . As  decreases,  increases. As  increases,  decreases. So why not always choose a small  (like  =.05 or  =.01)? Selecting a significance level  =.05 results in a test procedure that, used over and over with different samples, rejects a true H 0 about 5 times in 100.

The EPA has adopted what is known as the Lead and Copper Rule, which defines drinking water as unsafe if the concentration of lead is 15 parts per billion (ppb) or greater. The manager of a community water system might use lead level measurements from a sample of water specimens to test the following hypotheses: H 0 :  = 15 versus H a :  < 15 State a Type I error in context. A Type II error leads to the conclusion that a water source does NOT meet EPA standards when the water is really safe. State a Type II error in context. A Type I error leads to the conclusion that a water source meets EPA standards when the water is really unsafe. What is a consequence of a Type I? The community might lose a good water source. What is a consequence of a Type II? There are possible health risks to the community Which type of error has a more serious consequence? Since most people would consider the consequence of the Type I error more serious, we would want to keep  small – so select a smaller significance level of  =.01.

Homework Pg.586: #10.12, 14, 15, 20, 22