1. AP Test Practice

2. After a frost warning was issued, the owner of a large orange grove asked his workers to spray all his trees with water. The water was supposed to freeze and form a protective covering of ice around the orange blossom. Nevertheless, the owner suspected that some trees suffered considerable damage due to the frost. To estimate the proportion of trees that suffered more than 50 percent damage due to the frost, he took a random sample of 100 trees from his grove. What is the response variable in this experiment? (A) The proportion of trees that suffered more than 50 percent damage due to frost (B) The number of trees affected by the frost (C) The number of trees sampled from the grove (D) For each sampled tree, whether it was sprayed with water or not sprayed with water (E) For each sampled tree, whether it suffered more than 50 percent damage or at most 50 percent damage E

3. For each sampled tree, one of two possible outcomes will be noted: (1) the tree suffered more than 50 percent damage, or (2) the tree suffered at most 50 percent damage.

4. For which of the following purposes would it be most unreasonable to use a census? (A) To determine the proportion of students with a learning disability in a small rural area high school (B) To determine the proportion of red snappers with a high mercury level in the Gulf of Mexico (C) To determine the difference between the proportion of engineering professors and the proportion of business professors in favor of the new teaching initiative at a large university (D) To determine the mean wage earned by construction workers in a small town (E) To determine the mean selling price of houses in your neighborhood B

5. It would be impossible to catch all the red snappers in the Gulf of Mexico and measure their mercury levels.

6. AP STATISTICS 10.2 Errors in Hypothesis Testing

7. 10.2 Objectives:  Understand the definitions Type I and Type II error and the role these concepts play in hypothesis testing.  Be able to identify and discuss the relevance of both types of error in problem contexts.  Have a sense that error is inevitable because we are sampling, but the error can be quantified and managed.

8. 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! That you made an error!

9. 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 significance level of the test This is the lower-case Greek letter “alpha”.

10. 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  This is the lower-case Greek letter “beta”.

11. H 0 is true H 0 is false Reject H 0 Fail to reject H 0 Type I error Correct 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:

12. 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 for 2009 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 2009 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.

13. 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.

14. 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.

15. 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. Standard test procedures allow us to select , the significance level of the test, but we have no direct control over . 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. So why not always choose a small  (like  =.05 or  =.01)?

16. The relationship between  and  Let’s consider the following hypotheses: H 0 : p =.5 H a : p >.5 Let  =.05.5 This is the part of the curve that represents  or the Type I error. This tail would represent , the probability of failing to reject a false H 0. Suppose this normal curve represents the sampling distribution for p when the null hypothesis is true. If the null hypothesis is false and the alternative hypothesis is true, then the true proportion is believed to be greater than.5 – so the curve should really be shifted to the right.

17. Reject H 0 FTR H 0

18. How does one decide what  level to use? After assessing the consequences of type I and type II errors, identify the largest  that is tolerable for the problem. Then employ a test procedure that uses this maximum acceptable value –rather than anything smaller – as the level of significance. Remember, using a smaller  increases .

19. 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 or if the concentration of copper is 1.3 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.

20. AP* Tips If an AP exam question asks you to describe possible errors and possible consequences of the errors, be sure to do both. The error and a consequence of the error are different things.

21. 10.2 Objectives: Understand the definitions Type I and Type II error and the role these concepts play in hypothesis testing. Be able to identify and discuss the relevance of both types of error in problem contexts. Have a sense that error is inevitable because we are sampling, but the error can be quantified and managed.

22. HW: Read 10.2: Errors in Hypothesis Testing Kaplan 271-272 For Tonight: P586: 10.12, 10.14, 10.13, 10.17, 10.20, 10.22