550.111, Fall 2016: Homework 10 This homework is due at the beginning of lecture on Friday, December 9, 2016 by 8 p.m. You can submit your homework in lecture at on Friday, December 9, 2016, or you can submit your homework to the homework submission box outside Avanti Athreya's office, 306-D Whitehead Hall. The submission box will be available starting at 4:00 p.m. on Friday, December 9, 2016, so you can submit your homework from 4 p.m. on Friday until the deadline of 8:00 p.m. on Friday. TO ENSURE THAT YOUR HOMEWORK IS COLLECTED AND GRADED, PLEASE SUBMIT IT EITHER IN LECTURE OR TO THE SUBMISSION BOX. PLEASE DO NOT SUBMIT IT TO ANY OTHER LOCATION. Before beginning this homework, you should read through Sections 9.1, 9.2, 9.3, and 9.5 in the MBB text and watch the lecture video on hypothesis testing. Problem 1. Exercise 8.74 and 8.75, pages 316-317. Problem 2. MBB, Exercises 9.3, 9.4, 9.6, 9.13, pages 339-340. Problem 3. MBB, Exercises 9.8 and 9.16, pages 340-341. Problem 4. You are at a magic show. The magician has a coin and you, as a member of the audience, do not know whether the coin has heads on both sides or heads on one side and tails on the other. The null hypothesis is that the coin has heads on one side and tails on the other (with both heads and tails being equally likely), and the alternative hypothesis is that the coin has heads on both sides (so that the coin will always land heads). The magician tosses the coin five times, and you use the test statistic X = number of heads in the five tosses You decide that you will reject the null if all five tosses end up being heads, and otherwise you will accept the null hypothesis. Answer the following questions. Hint: you do NOT need Z statistics or T statistics to answer the questions below! You just need to think and use basic probability. (a) Identify Type I and Type II errors for this test procedure. (b) Calculate the probability of a Type I error. (c) For what values of X will you accept the null hypothesis? (d) Calculate the probability of Type II error for this test. (e) Calculate the power of this test. Problem 5. Suppose that in a certain county, 1% of the population has a certain disease. Two different tests, Test A and Test B, are available for diagnosing the disease. We have the following information about these tests. If a person has the disease, then with probabilty 0.9, Test A will be positive. If the person has the disease, then with probability 0.95, Test B will be positive. Finally, if a person has the disease, then with probability 1 0.98, at least one of Test A or Test B will be positive. If a person does not have the disease, then Test A will be negative with probability 0.92; if the person does not have the disease, Test B will be negative with probability 0.93, and if the person does not have the disease, both tests will be negative with probability 0.90. Let D denote the event that a randomly chosen person has the disease and H the event that a randomly chosen person is healthy. For a randomly chosen person, let A+ and B+ represent the events that Tests A and B are positive, respectively, and let A and B represent the events that Tests A and B are negative, respectively. Using set operations, i.e. unions and intersections, describe the following events: (i) Using set operations, i.e. unions and intersections, describe the event that at least one of A and B are positive; (ii) Compute the probability that a randomly selected person obtains a positive test result from Test A; that is, compute P (A+). You may leave sums, products, and ratios in your answer. (iii) As described above, let D represent the event that a person has the disease. Given that a person has the disease, compute the probability that both tests will be positive. You may leave sums, products, and ratios in your answer. (Hint: use the initial information at the start of the problem. In particular, recall that the initial information in the problem specifies that given that a person has the disease, the probability of at least one test being positive is 0.98. What other information is also provided there, and how can you use it?) (iv) As described above, let H represent the event that a person is healthy. Given that a person is healthy, calculate the probability that both tests will be positive. You may leave sums, products, and ratios in your answer. (Hint: given that a person is healthy, can you relate the probability that both tests are positive to the probability that at least one test is negative?) Now, you conduct the following test of hypothesis on the disease status of an individual. The null hypothesis is that the individual is healthy, and the alternative hypothesis is that the individual has the disease. If both Test A and Test B are positive, you will reject the null in favor of the alternative; otherwise, you will not reject the null. (i) Define Type I and Type II error probabilities for this test. (ii) Based on the conditional probabilities that are given in the problem and the conditional probabilities that you have calculated in the parts above, determine the probability of Type I error and the probability of Type II error. You may leave sums, products, and ratios in your answer. Note: AGAIN, YOU DO NOT NEED Z-SCORES OR THE CENTRAL LIMIT THEOREM TO CALCULATE THE PROBABILITIES OF TYPE I AND TYPE II ERROR! YOU JUST NEED TO WORK FROM THE DEFINITIONS. Problem 6. MBB, Exercises 9.31, 9.35, and 9.40 page 350-351. Problem 7. Suppose the weights of newborn infants are normally distributed with unknown mean and known variance 2 = 9. Suppose you want to test H0 : = 8 versus the alternative Ha : 6= 8 = 7.9 for this sample data. Let n = 10. Suppose X 2 (a) Conduct a test of hypothesis at the = 0.05 level of significance: determine the statistic, the rejection region, and decide whether or to accept the null with this sample data. Do you need the Central Limit Theorem to deduce the distribution of the test statistic? (b) Calculate , the probability of Type II error, for the particular alternative Ha : = 7.5, and calculate the power of the test for this particular alternative. (c) Now consider the alternative Ha : = a = 8.2. Calculate for this value of a . How does this compare to the value of in part (b)? (d) Suppose = 0.01 and consider the particular alternative = a = 7.5. Calculate for this case. How does this compare to the value of you computed in part (b)? (e) For a fixed value of and a fixed particular alternative Ha : = a , how does depend on the sample size n? 3