Consider the data in the scatterplot below on the number of points scored (Y) and the number of goals scored (r) by Newcastle United in their last n = 10 seasons in the Premier League.

The objectives of this experiment are to: • define a refrigeration cycle and quantitative measures of its performance. • evaluate the performance of a cycle based on experimental measurements.

1. To evaluate the performance of a new diagnostic test for disease X, the developer performed a trial on 100 people known to have disease X and 95 of them have positive test results. The test is also performed on 200 control cases known to be free of disease X and 30 of them have positive test results. Based on these data, what is the sensitivity and specificity of the test? (10 marks) 2. Based on all the information currently available, you estimate that Mr. Chen has a 0.2 probability of having disease Y. You order a diagnostic test for disease Y which has a sensitivity of 95% and a specificity of 85%. (a) Suppose that the result comes back positive. Based on all the information now available, what is the probability that Mr. Chen has disease Y? Compute your answer to at least 4 decimal places. (5 marks) (b) On the other hand, suppose that the result comes back negative. Based on all the information now available, what is the probability that Mr. Chen has disease Y? Compute your answer to at least 4 decimal places. (5 marks) 3. A test with 95% sensitivity and 85% specificity is used to screen a population of 10,000 people for disease Z. Previous studies indicated that this population has a 10% prevalence rate for disease Z. What is the expected number of people in the population that will have test results that are correct? (5 marks)

The objectives of this experiment are to: • define a refrigeration cycle and quantitative measures of its performance. • evaluate the performance of a cycle based on experimental measurements.

Problem 5: A traffic control engineer believes that the cars passing through a particular intersection arrive at a mean rate equal to either 3 or 5 for a given time interval. Prior to collecting any data, the engineer believes that it is much more likely that the rate λ = 3 than λ = 5. In fact, the engineer believes that the prior probabilities are: P(λ = 3) = 0.7 and P(λ = 5) = 0.3 One day, during a randomly selected time interval, the engineer observes x = 7 cars pass through the intersection. In light of the engineer's observation, what is the probability that λ = 3? And what is the probability that λ = 5?

Problem 4: A coin is thrown independently 10 times to test the hypothesis that the probability of heads is 0.5 versus the alternative that the probability is not 0.5. The test rejects if either 0 or 10 heads are observed. What is the significance level of the test?

Problem 5: The intensity of light reflected by an object is measured. Suppose there are two types of possible objects, A and B. If the object is of type A, the measurement is normally distributed with mean 100 and standard deviation 25; if it is of type B, the measurement is normally distributed with mean 125 and standard deviation 25. A single measurement is taken with the value X = 120. Consider the test Ho: Item is of type A versus Ha: Item is of type B. a) Calculate the likelihood ratio statistic of this test. b) If the prior probabilities of A and B are equal (1/2 each), what is the posterior probability that the item is of type B? c) Suppose that a decision rule has been formulated that declares the object to be of type B if X > 125. What is the significance level of this test? d) What is the power of this test? (Hint: Compute the power using its definition.) e) What is the p-value when X = 120? (Hint: Values as extreme or more extreme than the observed value are X ≥ 120.)

Problem 3: Suppose that X is a discrete random variable with 2 P(X= 0) = 0 P(X= 1) = 2 P(X= 2) = (1-0) P(X = 3) = (1-0) where 0 ≤ 0 ≤ 1 is a parameter. The following 10 independent observations were taken from such a distribution: (3, 0, 2, 1, 3, 2, 1, 0, 2, 1). a) Find the method of moments estimate of 0. b) What is the maximum likelihood estimate of 0? c) If the prior distribution of 0 is uniform on [0, 1], what is the posterior density function? d) Find the Bayes Posterior Mean Estimate, the MAP estimate, the Posterior Median Estimate, and a 90% Bayesian Confidence Interval for the parameter 0.

Problem 3: Let X be a binomial random variable with n trials and probability p of success. What is the generalized likelihood ratio for testing Ho: p = 0.5 versus H₁:p # 0.5?

Problem 4: Suppose that y follows a binomial distribution with parameters n and p, so that the pmf of Y given 80 is: g(y|0) = C(y,n)0 (1 - 0)¹-y for y = 0,1,..., n. Suppose that the prior pdf of the parameter is the Beta pdf, that is r(a +ß) Γ(α)Γ(β) for 0 < 0 < 1. h(0) = Find the posterior pdf of 0, given that Y = y. ·0⁰-¹ (1-0)B-1

Problem 1: Suppose that 100 items are sampled from a manufacturing process and 3 are found to be defective. A beta prior is used for the unknown proportion of defective items. Consider two cases: (1) a = b = 1, and (2) a = 0.5 and b = 5. Plot the two posterior distributions and compare them. Find the two posterior means and compare them. Explain the differences.