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Let X1 . .., X, be a random sample from Exponential distribution with parameter B > 0. Let T = ELIXi 1. [1.] We want to construct a confidence interval for the parameter B o [(a)] Find the distribution of T. Hint: Exponential(B) has the same distribution as Gamma(a = 1, B) 2 o [(b)] Find the distribution of the statistic T and write it as a x2- distribution. B o [(c)] If n = 15 and T' = 45, construct a 95% confidence interval for B. Hint: Find two constants a and b satisfying: P (a s T , XSc} o [(b)] If n = 14 find the value of c, at the level of significance a = 0.05.Let X denote the weight of patients admitted at SQU Hospital. Based on the recorded data, a medical staff claimed that the mean of the weight is greater than 77 Kg. Suppose that X follows Normal distribution N(/, 02 - 25). A random sample of size n = 36 patients has been selected. 1. [1.] Write the null and alternative hypothesis for testing the above claim 2. [2.] Suppose now the critical region of the test: C = {(1, $2, . . ., 236) : X _79} . [(a)] Compute the power function K(/) of this test. o [(b)] Compute the significance level of this test. [(c)] Calculate the values K(79) and K(81). . [(d)] Calculate the probability of committing type II error when u = 80. o [(e)] Calculate the corresponding p-value to X - 78.5.For each of the following series of measurement values, use the Q-test to determine whether any measurement should be rejected at the 90% confidence level. After rejecting any true "outliers," calculate x, S and Sm, and use the Student t-test to calculate the error range . at the 95% confidence level. Then report the final result as x x. (a) 2.8, 2.7, 2.7, 2.5, 2.9, 2.6, 3.0, 2.6 (b) 97.13, 97.10, 97.20, 97.35, 97.10, 97.19 (c) 0.134, 0.120, 0.109, 0.124, 0.131, 0.119, 0.135, 0.132, 0.132b) A pollution control inspector suspected that a river community was releasing amounts of semi-treated sewage into a river. To check his theory, he drew five randomly selected specimens of river water at a location above the town, and another five below. The dissolved oxygen readings (in parts per million) are as follows: Above Town 4.8 5.2 5.0 4.9 5.1 Below Town 5.0 4.7 4.9 4.8 4.9 i) Is it reasonable to assume that the populations sampled have common variance? Justify your answer by conducting a test of Ho : of = of against an appropriate alternative hypothesis using a = 0.1 level of significance. if) Do the data provide sufficient evidence to indicate that the mean oxygen content below the town is less than the mean oxygen content above? Test using a = 0.05. in) Place bounds on the associated p-value. 12 The summary statistics for the data in the above table are: 1 =5, Fy = 4.86. s = 0.025. = = 0.0133. Alberto Electronics is an electronic manufacturer and has designed three new products: AE-I (h,), AE-II (h2) and AE-III (h3). Alberto considers selecting one of the three new products to manufacture and sell. Demand for each of the products is uncertain and could be in any of the following three possible states: Bad demand (di), good demand (d2) and excellent demand (da). Alberto has estimated the profit (*[1m) of each product with respect to each level of demand as shown in the following table: Payoff Table Type of State of demand product Bad (d1) Good (dz) Excellent (d3) AE-1 (h1) 30 4 28 AE-II (hz) 2 55 60 AE-III (h3) -26 35 80 a) Alex, Tom and Ben are the three managers of the company. Although the current economic situations is uncertain, Tom is optimistic about the demand of electronic products; however, Ben is pessimistic about the demand; while also cautious, Alex is keen to avoid losing opportunity to make profit for the company. Based on their individual views, what advice would you give to each of the three managers respectively on which product should be chosen and why? Having known Tom's and Ben's opposite views, as the managing director of Alberto, Alex intends to combine Tom's and Ben's views to make the final choice. What new advice would you give to Alex for his new analysis and final choice and why? [15% marks] Given the uncertainty and importance of the planning, Alberto has commissioned a consulting firm to estimate the probabilities of demand for each type of product. The firm's estimation is as follows (i.e. p(d,)): (1) for AE-1, p(d;) = 0.3, p(d2) = 0.4 and p(d3) = 0.3; (2) for AE-II, d2 is the most likely and is about twice as likely as d; and about three times as likely as d1 : (3) for AE-III, d, is more or less as likely as dz and is about twice as likely as dy . b) Help Alberto to estimate the probabilities of demands dj , dz and d; for AE-II and AE-III respectively, and briefly discuss any limitations of the estimation. [5% marks] c) Draw and properly label a decision tree for the problem. Use the decision tree and the above information to help Alberto analyse which product should be chosen. Explain any uncertainty or assumptions made in the above process. What can you do to help improve the credibility of the choice? [15% marks]