 # Homework Assignment 4 (1) A random sample of size 81 will be taken...

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Homework Assignment 4 (1) A random sample of size 81 will be taken from a population with mean u = 128 and standard deviation 0 = 6.3. What is the approximate probability that the sample mean will fall between 126.6 and 129.4? (2) A random sample of size 64 will be taken from a population with mean u = 51.4 and standard deviation 0 = 6.8. (a) What is the approximate probability that the sample mean will exceed 52.9? (b) What is the approximate probability that the sample mean will be less than 50.6? (3) If 23 percent of all patients with high blood pressure have bad side effects from a certain kind of medicine, use the normal approximation to find the probability that among 120 patients with high blood pressure treated with this medicine more than 32 will have bad side effects. The median age of residents of the United States is 31 years. If a survey of 100 randomly selected U.S. residents is to be taken, what is the approximate probability that at least 60 will be under 31 years of age? (1) Let Y1, , Y2 , Yn denotes a random of size n from a population whose density function is given by: a-1 ay f(y)= , 0 y &lt; 0 0° 0, elsewhere Where a &gt; 0 is known, fixed value, but 0 is unknown. (This is the power distribution.) Consider the estimator 0 = max {Y , Y2 , Yn } , the largest-order statistic. (a) Show that 0 is a biased estimator for O. (b) Find a multiple of 0 that is unbiased estimator for O. (c) Derive MSE ( 0 ). . (2) Suppose that Y1YY Y, n denotes a random of size n from a population with an exponential distribution whose density function is given by: y e 0. f(y = , y &gt; 0 0, elsewhere If Y(1) = min{Y1,Y2 , Y, n } denote the smallest-order statistic, show that 0 = nY (1) is an unbiased estimator for 0 and MSE (0 . (3) We can place a 2 - -standard-deviatior bound on the error of estimation with any estimator for which we can find a reasonable estimate of the standard error. Suppose that Y1 , Y2 Yn represent a random sample from a Poisson distribution with mean A. We know that Var(Yi) = A, and hence E(Y) = 2 Var(Y) =- 2 and n (a) How would you employ Y1,Y2 Yn to estimate A? (b) How would you estimate the standard error of your estimator? (4) If Y1,Y2, , , Yn denotes a random of size n from an exponential distribution with mean 0, then and Var(Yi) = 02. Thus, E(Y) = 0 and Var(Y) =- of , or of = 0 Suggest an unbiased estimator for 0 and n In provide an estimate for the standard error of your estimator.

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