I would like to get the help from Question 1-12

Thanks for helping

Directions: Submit solutions to the following problems in beginning of lecture on the due date. Make sure that your solutions are legible, well-organized, and nished. Messy or incomplete work will be marked down. Please staple your homework. You are encouraged to work with other students on these problems, but the actual solution must be written individually. Bonus points will be awarded for organization and nectness. 1. The amount of soda dispensed by a soda machine is normally distributed with o- = 2 ounces. (a) If n = 16 cups are randomly selected from the output of the machine, what is P(|X pal g 0.3)? (b) Find P(|X pil S 0.3) when X is to be computed using samples of sizes a = 25, n = 36, n = 49, n = 64. (c) What pattern do you observe among the values for P(|}_{ ,u.| S 0.3) that you observed for various values of n? ((1) Compute P(|}_( ,u.| S 0.3) when 0' = 1 and n = 16. How does this probability compare to the probability obtained in part (a) (where o- = 2)? (9) Find a number I) such that P(.S'2 S b) = .975 when 02 = 1.4 and n = 20. 2. Find IE(S2) and viii-(52) when Y1, 1/2, ...,Y., is a random sample from a normal distribution 2 with mean pi and variance (:2 (Hint: What is the distribution of $3.13 5'2 a consistent estimate of 02? 3. Let Y1, Y2, . . . ,Y5 be a random sample of size 5 from a normal p0pulation with mean 0 and variance 1 and let 5 1] = Ei=1 Y; . 5 Let Y5 be another independent observation from the same p0pulation. What is the distribution of the random variables below and why? (a) 317Yo+1 (b) W=Z=132 and U=Z=1(l',Y)2 (c) V=U~|~Y2 (a) :3; 0- (a) Show that 91 = iii/(1) is an unbiased estimator for s and nd MSE((1). Hint: What is the distribution of YO)? (b) Show that 52 = Y is an unbiased estimator for 6 and nd MSE(2). (c) Find the efciency of 91 relative to 032. Which estimate is \"better\" (i.e., more efcient)? . If Y has a binomial distribution with n trials and success probability 3;, show that Y is a consistent estimator of p. Let 171,172,. . . ,Yn denote a random sample from the probability density function: 1'14?!) = 9y9_111{0 0. Show that Y 1s a consistent estimator of 3 +1 Let Y1,Y2, ...,Yn, denote a random sample from pdf given by (393) e'yZ/g, y > 0 0, otherwise f(y|9) ={ with parameter 0. Show that 2:1 YE is sufcient for 9. Let Y1, Y2, . . . ,Yn denote a random sample from the probability density function f(y|9) = rs'5'). y 2 9. Show that Y0) = min