STAT 400 1 Spring 2015 Discussion 10 Let 0 and let X 1 X 2 X n be a random sample from the distribution with the probability density function f X x f X x 2 x e x x 0 a Find the method of moments estimator of b Suppose n 4 and c Find the maximum likelihood estimator of d Suppose n 4 and x 1 0 01 x 2 0 04 Find the method of moments estimate of x 1 0 01 x 2 0 04 x 3 0 09 x 4 0 36 x 3 0 09 x 4 0 36 Find the maximum likelihood estimate of 1 2 e Find E X k k 2 Let X 1 X 2 X n be a random sample from the distribution with probability density function f x Hint Consider u 2 x 2 0 x 0 a Obtain the method of moments estimator of b Is an unbiased estimator for c Find Var x 3 A supermarket selected a random sample of 196 customers which showed the sample mean bill amount of 65 40 Suppose the overall population standard deviation of the bill amounts is 12 30 a Construct a 93 confidence interval for the overall mean bill amount at this supermarket b Suppose the supermarket puts Alex in charge of computing the confidence interval and he gets the answer 63 52 67 28 Alex says that he used a different confidence level but other than that did everything correctly Find the confidence level used by Alex c What is the minimum sample size required for estimating the overall mean bill amount to within 1 with 93 confidence 4 5 A random sample of size n 9 from a normal distribution is obtained 4 4 4 a 3 7 5 1 4 3 4 7 3 7 3 5 4 6 4 7 Compute the sample mean x and the sample standard deviation s Do NOT use a computer You may only use and on a calculator Show all work 5 b Construct a 95 two sided confidence interval for the overall population mean c Construct a 90 one sided confidence interval for that provides an upper bound for d Construct a 95 one sided confidence interval for that provides a lower bound for 1 Let 0 and let X 1 X 2 X n be a random sample from the distribution with the probability density function f X x f X x a 2 x e x x 0 Find the method of moments estimator of E X 0 0 x f x dx x 2 e x dx x y dy x dx 2 x E X y 2 e y dy 0 Integration by parts Choice of u b u dv u v a u y 2 2 E X y e u y b a L A T E b v du a dv e y dy y 0 2 y e 0 dv e y dy y ogarithmic lgebraic rigonometric xponential v e y du 2 y dy dy 2 y e y dy 0 du dy 1 v e y 1 y 1 2 E X 2 y e y e dy e y dy 0 0 0 2 2 1 e y 2 0 OR E X 0 0 x f x dx x 2 x e y y E X 2 x dx dy x dx 2 x e y dy E Y 2 0 where Y has Exponential distribution with mean E X X 1 E Y 2 Var Y n 2 X i 2 n i 1 b E Y Suppose n 4 and 2 x 2 0 04 Find the method of moments estimate of n X i i 1 0 50 x 1 0 01 x 0 125 1 1 2 1 2 2 X x 3 0 09 2 2 2 n n X i i 1 x 4 0 36 2 0 125 4 c Find the maximum likelihood estimator of L n i 1 2 x i e n ln L n ln ln 2 i 1 ln L n xi n xi xi i 1 n i 1 n xi 0 n i 1 d Suppose n 4 and x 1 0 01 x 2 0 04 x 3 0 09 Xi x 4 0 36 Find the maximum likelihood estimate of n i 1 e x i 1 2 Find E X k k E Xk xk 0 u 0 For example 1 2 2 x 2k e u 4 10 3 3333 3 1 2 Hint Consider u e x dx du 1 2k x u x u e u du 2k 0 E X E X1 1 2 3 1 2k 2 2 du 2 k 1 2 2 2 x dx 2 Let X 1 X 2 X n be a random sample from the distribution with probability density function 2 x f x a 0 x 2 Obtain the method of moments estimator of 0 2 E X x f x dx x X b 3 2 2 x2 2 x3 x dx 3 2 0 3 1 n 3 X 3 X i n i 1 Is an unbiased estimator for E E 3 X 3 E X 3 3 3 an unbiased estimator for c 0 Find Var E X2 2 2 2 x f x dx x 2 x dx 6 0 2 Var X 2 2 6 2 9 2 2 18 Var Var 3 X 9 Var X 9 2 2 2 9 n 18 n 2n 3 A supermarket selected a random sample of 196 customers which showed the sample mean bill amount of 65 40 Suppose the overall population standard deviation of the bill amounts is 12 30 12 30 X 65 40 a n 196 Construct a 93 confidence interval for the overall mean bill amount at this supermarket is known n 144 large The confidence interval for X z 2 0 035 2 0 07 65 40 1 81 b n 12 30 z z 0 035 1 81 63 81 66 99 65 40 1 59 196 2 Suppose the supermarket puts Alex in charge of computing the confidence interval and he gets the answer 63 52 67 28 Alex says that he used a different confidence level but other than that did everything correctly Find the confidence level used by Alex n X z 2 1 88 z 2 2 67 28 65 40 65 40 63 52 1 88 12 30 z 196 2 2 14 2 0 0162 0 0324 Area to the right of 2 14 0 0162 Confidence level 100 1 96 76 c What is the minimum sample size required for estimating the overall mean bill amount to within 1 with 93 confidence 1 n 12 30 1 81 2 12 30 2 12 0 07 495 641169 0 035 2 Round up z 2 n 496 z 0 035 1 81 4 5 A random sample of size n 9 from a normal distribution is obtained 4 4 4 …
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