(a) Show that X² is not an unbiased estimator for the area of the square plot p². [Hint: for any rv Y, E[Y²] = V[Y] + E[Y]². Apply this for Y = X.] (b) For what value of k is the estimator X² – kS² unbiased for µ²?
(a) Show that X² is not an unbiased estimator for the area of the square plot p². [Hint: for any rv Y, E[Y²] = V[Y] + E[Y]². Apply this for Y = X.] (b) For what value of k is the estimator X² – kS² unbiased for µ²?
Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
18th Edition
ISBN:9780079039897
Author:Carter
Publisher:Carter
Chapter10: Statistics
Section10.1: Measures Of Center
Problem 9PPS
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![- (Sec. 6.1) Using a long rod that has length µ (unknown), you are going to lay out a square plot in
which the length of each side is µ. Thus the area of the plot will be µ². However, because you do
not know the value of µ, you decide to make n independent measurements X1,...,X, of the length.
Assume that each X; has mean µ and variance o².
(a) Show that X² is not an unbiased estimator for the area of the square plot µ². [Hint: for any rv
Y, E[Y²] = V[Y] + E[Y]². Apply this for Y = X.]
(b) For what value of k is the estimator X² – kS² unbiased for µ²?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff4f8485b-d935-478d-91e7-b806502d9702%2F956ca2fa-b50b-4caa-9b5e-39554cd27dd5%2Fgkqcd2_processed.png&w=3840&q=75)
Transcribed Image Text:- (Sec. 6.1) Using a long rod that has length µ (unknown), you are going to lay out a square plot in
which the length of each side is µ. Thus the area of the plot will be µ². However, because you do
not know the value of µ, you decide to make n independent measurements X1,...,X, of the length.
Assume that each X; has mean µ and variance o².
(a) Show that X² is not an unbiased estimator for the area of the square plot µ². [Hint: for any rv
Y, E[Y²] = V[Y] + E[Y]². Apply this for Y = X.]
(b) For what value of k is the estimator X² – kS² unbiased for µ²?
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