(Q2) Suppose Y₁,..., Y, are mutually independent random variables such that for a given Xi Y~N(a+BXi, o²), i=1,...,n. where a, ẞ and the variance o² are unknown. Unbiased estimators â, 3, 52 of a, 3, 02 are â=Y-BX, ẞ=S, S²=; B= Sxx' 1 n-2 i=1 (X-&-BX;)², where we have denoted Sry=XY(X) (Y). It is known that - T= B-B S/√√STE ~tn-2- Let c>0 be a constant such that P(|T|> c) = 0.05 where T~tn−2. (i) Show that the 95% confidence interval for ẞ is given by S (ii) Suppose we observe the following data: X 70, 74, 72, 68, 58, 54, 82, 64, 80, 61 Y 77, 94, 88, 80, 71, 76, 88, 80, 90, 69 Calculate, ẞ, S² from the data. (iii) Calculate the 95% confidence interval for ẞ.
(Q2) Suppose Y₁,..., Y, are mutually independent random variables such that for a given Xi Y~N(a+BXi, o²), i=1,...,n. where a, ẞ and the variance o² are unknown. Unbiased estimators â, 3, 52 of a, 3, 02 are â=Y-BX, ẞ=S, S²=; B= Sxx' 1 n-2 i=1 (X-&-BX;)², where we have denoted Sry=XY(X) (Y). It is known that - T= B-B S/√√STE ~tn-2- Let c>0 be a constant such that P(|T|> c) = 0.05 where T~tn−2. (i) Show that the 95% confidence interval for ẞ is given by S (ii) Suppose we observe the following data: X 70, 74, 72, 68, 58, 54, 82, 64, 80, 61 Y 77, 94, 88, 80, 71, 76, 88, 80, 90, 69 Calculate, ẞ, S² from the data. (iii) Calculate the 95% confidence interval for ẞ.
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Transcribed Image Text:(Q2) Suppose Y₁,..., Y, are mutually independent random variables such that for a given Xi
Y~N(a+BXi, o²), i=1,...,n.
where a, ẞ and the variance o² are unknown. Unbiased estimators â, 3, 52 of a, 3, 02
are
â=Y-BX, ẞ=S, S²=;
B=
Sxx'
1
n-2
i=1
(X-&-BX;)²,
where we have denoted Sry=XY(X) (Y). It is known that
-
T=
B-B
S/√√STE
~tn-2-
Let c>0 be a constant such that P(|T|> c) = 0.05 where T~tn−2.
(i) Show that the 95% confidence interval for ẞ is given by
S
(ii) Suppose we observe the following data:
X
70, 74, 72, 68, 58, 54, 82, 64, 80, 61
Y
77, 94, 88, 80, 71, 76, 88, 80, 90, 69
Calculate, ẞ, S² from the data.
(iii) Calculate the 95% confidence interval for ẞ.
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