Q 4. (a) (i) Show that if U and V are independent random variables having chi-square distributions with vị andv2 degrees of freedom respectively, thenX =is a random variable having an F distribution with vị and v2 degrees of freedom , i.e. a randomvariable whose probability density function is given byf (x) =I (4)r (4) (v½.1+-x(ii) Using (i), or otherwise, show that if S? and S are respective variances of independent randomsamples of sizes nį and n2 from normal populations with variances a? and ož, thenY%3Dis a random variable having an F distribution with n1 – 1 and n2 – 1 degrees of freedom.(b) Let X be a random variable modeling the heights of all adults in the population where o² is known.(i) For a random sample X1, X2, ..., Xn, state the sampling distribution of Š when n > 30.(ii) A random sample of 50 adults gives a mcan of 174.5 cm. Given that o = 6.9 construct a 98%confidence interval for µ, the mean height of all adults. Based on the confidence interval obtained,is it likely that µ=170 cm? Explain.You may use the follouing resulls:20.01 = 2.326 :20.02 = 2.054 :to.01,49 = 2.4049 ; to.02,49 = 2.1099.%3D

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ow that if U and degrees of freedom respectively, then are independent random variables having chỉ- X = a random variable having an F distribution with v and v2 deg riable whose probability density function is given by (2) f (2) 17-1 (1+ I (4)r()

ow that if U and degrees of freedom respectively, then are independent random variables having chỉ- X = a random variable having an F distribution with v and v2 deg riable whose probability density function is given by (2) f (2) 17-1 (1+ I (4)r()

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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Question

Q 4. (a) (i) Show that if U and V are independent random variables having chi-square distributions with vị and
v2 degrees of freedom respectively, then
X =
is a random variable having an F distribution with vị and v2 degrees of freedom , i.e. a random
variable whose probability density function is given by
f (x) =
I (4)r (4) (v½.
1+-x
(ii) Using (i), or otherwise, show that if S? and S are respective variances of independent random
samples of sizes nį and n2 from normal populations with variances a? and ož, then
Y
%3D
is a random variable having an F distribution with n1 – 1 and n2 – 1 degrees of freedom.
(b) Let X be a random variable modeling the heights of all adults in the population where o² is known.
(i) For a random sample X1, X2, ..., Xn, state the sampling distribution of Š when n > 30.
(ii) A random sample of 50 adults gives a mcan of 174.5 cm. Given that o = 6.9 construct a 98%
confidence interval for µ, the mean height of all adults. Based on the confidence interval obtained,
is it likely that µ=170 cm? Explain.
You may use the follouing resulls:
20.01 = 2.326 :
20.02 = 2.054 :
to.01,49 = 2.4049 ; to.02,49 = 2.1099.
%3D

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