For a mechanical system, the (independent) inputs are represented by random vari

Question

For a mechanical system, the (independent) inputs are represented by random variables X and Y, and the output is represented by random variable Z. Theoretically they are related by
Z = -5X + 2Y + 4
It is known that the random variables are normal distributed and X has a variance of 4. The variance of Y is unknown and cannot be measured due to technical difficulties. Instead, measurements of Z are made and the following data are obtained
220 , 193 , 212 , 201
Do the data support the conjecture that the variance of Y is 10?

Explanation / Answer

Var(z) = 1/n * (sum(z^2) - n* mean^2) = 1/4 * ( 170994 - 4*206.5^2) = 106.25

z = -5x + 2y + 4

var(z) = var(-5x +2y+4)

var(z) = 25 var(x) + 4 var(y) + 0

106.25 = 25*4 + 4var(y)

4 var(y) = 6.25

var(y) = 1.56

actual varience of y = 1.56

Hence this data is not support the that vairence of y is 10.

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