Question

We want to measure the perimeter of a rectangle ABCD. The error made in measuring the sides AB and CD has standard deviation 3cm. The error made in measuring the sides BC and AD has standard deviation 2cm. Assume that the errors made in measuring the four sides are independent random variables.

(a) What is the standard deviation for the error of measurement of the perimeter of the rectangle if we measure sides AB and BC and multiply the sum of the two measurements by two?

(b) What is the standard deviation for the error of measurement of the perimeter of the rectangle if we measure each four sides and add up the measurements together?

Answer 1

We know that perimeter of a rectangle is the sum of lengths of all sides.

a) In this case perimeter is given by

P = 2(AB BC)

As Varinace has the property ​​​​​​

Var(X + y) = Var(X) + Var(Y) Var(aX) = a2Var(X) ,

And standard deviation is the positive square root of Varinace

So

Var(P)-4|Var(AB) + Var(BC)-49 + 4) 52

So that standard deviation is

$\sigma _P=\sqrt {52} =2\sqrt {13}$

b) In this case we have perimeter given by

P=AB + CD+BC + AD

Hence.

Var(P) Var(AB)Var(CD) +Var(BC)Var(AD) =94944+4

So, $\sigma _P=\sqrt {26}$