A random variable X is Normally distributed with mean = 75 and = 8.Let Y be a second Normally distributed random variable with mean = 70 and = 12.It is also known that X and Y are independent of one another.Let W be a random variable that is the difference between X and Y (i.e. ,W = X - Y).What is the probability that W is greater than -5.5 (i.e. ,P{W -5.5})?
A)0.233
B)0.512
C)0.767
D)0.488
E)Not within ± 0.005 of any of the above

Question

A random variable X is Normally distributed with mean  = 75 and  = 8.Let Y be a second Normally distributed random variable with mean  = 70 and  = 12.It is also known that X and Y are independent of one another.Let W be a random variable that is the difference between X and Y (i.e. ,W = X - Y).What is the probability that W is greater than -5.5 (i.e. ,P{W > -5.5})?
A)0.233 
B)0.512 
C)0.767 
D)0.488 
E)Not within ± 0.005 of any of the above

Quizplus · Accepted Answer

The mean of W is (the mean of X) - (the mean of Y) = 75 - 70 = 5. 
The variance of W is (the variance of X) + (the variance of Y) = 8^2 + 12^2 = 208. 
Therefore, W ~ N(5, 208).
To find P{W > -5.5}, we first standardize: 
P{W > -5.5} = P{(W - 5)/sqrt(208) > (-5.5 - 5)/sqrt(208)} = P{Z > -0.479} 
= 1 - P{Z <= -0.479} = 1 - 0.317 (from standard normal table) = 0.683.
Therefore, the answer is C.

A Random Variable X Is Normally Distributed with Mean