Problem 1 Suppose X is a normal random variable having mu 2

Problem 1. Suppose X is a normal random variable having mu = 270 and sigma = 8. Compute the following. (a) P(X > 292.48) (b) P(X > 248.88) (c) P(X 268.5) (e) P(254.16

Solution

for normal distribution

P(X <x) = P(Z < x -mean/std)

a)

P(X>292.48) = P(Z>(292.48-270)/8) = P(Z > 2.81) = 0.0025

b)

P(X>248.88) = P(Z>(248.88-270)/8) = P(Z > -2.64) = 0.9959

c)

P(X<257.04) = P(Z<(257.04-270)/8) = P(Z <-1.62) = 0.0526

d)

P(254.16<= X <=274.96) = P(X<=274.96) - P(X<254.16)

= P(Z<0.62) - P(Z<-1.98)

= 0.7324 - 0.0239

= 0.7085

f)

P(|X-268|>16) = P( X < 252 or X>284) = P(X>284) + P(X<252) = P(Z>1.75) + P(Z<-2.25)

= 0.0401 + 0.0122

= 0.0523

g)

P(X<=x) = 0.1

=>

x- 270 /8 = invnorm(0.1) =-1.282

x = 259.744

h)

P(X>x) = 0.975

=> P(X<=x) = 1-0.975 = 0.025

x-270 /8 = invnorm(0.025)= -1.96

=>x = 254.32