Suppose that 60 of smurfs are female and 40 are male A rando

Suppose that 60% of smurfs are female and 40% are male. A randomly chosen female

smurf has a height which is normally distributed with mean 1 feet and standard deviation 0.1 feet,

whereas a male smurf has a height which is normally distributed with mean 1.5 feet and standard

deviation 0.3 feet. Find the probability that the height of a randomly chosen smurf is between 1

and 1.3 feet.

Probability of a female smurf having height between 1 and 1.3 feet=P(z2)-P(z1) where

z2=(1.3-1)/0.1 = 3

and z1=(1-1)/0.1=0

So, Probability of a female smurf having height between 1 and 1.3 feet = P(z=3)-P(z=0) = 0.99-0.5= 0.49

Similarly

Probability of a mmale smurf having height between 1 and 1.3 feet=P(z2)-P(z1) where

z2=(1.3-1.5)/0.3 = -0.667

and z1=(1-1.5)/0.3=-1.667

So, Probability of a male smurf having height between 1 and 1.3 feet = P(z=-0.667)-P(z=-1.667) = 0.25-0.05= 0.2

probability that the height of a randomly chosen smurf is between 1

and 1.3 feet = 0.6*0.49+0.4*0.2 = 0.37