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.

Solution

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 male 0.4, mean 1.5 and sd 0.3

Probability of female 0.6, mean 1 and sd 0.1

For a randomly chosen smurf, mean =0.4*1.5+0.6*1=1.2

For a randomly chosen smurf, variance =0.4²*0.3²+0.6²*0.1²=0.018

Sd=sqrt(0.018)=0.1342

Z value of 1, z = (1-1.2)/0.1342 = -1.49

Z value of 1.3, z = (1.3-1.2)/0.1342 =0.75

P(1<x<1.3)=P(-1.49 <z<0.75)

P( z<0.75) –P(z <-1.49)

=0.7734 - 0.0681

=0.7053

The required probability =0.7053