Question

The height of grade 1 pupils are approximately normally distributed with µ = 45 inches and s = 2.

1. If an individual pupil is selected at random, what is the probability that he or she has a height of 42 and 47?
2. A class of 30 of these pupils is used as a sample. What is the probability that the class mean is between 42 and 47?
3. If a pupil is selected at random, what is the probability that is taller than 46 inches?
4. A class of 30 of these pupils is used as sample. What is the probability that the class mean is greater than 46 inches?

Answer 1

X ~ $N(45, 2^2)$

1)$P(42<X<47)=P(42<N(45,2^2)<47)=P(42<45+2N(0,1)<47)=P(-1.5<N(0,1)<1)=P(N(),1)<1)-P(N(0,1)<-1.5)=0.84134-0.06681=0.77453$

2) Y~$N(45, ({\frac {2^2} {\sqrt{30}}}))$

$P(42<Y<47)=P(42<N(45, ({\frac {2^2} {\sqrt{30}}}))<47)=P(42<45+0.73N(0,1)<47)=P(-4.11<N(0,1)<2.74)=P(N(0,1)<2.74)-P(N(0,1)<-4.11)=0.99693-0.00002=0.99691$

3)

$P(X>46)=P(N(45,2^2)>46)=P(45+2N(0,1)>46)=P(N(0,1)>0.5)=0.30854$

4)$P(Y>46)=P(45+0.73N(0,1)>46)=P(N(0,1)>1.37)=0.08534$