Answer to Question #173091 in Statistics and Probability for junkyu

Question

1. The height of grade 1 pupils is are approximately normally distributed with µ = 45 inches and σ
2. σ
3. σ
4. σ
5.  = 6.
6. If an individual pupil is selected at random, what is the probability that he or she has a height of 42 and 47?
7. 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?
8. If a pupil is selected at random, what is the probability that is taller than 46 inches?
9. 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

6) $P(42<X<47)=P(\frac{42-45}{6}<Z<\frac{47-45}{6})=P(-0.5<Z<0.33)=$

$=P(Z<0.33)-P(Z<-0.5)=0.3208.$

7) $P(42<\bar X<47)=P(\frac{42-45}{\frac{6}{\sqrt{30}}}<Z<\frac{47-45}{\frac{6}{\sqrt{30}}})=P(-2.74<Z<1.83)=$

$=P(Z<1.83)-P(Z<-2.74)=0.9633.$

8) $P(X>46)=P(Z>\frac{46-45}{6})=P(Z>0.17)=1-P(Z<0.17)=0.4325.$

9) $P(\bar X>46)=P(Z>\frac{46-45}{\frac{6}{\sqrt{30}}})=P(Z>0.91)=1-P(Z<0.91)=0.1814.$