Answer to Question #312714 in Statistics and Probability for chichi

Question

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

a. If an individual pupil is selected at random, what is the probability that he or she has a height of 42 and 47?

b. 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?

c.  If a pupil is selected at random, what is the probability that is taller than 46 inches?

d. 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

Population mean$~\mu=45$

Standard deviation$~\sigma=2$

(a) $P(42~and~47)$

$Z=\dfrac{X-\mu}{\sigma}$

$Z_{42}=\dfrac{42-45}{2}=-1.5$

$P_{42}=0.06681$

$Z_{47}=\dfrac{47-45}{2}=1.0$

$P_{47}=1-0.84134$

$P_{47}=0.15866$

$=0.06681 ~~and~~0.15866$

(b) $P(42<\bar X<47)$

$=P(\frac{42-45}{2}<Z<\frac{47-45}{2})$

$=P(-1.5<Z<1.0)$

$=P(Z<1.0)-P(Z<-1.5)$

$=0.77453$

(c) $P(X>46)=P(Z>\frac{46-45}{2})$

$=P(Z>0.5)$

$=1-P(Z<0.5)$

$=0.30854$

(d) $P(\bar X>46)=P(Z>\frac{46-45}{2})$

$=P(Z>0.5)$

$=1-P(Z<0.5)$

$=0.30854$