Answer to Question #110578 in Statistics and Probability for judith

Question

The heights of a group of male students follow the normal distribution with mean 178 cm and
standard deviation of 8 cm.
(a) What percentage of the students would you expect to be shorter than 173 cm? (3)
(b) What proportion of students have a height between 179 cm and 184 cm? (4)
(c) Suppose there are 800 students, how many students are expected to be taller than
187 cm? (4)
(d) What height are 31% of the students shorter than and what height are 69% of them taller
than?

Answer 1

(a) $P(X<173)=P(Z<\frac{173-178}{8})=P(Z<-0.63)=0.2643.$

(b) $P(179<X<184)=P(\frac{179-178}{8}<Z<\frac{184-178}{8})=P(0.13<Z<0.75)=$

$=P(Z<0.75)-P(Z<0.13)=0.2217.$

(c) $P(X>187)=P(Z>\frac{187-178}{8})=P(Z>1.13)=1-P(Z<1.13)=0.1292.$

$n=800*0.1292=103$ students.

(d) $P(Z<z)=0.31\to z=-0.4959.$

$\frac{x-\mu}{\sigma}=z\to x=\mu+z*\sigma=178-0.4959*8=174.0 \;cm.$

$P(Z>z)=0.69\to P(Z,z)=0.31)\to z=-0.4959\to x=174.0\;cm.$

31% of students are shorter than 174.0 cm and 69% of students are taller than 174.0 cm.