Answer to Question #312139 in Statistics and Probability for rian

Question

Supposed that the heights of Filipino high school students are normally distributed with a mean of 158 cm and a standard deviation of 3 cm.

a. what is the area and probability of students with at least 147 cm and at most 168 cm height?

b. what is the area and probability of students with 152 cm height from the mean?

c. what are the heights of students who belong to the 80th percentile?

Answer 1

X ~ N$(158, 3^2)$. The probability and corresponding area are equal

a) $P(147<X<168)=P(147<N(158,3^2)<168)=P(147<158+3Z<168)=P(-3.67<Z<3.33)=P(Z<3.33)-P(Z<-3.67)=0.99957-0.00012=0.99945$

b) $P(152<X<158)=P(152<158+3Z<158)=P(-2<Z<0)=P(Z<0)-P(Z<-2)=0.5-0.02275=0.47725$

c)Let a be a height of students belong to the 80th percentile, then $P(X<a)=0.8\implies P(158+3Z<a)=0.8\implies P(Z<{\frac {a-158} 3})=0.8\implies {\frac {a-158} 3}=0.84\implies a=160.52$

So, their height is below 160.52 cm.