Question

A distribution is normal with a mean of 25 and a standard deviation of 3.

11. What is the median of the distribution?

12. What percent of the distribution lies between 22 and 28?

13. What percent of the distribution lies below 16?

14. What percent of the distribution lies above 28?

Answer 1

A distribution is normal with a mean of 25 and a standard deviation of 3.

So, (X-25)/3~Normal(0,1)

11) What is the median of the distribution ?

Median is the value of the random variable, below which the probability is 1/2.

We know that for a normal distribution with mean m and standard deviation s, the median is m.

So, median of the distribution is 25.

12) What percent of the distribution lies between 22 and 28?

Let, X be the random variable in this case.

P(22<X<28)

=P(22-25<X-25<28-25)

=P(-3<X-25<3)

=P(-3/3<(X-25)/3<3/3)

=P(-1<(X-25)/3<1)

=P(-1<Z<1)

Where, Z is the standard normal variate.

=phi(1)-phi(-1)

Where, phi is the distribution function of the standard normal variate.

=2*phi(1)-1

=2*0.8413-1

=1.6826-1

=0.6826

13) What percentage of the distribution lies below 16?

P(X<16)

=P(X-25<16-25)

=P(X-25<-9)

=P((X-25)/3<-9/3)

=P(Z<-3)

Where, Z is the standard normla variate.

=phi(-3)

=1-phi(3)

Where, phi is the distribution function of the standard normal variate.

=1-0.9987

=0.0013

14) What percentage of the distribution lies above 28?

P(X>28)

=P(X-25>28-25)

=P(X-25>3)

=P((X-25)/3>3/3)

=P(Z>1)

Where, Z is the standard normla variate.

=1-phi(1)

Where, phi is the distribution function of the standard normal variate.

=1-0.8413

=0.1587