Question

Household Income in Maryland:

According to Money magazine, Maryland had the highest median annual household income of an state in 2018 at $75,847 (Time.com website). Assume that annual household income in Maryland follows a normal distribution with a median of $75,847 and a standard deviation of $33,800.

a. What is the probability that a household in Maryland has an annual income of $100,000 or more?

b. What is the probability that a household in Maryland has an annual income of $40,000 or less?

c. What is the probability that a household in Maryland has an annual income between $50,000 and $70,000?

d. What is the annual income of a household in the 90th percentile of annual household income in Maryland?

Answer 1

For a normal distribution:

median = = 75847

= 33800

Let X be the random variable denoting annual household income in Maryland which is normally distributed.

Std. variable, Z = (X-)/

A.

P(X 100000) = P(Z>(100000-75847)/33800)

= P(Z>0.7146)

= 0.2374

B.

P(X 40000) = P(Z(40000-75847)/33800)

= P(Z -1.0605)

= 0.1444

C.

P(50000 < X < 70000) = P(X<70000) - P(X<50000)

= P(Z<(70000-75847)/33800) - P(Z<(40000-75847)/33800)

= P(Z<-0.173) - P(Z -0.7647)

=0.43133-0.2222

= 0.2091

D.

P(Xx) = 0.90

P(Z (x-)/) = 0.90

Z for this probability is = 1.2816

(x-)/ = 1.2816

x = 1.2816*33800 + 75847

= $119163.4