Question

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

(a) What is the probability that a household in Maryland has an annual income of $90,000 or more? (Round your answer to four decimal places.)

(b) What is the probability that a household in Maryland has an annual income of $50,000 or less? (Round your answer to four decimal places.)

(c) What is the probability that a household in Maryland has an annual income between $40,000 and $70,000? (Round your answer to four decimal places.)

(d) What is the annual income (in $) of a household in the eighty-sixth percentile of annual household income in Maryland? (Round your answer to the nearest cent.)

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 90000) = P(Z>(90000-75847)/33800)

= P(Z>0.418)

= 0.3377

B.

P(X 50000) = P(Z(50000-75847)/33800)

= P(Z -0.7647)

= 0.2222

C.

P(40000 < X < 70000) = P(X<70000) - P(X<40000)

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

= P(Z<-0.173) - P(Z<-1.0606)

= 0.2869

D.

P(Xx) = 0.86

P(Z (x-)/) = 0.86

Z for this probability is = 1.08032

(x-)/ = 1.08032

x = 1.08032*33800 + 75847

= $112,361.816

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