Suppose there were reason to believe that incomes in some population were normal

Question

Suppose there were reason to believe that incomes in some population were normally distributed, with a mean of $47,000 per year, and a standard deviation of $5,500 per year.

a. What fraction of the population would make between $40,000 and $55,000 per year?

b. What fraction of the population would be described as below the poverty level, if that number is $22,350 (assuming a single wage-earner supporting a family of 4)?

c. If used blindly, this distribution would suggest that some people earn negative income per year. What fraction would that be?

d. We hear a lot these days about 1% / 99%. In this distribution, what is the minimum income of someone in the top 1%?

Explanation / Answer

a. What fraction of the population would make between $40,000 and $55,000 per year?

P(40000<X<55000) = P((40000-47000)/5500 <(X-mean)/s <(55000-47000)/5500)

=P(-1.27<Z<1.45)

=0.8244 (from standard normal table)

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b. What fraction of the population would be described as below the poverty level, if that number is $22,350 (assuming a single wage-earner supporting a family of 4)?

P(X<22350) = P(Z<(22350-47000)/5500)

=P(Z< -4.48)

= 0 (from standard normal table)

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c. If used blindly, this distribution would suggest that some people earn negative income per year. What fraction would that be?

P(X<0) = P(Z<(0-47000)/5500)

=P(Z<-8.55)

=0 (from standard normal table)

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d. We hear a lot these days about 1% / 99%. In this distribution, what is the minimum income of someone in the top 1%?

P(X<c) = 0.99

--> P(Z<(c-47000)/5500) = 0.99

--> (c-47000)/5500 =2.33 (from standard normal table)

So c = 47000+2.33*5500 = 59815

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