The Federal Reserve System publishes data on family income based on its Survey o

Question

The Federal Reserve System publishes data on family income based on its Survey of Consumer Finances. When the head of the household has a college degree, the mean before-tax family income is $ 84,750. Suppose that 66% of the before-tax family incomes when the head of the household has a college degree are between $74,600 and $94,900 and that these incomes are normally distributed. What is the standard deviation of before-tax family incomes when the head of the household has a college degree? (Round the value of z to 2 decimal places. Round your answer to 2 decimal places.)

Explanation / Answer

Answer: We have,

Let X be the random variable denoting before-tax family income and the s.d. be

Then, P( 74600 X 94900) = 0.66
Or, P( [ 74600 - ] / [X - ] / [ 94900 - ] / ) = 0.66
or, P( [ 74600 - 84750 ] / z [ 94900 - 84750 ] / ) = 0.66 ; z = [X - ] / is the standard normal variable
Or, P( -10150 / z 10150/ ) = 0.66
Or, P(z 10150 / ) - P( z -10150/ ) = 0.66 ……[1]
As X~ Normal distribution and it is a symmetric distribution we can write as P( z -10150 / ) = P(z 10150 / )
And P(z 10150 / ) = 1 - P(z 10150 / )
So equation [1] can be restated as:
Or, P(z 10150 / ) - P( z -10150 / ) = 0.66
Or, P(z 10150 / ) - { 1 - P(z 10150 / ) } = 0.66
Or, P(z 10150 / ) + P(z 10150 / ) – 1 = 0.66
Or, 2 P(z 10150 / ) = 1.66
Or, P(z 10150 / ) = 1.66/2 = 0.83
Or, P(z 10150 / ) = P(z 0.9542) {from standard Normal table we get P(z 0.9542) = 0.83 }
Or, 10150/ = 0.9542
Or, = 10150 / 0.9542 = 10637.18
Ans:
The standard deviation of before-tax family incomes when the head of the household has a college degree is $10627.18

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