Question

In a certain state, the tax returns of everyone whose income is in

the top 11% of incomes are audited. Assume that income in this

state is normally distributed with mean $ 68,886 and standard

deviation $ 8,091.

(a) Find the minimum income in the audited group.

(b) If the political decision of auditing anyone earning more

than $ 80,000 were to be made, what is the percent of

those who would not be audited under this scheme?

Answer 1

a )

X ~ N ( µ = 68886 , σ = 8091 )
P ( X > x ) = 1 - P ( X < x ) = 1 - 0.11 = 0.89
To find the value of x
Looking for the probability 0.89 in standard normal table to calculate critical value Z = 1.23
Z = ( X - µ ) / σ
1.23 = ( X - 68886 ) / 8091
X = 78837.93 = 78838

b)

X ~ N ( µ = 68886 , σ = 8091 )
We convert this to standard normal as
P ( X < x ) = P ( Z < ( X - µ ) / σ )
P ( ( X < 80000 ) = P ( Z < 80000 - 68886 ) / 8091 )
= P ( Z < 1.37 )
P ( X < 80000 ) = 0.9147
= 91.47%