For a certain income level the IRS knows that the amounts cl

For a certain income level, the IRS knows that the amounts claimed for Medical Deductions (X₁), Charitabel Contributions (X₂), and Miscellaneous Deductions (X₃), are Independent Normally distributed variables with means $400, $800, and $100 and standard deviatiosn $100, $250, and $40 repectively.

A)What is the probability that the amount claimed for these 3 deductions is less than $1600?

B) If someone, from this income level, reports on his return a total of $2100 for these deductions, how likely is this or a larger amount? Justify

Solution

For a certain income level, the IRS knows that the amounts claimed for Medical Deductions (X₁), Charitabel Contributions (X₂), and Miscellaneous Deductions (X₃), are Independent Normally distributed variables with means $400, $800, and $100 and standard deviations $100, $250, and $40 repectively.

A)What is the probability that the amount claimed for these 3 deductions is less than $1600?

E(X+Y+Z) = E(X)+E(Y)+E(Z)

V(X+Y+Z) = V(X)+V(Y)+V(Z)

Mean of 3 deductions = 400+800+100 =1300

Variance of the three deductions = 100² +250² +40² =74100

standard deviations of the three deductions =sqrt(74100) = 272.2132

z value for 1600, z=( 1600-1300)/272.2132 =1.1

p( x <1600) = p( z <1.1) = 0.8643

B) If someone, from this income level, reports on his return a total of $2100 for these deductions, how likely is this or a larger amount? Justify

z value for 2100, z=( 2100-1300)/272.2132 =2.94

p( x > 2100) = p( z > 2.94 ) = 0.0016

This probability is small

If someone, from this income level, reports on his return a total of $2100 for these deductions, it is unlikely is this or a larger amount.