Question

Show all work for (a), (b), and (c).

Suppose the Internal Revenue Service reported that the mean tax refund for the year 2016 was $2,800 and the standard deviation was $450, and that the amounts refunded follow a normal distribution.

(a) What percent of the refunds are more than $3,100?

(b) What percent of the refunds are more than $3,100 but less than $3,500?

(c) What percent of the refunds are more than $2,250 but less than $3,500?

Answer 1

a)

X ~ N ( µ = 2800 , σ = 450 )
We covert this to standard normal as
P ( X < x) = P ( (Z < X - µ ) / σ )
P ( X > 3100 ) = P(Z > (3100 - 2800 ) / 450 )
= P ( Z > 0.67 )
= 1 - P ( Z < 0.67 )
= 1 - 0.7486
= 0.2514

= 25.14%

b)

P ( 3100 < X < 3500 ) = P ( Z < ( 3500 - 2800 ) / 450 ) - P ( Z < ( 3100 - 2800 ) / 450 )
= P ( Z < 1.56) - P ( Z < 0.67 )
= 0.9406 - 0.7486
= 0.1920

= 19.20%

c)

P ( 2250 < X < 3500 ) = P ( Z < ( 3500 - 2800 ) / 450 ) - P ( Z < ( 2250 - 2800 ) / 450 )
= P ( Z < 1.56) - P ( Z < -1.22 )
= 0.9406 - 0.1112
= 0.8294

= 82.94%