Question

# Show all work for (a), (b), and (c). Suppose the Internal Revenue Service reported that the...

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?

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%

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