Question

Based on information from California Revenue Board spokesperson, Aisha, the mean tax refund for the year 2015 was $5,000. Assume the standard deviation is $750 and that the amounts refunded follow a normal probability distribution. What percent of the refunds are more than $3,500? What percent of the refunds are more than $1250 but less than $2,000? What percent of the refunds are more than $2,750 but less than $3,500? The Board whats to find out the refund paid to 68% of Californians. What is the range of refund paid by 68% of Californians?

Answer 1

a)

X ~ N ( µ = 5000 , σ = 750 )
P ( X > 3500 ) = 1 - P ( X < 3500 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 3500 - 5000 ) / 750
Z = -2
P ( ( X - µ ) / σ ) > ( 3500 - 5000 ) / 750 )
P ( Z > -2 )
P ( X > 3500 ) = 1 - P ( Z < -2 )
P ( X > 3500 ) = 1 - 0.0228
P ( X > 3500 ) = 0.9772
= 97.72%

b)

P ( 1250 < X < 2000 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 1250 - 5000 ) / 750
Z = -5
Z = ( 2000 - 5000 ) / 750
Z = -4
P ( -5 < Z < -4 )
P ( 1250 < X < 2000 ) = P ( Z < -4 ) - P ( Z < -5 )
P ( 1250 < X < 2000 ) = 0 - 0
P ( 1250 < X < 2000 ) = 0

c)

P ( 2750 < X < 3500 ) = ?
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 2750 - 5000 ) / 750
Z = -3
Z = ( 3500 - 5000 ) / 750
Z = -2
P ( -3 < Z < -2 )
P ( 2750 < X < 3500 ) = P ( Z < -2 ) - P ( Z < -3 )
P ( 2750 < X < 3500 ) = 0.0228 - 0.0013
P ( 2750 < X < 3500 ) = 0.0214

= 2.14%

d)

According to empirical (68 - 95 - 99.7) rule,

approximately 68% of the data falls in 1 standard deviation of the mean.

Range = µ - σ to µ - σ

= 5000 - 750 to 5000 + 750

= $4250 to $5750