Question

IRS data indicates that the tax refunds it issued this year follow the normal distribution with μ = 1,200 and σ = 200. Based on this information calculate the following probabilities.

1. Probability of selecting a tax return, the refund for which will fall between $1,170 and $1,200:
   
   
   
2. Probability of selecting a tax return, the refund for which will be less than $1,406:
   
   
   
3. Probability of selecting a tax return, the refund for which will be more than $1,598:
   
   
   
4. Probability of selecting a tax return, the refund for which will fall between $1,132 and $1,354:

Answer 1

Given,

= 1200, = 200

We convert this to standard normal as

P( X < x) = P( Z < x - / )

a)

P( 1170 < X < 1200) = P( X < 1200) - P( X < 1170)

= P( Z < 1200 - 1200 / 200) - P( Z < 1170 - 1200 / 200)

= P( Z < 0) - P( Z < -0.15)

= P( Z < 0) - ( 1 - P( Z < 0.15) )

= 0.5 - ( 1 - 0.5596)

= 0.0596

b)

P( X < 1406) = P (Z < 1406 - 1200 / 200)

= P( Z < 1.03)

= 0.8485

c)

P( X > 1598) = P( Z > 1598 - 1200 / 200)

= P( Z > 1.99)

= 1 - P( Z < 1.99)

= 1 - 0.9767

= 0.0233

d)

P( 1132 < X < 1354) = P( X < 1354) - P( X < 1132)

= P( Z < 1354 - 1200 / 200) - P( Z < 1132 - 1200 / 200)

= P( Z < 0.77) - P( Z < -0.34)

= P( Z < 0.77) - ( 1 - P( Z < 0.34) )

= 0.7794 - ( 1 - 0.6330)

= 0.4124