Question

According to the Internal Revenue Service, the mean tax refund for the year 2014 was $2,800. Assume the standard deviation is $450 and that the amounts refunded follow a normal probability distribution.

a). What percent of the refunds are more than $3,100? (Round the intermediate values to 2 decimal places. Round your answer to 2 decimal places.)

b). What percent of the refunds are more than $3,100 but less than $3,500? (Round the intermediate values to 2 decimal places. Round your answer to 2 decimal places.)

c). What percent of the refunds are more than $2,250 but less than $3,500? (Round the intermediate values to 2 decimal places. Round your answer to 2 decimal places.)

Answer 1

Solution :

Given that ,

mean = = 2800

standard deviation = = 450

(a)

P(x > 3100) = 1 - P(x < 3100)

= 1 - P((x - ) / < (3100 - 2800) / 450)

= 1 - P(z < 0.67)

= 1 - 0.7486

= 0.2514

Answer = 25.14%

(b)

P(x < 3500) = P((x - ) / < (3500 - 2800) / 450)

= P(z < 1.56)

= 0.9406

Answer = 0.9406 - 0.7486 = 19.2%

(c)

P(2250 < x < 3500) = P((2250 - 2800)/ 4500) < (x - ) /  < (3500 - 2800) / 450) )

= P(-1.22 < z < 1.56)

= P(z < 1.56) - P(z < -1.22)

= 0.9406 - 0.1112

= 0.8294

Answer = 82.94%