Question

In addtion to the questions, make sure to display the regions under the normal distribution curve.

The monthly income of residents of Big City is normally distributed with a mean of $3000 and a standard deviation of $500.

a. The mayor of Big City makes $2,250 a month. What percentage of Big City's residents has incomes that are more than the mayor's?

b. Individuals with incomes of less than $1,985 per month are exempt from city taxes. What percentage of residents is exempt from city taxes?

c. What are the minimum and the maximum incomes of the middle 95% of the residents?

d. Two hundred residents have incomes of at least $4,440 per month. What is the population of Big City?

Answer 1

Solution:

Given that,

mean =  = 3000

standard deviation =  = 500

A ) p ( x > 2250 )

= 1 - p (x < 2250 )

= 1 - p ( x -  / ) < ( 2250 - 3000 / 500)

= 1 - p ( z < -750 / 500 )

= 1 - p ( z < -1.5)

Using z table

= 1 - 0.0668

= 0.9332

Probability = 93.32%

B ) p ( x < 1985 )

= p ( x -  / ) < ( 1985 - 3000 / 500)

= p ( z < - 1050 / 500 )

= p ( z < - 2.03 )

Using z table

= 0.0212

Probability = 2.12%

C ) Using standard normal table,

P(-z < Z < z) = 95%
P(Z < z) - P(Z < z) = 0.95
2P(Z < z) - 1 = 0.95
2P(Z < z ) = 1 + 0.95
2P(Z < z) = 1.95
P(Z < z) = 1.95 / 2
P(Z < z) = 0.975
z = 1.96 znd z = - 1.96

Using z-score formula,

x = z * +

x = 1.96 * 500 + 3000

= 3980

Maximum incomes = 3980

x = z * +

x = - 1.96 * 500 + 3000

= 2020

The minimum = 2020

D ) p ( x    4440 )

= 1 - p (x   4440 )

= 1 - p ( x -  / ) ( 4440 - 3000 / 500)

= 1 - p ( z 1440 / 500 )

= 1 - p ( z 2.88)

Using z table

= 1 - 0.9980

= 0.0020

Probability = 0.0020