Find the probability

Question

The per capita disposable income for residents of a U.S. city in a recent year is normally distributed, with a mean of approximately $44,000 and a standard deviation of about $2450. Use this information to complete the following.

5. Find the probability that the disposable income of a resident is more than $45,000. Is this an unusual event? Explain.

 6. Out of 800 residents, about how many would you expect to have a disposable income between $40,000 and $42,000?

Answer 1

First we can assign the variables we will use in this problem:

𝛍 = 44000

𝛔 = 2450

5. x = the per capita disposable income for residents of a U.S. city

= P(x > 45000)

= P(x-𝛍/𝛔 > 45000 - 44000/ 2450)

= P(z > .4082)

= 1 - P(z < .4082)

= 1 - .6584 = .3416

6. For this problem we have a sample size of n = 800

x̄ = 44000

𝞼 = 2450

= P(40000 < x < 42000)

= P(40000 - 44000/2450 < x - 𝛍/s < 42000 - 44000/2450

= P(-1.63 < z < -.8163)

= P(z < -.8163) - P(z < -1.63)

= .2072 - .0516 = .1556

800(.1556) = 124.48 or 125 people