Household Income in Maryland:
According to Money magazine, Maryland had the highest median annual household income of an state in 2018 at $75,847 (Time.com website). Assume that annual household income in Maryland follows a normal distribution with a median of $75,847 and a standard deviation of $33,800.
a. What is the probability that a household in Maryland has an annual income of $100,000 or more?
b. What is the probability that a household in Maryland has an annual income of $40,000 or less?
c. What is the probability that a household in Maryland has an annual income between $50,000 and $70,000?
d. What is the annual income of a household in the 90th percentile of annual household income in Maryland?
Solution :
Given that ,
mean = = 75847
standard deviation = = 33800
a) P(x 100000) = 1 - P(x 100000)
= 1 - P[(x - ) / (100000 - 75847 ) / 33800]
= 1 - P(z 0.71)
Using z table,
= 1 - 0.7611
= 0.2389
b) P(x 40000)
= P[(x - ) / (40000 - 75847) / 33800]
= P(z -1.06)
Using z table,
= 0.1446
c) P(50000 < x < 70000) = P[(50000 - 75847)/ 33800) < (x - ) / < (70000 - 75847) / 33800) ]
= P(-0.76 < z < -0.17)
= P(z < -0.17) - P(z < -0.76)
Using z table,
= 0.4325 - 0.2236
= 0.2089
d) Using standard normal table,
P(Z < z) = 90%
= P(Z < z) = 0.90
= P(Z < 1.28) = 0.90
z = 1.28
Using z-score formula,
x = z * +
x = 1.28 * 33800 + 75847
x = $ 1,19,111
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