Question

Section 5.4 Exercise 16

The mean household income in a country in a recent year was about $69,953 and the standard deviation was about $90,000. (The median income was $55,517)
a) If a Normal model was used for these incomes, what would be the household income of the top 4%?
b) How confident should one be in the answer in part a?
c) Why might the Normal model not be a good one for incomes?
a) The income would be s
(Round to the nearest dollar as needed)

Answer 1

Explanation:

a) To find the household income of the top 4%, we can use the standard Normal distribution table or calculator to find the Z-score associated with the 96th percentile.

The Z-score corresponding to the 96th percentile is approximately 1.75. We can then use the Z-score formula to find the income value:

Z = (x - mean) / standard deviation

1.75 = (x - 69953) / 90000

x - 69953 = 1.75 * 90000

x - 69953 = 157500

x = 227453

Therefore, the household income of the top 4% is about $227,453.

b) The answer in part a should be taken with some caution as the Normal model assumes that the incomes follow a bell-shaped distribution, which may not be the case for all income distributions.

c) The Normal model may not be a good one for incomes because incomes often have a skewed distribution with a long tail to the right, meaning there are a small number of individuals with very high incomes that can greatly affect the mean and standard deviation. In addition, incomes may have outliers or gaps that are not well captured by a Normal distribution.