Question

A real estate company is interested in the ages of home buyers. They examined the ages of thousands of home buyers and found that the mean age was 48 years old, with a standard deviation of 9 years. Suppose that these measures are valid for the population of all home buyers. Complete the following statements about the distribution of all ages of home buyers.

(a) According to Chebyshev's theorem, at least of the home buyers' ages lie between 30 years and 66.
(b) (a) According to Chebyshev's theorem, at least of the home buyers' ages lie between 25.5 years and 70.5.
(c) Suppose that the distribution is bell-shaped. According to the empirical rule, approximately of the home buyers' age lie between 30 years and 66.
(d) Suppose that the distribution is bell-shaped. According to the empirical rule, 99.7% of the home buyers' ages lie between years and years.

Answer 1

Between 21 years and 75 years

Explanation:

Given that a real estate company is interested in the ages of home buyers. They examined the ages of thousands of home buyers and found that the mean age was 48 years old, with a standard deviation of 9 years.

X the ages of home buyers is N(48, 9)

a)
`30\leq x\leq 66\\|x-48|\leq 18\\|x-48|\leq 2\sigma`

Hence using Cheby chev inequality

`P(|x-48|\leq 2\sigma)\geq 1-(1)/(2^2) \\=0.75`

b)
`25.5\leq x\leq 70.5\\|x-48|\leq 22.5\\|x-48|\leq 1.5\sigma`

`P(|x-48|\leq 1.5\sigma)\geq 1-(1)/(1.5^2) \\=0.556`

c) Using normal distribution we have

`P(|x-48|\leq 2\sigma)=0.95`

d) z value is 2.97

Hence x lies between

`(48-9*2.97, 49+9*2.97)\\=(21.27,74.73)`

Between 21 years and 75 years