Question

The NAR estimates that 23% of all homes purchased in 2004 were considered investment properties. If a sample of 800 homes sold in 2004 is obtained find the probability that at least 175 homes are going to be used as investments.

Answer 1

Answer: 0.7752

Explanation:

Given : The proportion of all homes purchased in 2004 were considered investment properties estimated by NAR: p = 0.23

Sample size : n= 800

Required sample proportion :
`(175)/(800)=0.21875`

Now , the probability that at least 175 homes are going to be used as investments will be :

`P(p\geq 0.21875)=P(\frac{\hat{p}-p}{\sqrt{(p(1-p))/(n)}}\geq\frac{0.21875-0.23}{\sqrt{(0.23(1-0.23))/(800)}})\\\\=P(z\geq-0.756)\ \ [\because\ z=\frac{\hat{p}-p}{\sqrt{(p(1-p))/(n)}}]\\\\=P(z<0.756)\ \ [\because\ P(Z>-z)=P(Z<z)]\\\\=0.7751754\approx0.7752` [using p-value calculator or z-table]

Hence, the required probability = 0.7752