Answer:
a) 18.98%
b) -0.2142
c) (183000,407000)
d) (127000,463000)
Step-by-step explanation:
We are given the following in the question:
Mean, μ = $295,000
Standard Deviation, σ = $56,000
a) Coefficient of variation
![COV =\dfrac{\sigma}{\mu} = \dfrac{56000}{295000}\\\\=0.1898\\=18.98\%](https://tex.z-dn.net/?f=COV%20%3D%5Cdfrac%7B%5Csigma%7D%7B%5Cmu%7D%20%3D%20%5Cdfrac%7B56000%7D%7B295000%7D%5C%5C%5C%5C%3D0.1898%5C%5C%3D18.98%5C%25)
b) z-score
x = $283,000
Formula:
![z_{score} = \displaystyle\frac{x-\mu}{\sigma}](https://tex.z-dn.net/?f=z_%7Bscore%7D%20%3D%20%5Cdisplaystyle%5Cfrac%7Bx-%5Cmu%7D%7B%5Csigma%7D)
Putting the value, we get,
![z_{score} = \displaystyle\frac{283000-295000}{56000} = -0.2142](https://tex.z-dn.net/?f=z_%7Bscore%7D%20%3D%20%5Cdisplaystyle%5Cfrac%7B283000-295000%7D%7B56000%7D%20%3D%20-0.2142)
c) 95% interval
Empirical Rule:
- According to this rule almost all the data lies within three standard deviation of mean for a normal distribution.
- About 68% of data lies within one standard deviation of mean.
- About 95% of data lies within two standard deviation of mean.
![\mu \pm 2(\sigma)\\=295000 \pm 2(56000)\\=(183000,407000)](https://tex.z-dn.net/?f=%5Cmu%20%5Cpm%202%28%5Csigma%29%5C%5C%3D295000%20%5Cpm%202%2856000%29%5C%5C%3D%28183000%2C407000%29)
Thus, the price of 95% of the homes lies between $183,000 and $407,000.
d) 90% interval
Chebyshev's Rule:
- For a non normal data atleast
percent of data lies within k standard deviation of mean. - For k = 3
![1 - \dfrac{1}{(3)^2} = 88.89\% \approx 90\%](https://tex.z-dn.net/?f=1%20-%20%5Cdfrac%7B1%7D%7B%283%29%5E2%7D%20%3D%2088.89%5C%25%20%5Capprox%2090%5C%25)
Thus, 90% of prices will lies within three standard deviation of mean.
![\mu \pm 3(\sigma)\\=295000 \pm 3(56000)\\=(127000,463000)](https://tex.z-dn.net/?f=%5Cmu%20%5Cpm%203%28%5Csigma%29%5C%5C%3D295000%20%5Cpm%203%2856000%29%5C%5C%3D%28127000%2C463000%29)
According to Chebyshev's Theorem 90% of prices will lie between $127,000 and $463,000.