Answer
$\bar{x}$ =$\frac{∑X}{n}$
=$\frac{1264}{6}$
= 210.67
s^2 = $\frac{ n(∑X^2) - [(∑X)^2] }{n(n-1)}$
= $\frac{1637796 - 1597696}{6*5}$
=1336.667
s = $ \sqrt 1336.667 \ $
s =36.56
n = 6, s = 36.56 , df = 6-1=5, α = 1-0.90 = 0.10
To find χ2 right ,
α/2=0.05
From the table,
χ2 right = 11.071
To find χ2 left,
1-0.05=0.95
From the table,
χ2 left =1.145
The Confidence Interval for a Variance:
$ \frac{(n-1)s^2}{χ2 right}$ < $σ^{2}$< $ \frac{(n-1)s^2}{ χ2 left}$
$ \frac{5*36.56^2}{11.071}$ < $σ^{2}$ < $ \frac{5*36.56^2}{1.145}$
= 603.6793 dollars < $σ^{2}$ < 5836.972 dollars
The Confidence Interval for a Standard Deviation:
$\sqrt 603.6793 $ < σ< $\sqrt 5836.972$
24.57 dollars < σ < 76.4 dollars
Hence, we can be 90% confident that the true population variance for the monthly rates (in dollars) of brand-new randomly selected car models is between 603.6793 dollars to 5836.972 dollars, while the true population standard deviation is between 7.523g to 15.373g based on a sample of 6 randomly selected models.
