Answer
$\bar{x}$ =$\frac{∑X}{n}$
=$\frac{1780}{15}$
= 118.6667
s^2 = $\frac{ n(∑X^2) - [(∑X)^2] }{n(n-1)}$
= $\frac{3558000- 3168400}{15*14}$
=1855.238
s = $ \sqrt 1855.238 \ $
s =43.0725
n = 15, s = 43.0725, df = 15-1=14, α = 1-0.95 = 0.05
To find χ2 right ,
α/2=0.025
From the table,
χ2 right = 26.119
To find χ2 left,
1-0.025=0.975
From the table,
χ2 left =5.629
The Confidence Interval for a Variance:
$ \frac{(n-1)s^2}{χ2 right}$ < $σ^{2}$ < $ \frac{(n-1)s^2}{ χ2 left}$
$ \frac{14*43.0725^2}{26.119}$ < $σ^{2}$ < $ \frac{14*43.0725^2}{5.629}$
= 994.423 < $σ^{2}$ < 4614.2
The Confidence Interval for a Standard Deviation:
$\sqrt 994.423 $ < σ< $\sqrt 4614.2$
31.5345 < σ < 67.9279
Hence, we can be 95% confident that the true population variance for the cholesterol amounts in a random sample of grilled meats is between 994.423 mg to 4614.2mg , while the true population standard deviation is between 31.5345mg to 67.9279mg based on a sample of 15 grilled meats.
