Answer
$\bar{x}$ =$\frac{∑X}{n}$
=$\frac{3405}{14}$
= 243.2143
s^2 = $\frac{ n(∑X^2) - [(∑X)^2] }{n(n-1)}$
= $\frac{11697350- 11594025}{14*13}$
=567.7198
s = $ \sqrt 567.7198 \ $
s =23.8269
n = 14, s = 4.4483 , df = 14-1=13, α = 1-0.95 = 0.05
To find χ2 right ,
α/2=0.025
From the table,
χ2 right = 24.736
To find χ2 left,
1-0.025=0.975
From the table,
χ2 left =5.009
The Confidence Interval for a Variance:
$ \frac{(n-1)s^2}{χ2 right}$ < $σ^{2}$ < $ \frac{(n-1)s^2}{ χ2 left}$
$ \frac{13*23.8269^2}{24.736}$ < $σ^{2}$ < $ \frac{13*23.8269^2}{5.009}$
= 298.365 calories< $σ^{2}$ < 1473.419 calories
The Confidence Interval for a Standard Deviation:
$\sqrt 298.365$ < σ< $\sqrt 1473.419$
17.2733 calories < σ < 38.3851 calories
Hence, we can be 95% confident that the true population variance for the calories in a standard size candy bar is between 298.365 calories to 1473.419 calories , while the true population standard deviation is between 17.2733 calories to 38.3851 calories based on a sample of 14 randomly selected candy bars.
