Answer
$\bar{x}$ =$\frac{∑X}{n}$
=$\frac{530}{17}$
= 31.1765
s^2 = $\frac{ n(∑X^2) - [(∑X)^2] }{n(n-1)}$
= $\frac{308652 - 280900 }{17*16}$
=102.0294
s = $ \sqrt 102.0294 \ $
s =10.101
n = 24, s = 10.101 g , df = 17-1=16, α = 1-0.95 = 0.05
To find χ2 right ,
α/2=0.025
From the table,
χ2 right = 28.845
To find χ2 left,
1-0.025=0.975
From the table,
χ2 left =6.908
The Confidence Interval for a Variance:
$ \frac{(n-1)s^2}{χ2 right}$ < σ^2 < $ \frac{(n-1)s^2}{ χ2 left}$
$ \frac{16*10.101^2}{28.845}$ < σ^2 < $ \frac{16*10.101^2}{6.908}$
= 156.595 < σ^2 < 236.316
The Confidence Interval for a Standard Deviation:
$\sqrt 156.595 $ < σ< $\sqrt 236.316$
7.523 < σ < 15.373
Hence, we can be 95% confident that the true population variance for the number of carbohydrates per 8-ounce serving of yogurt is between 156.595 g to 236.316g, while the true population standard deviation is between 7.523g to 15.373g based on a sample of 17 8-ounce yogurt of different selection of brands.
