Answer
n = 24, s = 4.8 months , df = 24-1=23, α = 1-0.90 = 0.10
To find χ2 right ,
α/2=0.05
From the table,
χ2 right = 35.172
To find χ2 left,
1-0.05=0.95
From the table,
χ2 left =13.091
The Confidence Interval for a Variance:
$ \frac{(n-1)s^2}{χ2 right}$ < σ^2 < $ \frac{(n-1)s^2}{ χ2 left}$
$ \frac{23*4.8^2}{35.172}$ < σ^2 < $ \frac{23*4.8^2}{13.091}$
= 15.067 < σ^2 < 40.477
The Confidence Interval for a Standard Deviation:
$\sqrt 15.067 $ < σ< $\sqrt 40.477$
3.882 < σ < 6.362
Hence, we can be 90% confident that the true standard deviation for the lifetimes of inexpensive wristwatches is between 3.882 months to 6.362 months based on a sample of 24 watches.
The lifetimes are relatively inconsistent, as the range of the lifetimes are rather large. In order to obtain a more consistent value, we should assume the confidence internal to become 95% or 99%.
