Page 311 - Applied statistics and probability for engineers

Section 8-3/Conidence Interval on the Variance and Standard Deviation of a Normal Distribution 289

2
.
Conversely, a lower 5% point of chi-square with 10 degrees of freedom would be χ .0 95 10, = 3 94
(from Appendix A). Both of these percentage points are shown in Figure 8-9(b).
2
The construction of the 100(1 – α)% CI for σ is straightforward. Because
( n 1− ) S 2
X =
2
σ 2
is chi-square with n – 1 degrees of freedom, we may write
χ 2 ≤ 2 2
1
/ ,n−1
P( 1 −≠ / ,n−1 X ≤ χ α 2 ) = − α
2
so that ⎛ ( 2 ⎞
P χ 2 1 −≠ / ,n−2 1 ≤ n − ) 1 S ≤ χ 2 α 2 ⎟ = − α
1
⎜
/ ,n−1
⎝ σ 2 ⎠
This last equation can be rearranged as
( ⎛ n − ) 1 s 2 ( n − ) 1 s ⎞
2
2
1
P ⎜ 2 ≤ σ ≤ 2 ⎟ = − α
/ ,n ⎠
⎝ χ α 2 χ 1 −≠ 2 −1
/ ,n−1

2
This leads to the following deinition of the conidence interval for σ .

Coni dence Interval
2
on the Variance If s is the sample variance from a random sample of n observations from a normal dis-
2
2
tribution with unknown variance σ , then a 100(1 – `)% coni dence interval on r is
( n − ) 1 s 2 ( n − ) 1 s 2
2
≤ σ ≤ (8-19)
2 2
χ α / ,n−2 1 χ −α 2
1
/ ,n−1
2 2
where χ α/ ,n2 −1 and χ −≠1 2 / ,n −1 are the upper and lower 100α 2 percentage points of
the chi-square distribution with n – 1 degrees of freedom, respectively. A coni dence
interval for σ has lower and upper limits that are the square roots of the correspond-
ing limits in Equation 8-19.
It is also possible to i nd a 100(1 – α)% lower coni dence bound or upper coni dence
2
bound on σ .
One-Sided Coni dence
2
Bounds on the The 100(1 – α)% lower and upper conidence bounds on σ are

Variance ( n − ) 1 s 2 ( n − ) 1 s 2
2
≤ σ 2 and σ ≤ (8-20)
χ 2 α ,n−1 χ 1 2 −α ,n−1
respectively.
( S σ 2
2
The CIs given in Equations 8-19 and 8-20 are less robust to the normality assumption. The distribution of n − ) 1
can be very different from the chi-square if the underlying population is not normal.
Example 8-7 Detergent Filling An automatic illing machine is used to ill bottles with liquid detergent. A

random sample of 20 bottles results in a sample variance of i volume of s 2 = 0.0153 2 (l uid
ll

ounce). If the variance of ill volume is too large, an unacceptable proportion of bottles will be under- or overi lled. We

will assume that the ill volume is approximately normally distributed. A 95% upper conidence bound is found from

Equation 8-26 as follows: (n − ) 1 s 2
σ à
2
χ 2
,
. 0 95 19