Page 473 - Probability and Statistical Inference

450 9. Confidence Interval Estimation

In other words,

is a (1 – α) two–sided confidence interval estimator for µ. !
Example 9.2.9 Normal Variance: Suppose that X , ..., X are iid N(µ,
1
n
σ ) with both unknown parameters µ ∈ ℜ and σ ∈ ℜ , n ≥ 2. Given some α
2
+
∈ (0,1), we wish to construct a (1 – α) two–sided confidence interval for σ .
2
Let be the sample mean and be the sample
variance. The statistic T ≡ ( , S) is minimal sufficient for (µ, σ). Here, the
distributions of the X’s belong to the location–scale family. The pivot U = (n
2
– 1) S / σ has the Chi–square distribution with (n – 1) degrees of freedom.
2
Recall that is the upper 100γ% point of the Chi–square distribution with
v degrees of freedom.

Figure 9.2.6. The Area on the Right (or Left) of
is α/2
The distribution of S belongs to the scale family. So, we can claim that
2
See the Figure 9.2.6. Thus, we
claim that

In other words,

2
is a (1 – α) two–sided confidence interval estimator for σ . !