1. Estimates Made Using Sx
Two different statistical estimates can be made using Sx.
[1] the value of the next sample value, xi, where
[2] the value of the true mean, x‘, where

2. Example 8.5 [1] p range that contains the next measurement
with P = 95 %
[2] same but for N = 5
[3] for N = 19 and P = 50 %
[4] p range that contains the true mean

3. c2 and the c2 distribution
In the same way we used the student t distribution to estimate the range containing the population mean…
We can use the statistical variable c2 is and the c2 distribution to estimate the range containing the population standard deviation
The statistical variable c2 is defined as
since
and assuming the sample mean
equals the population mean…

4. The c2 distribution: how c2 behaves for normally distributed data
c2 = f(n)  infinite number of c2 distributions (like for Student’s t).
pdf
PDF
Figure 8.10
Figure 8.12
Determine Pr[c2≤10] for N = 11:
Determine Pr[c2≤10] for N = 5:
Determine c2 for P = 50 % and N = 5:

5. subscript α often used: c2a
“level of significance” P a
P+a=1
Figure 8.11

6. c2 Table Table 8.8 For N = 13, find a when c2 = 21.0 For P = 5 %, find
c2 if N = 20
Table 8.8

7. Uses of the c2 distribution
To infer s from Sx
To establish a rejection criterion (e.g., when to stop making something and fix the equipment; see ex 8.9)
To compare a sample to an assumed population (see ex 8.10)
Back to the top bullet, the true variance, s2, estimated with P % confidence, is in the range
noting a = 1 – P and n = N -1.

8. In-Class Example (x’ and s Inference)
Given the mean and standard deviation are 10 and 1.5, respectively, for a sample of 16, estimate with 95 % confidence the ranges within which are the true mean and true standard deviation (assuming a normally distributed population).
range for true mean = sample mean ± tn,PSx/√N
range for true variance: nSx2/ca/22 ≤ s2 ≤ nSx2/c1-a/22

9. In-Class Example (cont’d)
What happens to the range which contains the true standard
deviation when P is reduced from 95 % to 90 %?
To be less confident, we would expect the extent of the range to (increase or decrease?)
Let’s see what happens.