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

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

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…

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:

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

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

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.

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

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.