6
Chapter 1
A. Gaussian Distribution
1. Students t Distribution
A sample mean 2 is calculated from test samples with standard deviation
of s. It is desired to find the infinite sample mean ~ to a confidence of
95% (higher levels may also be used).
s
s
-- N 1
(1.21)
~: -- /0.975 N - l - -
-
Note that 2.5% is in each tail to make 95% therefore for each side t0.975 is
used. The degrees of freedom (d.f) is N-1 and the values of t for 2.5%
can be read in Table E.2.
2. Chi-Square Distribution
Ns 2
x 2 = 0
2
(1.22)
where N-1 is the degrees of freedom d.f. calculated from the test sample of
N. 0"2 is the infinite sample size standard deviation. Here again 95% confidence x0.975 and x0.025 are used so that
-- < r~ < -(1.23)
X0.025
X0.975
values for x~.975 and X~.o2 ~ are read from Table E.3. Here 0" is used for ~ in Eq.
(1.1)
3. One Sided Tolerance Limit
r~ = .7c - Ks
(1.24)
~, s are from limited sample. Fig. E.I allows selection of K when the
sample size and percent confidence is known. For example, choosing a
so that 90% of the experimental values are greater than ¢ with a 95% confidence limit.
4. Estimate of the Mean
The estimate of the mean,/~, and standard deviation 0" or ~ are discussed by
Dixon and Massey [1.8], where small sample size values are arranged in
ascending order xl, x2, ..., xn and number n < 20. The estimate for the
/~ and 0" or ~ are listed in Tables E.4-E.6. The values do not have a percent
confidence attached but are for the infinite sample size.
The following example shows the good and bad features of a small
sample size and is presented to show the variation in some calculations.