preface to fifteenth edition

2.128 SECTION 2
Example 10 Six samples from a bulk chemical shipment averaged 77.50% active ingredient
with s 1.45%. The manufacturer claimed 80.00%. Can this claim be supported?
A one-tailed test is required since the alternative hypothesis states that the population parameter
is equal to or less than the hypothesized value.
77.50 80.00
t p6 1 3.86
1.45
Since t 0.95 2.01, and t 0.99 3.36, the hypothesis is rejected at both the 0.05 and the 0.01
levels of significance. It is extremely unlikely that the claim is justified.
2.3.8 The Chi-square ( 2 ) Distribution
The 2 distribution describes the behavior of variances. Actually there is not a single 2 distribution
but a whole set of distributions. Each distribution depends upon the number of degrees of freedom
(designated variously as df, d.f., or f ) in that distribution. Table 2.28 is laid out so that the horizontal
axis is labeled with probability levels, while the vertical axis is listed in descending order of increasing
number of degrees of freedom. The entries increase both as you read down and across the
table. Although Table 2.28 does not display the values for the mid-range of the distributions, at the
50% point of each distribution, the expected value of 2 is equal to the degrees of freedom. Estimates
of the variance are uncertain when based only on a few degrees of freedom. With the 10 samples
in Example 11, the standard deviation can vary by a large factor purely by random chance alone.
Even 31 samples gives a spread of standard deviation of 2.6 at the 95% confidence level.
Understanding the 2 distribution allows us to calculate the expected values of random variables
that are normally and independently distributed. In least squares multiple regression, or in calibration
work in general, there is a basic assumption that the error in the response variable is random and
normally distributed, with a variance that follows a 2 distribution.
Confidence limits for an estimate of the variance may be calculated as follows. For each group
of samples a standard deviation is calculated. These estimates of possess a distribution called the
2 distribution:
s 2
2
(2.15)
2
/df
The upper and lower confidence limits for the standard deviation are obtained by dividing
(N 1)s 2 by two entries taken from Table 2.28. The estimate of variance at the 90% confidence
limits is for use in the entries
2
and
2
0.05 0.95
(for 5% and 95%) with N degrees of freedom.
Example 11 The variance obtained for 10 samples is (0.65) 2 . 2 is known to be (0.75) 2 .How
reliable is s 2 as an estimate of 2 ?
2 2
s (N 1) s (N 1)
2
2 2
0.975 0.025
(0.65)
2
(10 1) (0.65)
2
(10 1)
2
19.02 2.70
2
0.20 1.43
Thus, only one time in 40 will 9s 2 / 2 be less than 2.70 by chance alone. Similarly, only one time