Instant Notes: Analytical Chemistry

The application of a two-tailed F-test is demonstrated by the following

example.

Example 3

A proposed new method for the determination of sulfate in an industrial waste

effluent is compared with an existing method, giving the following results:

Method Mean/g dm-^3 No. of No. of degrees s/mg dm-^3

replicates of freedom

Existing 72 8 7 3.38

New 72 8 7 1.50

Is there a significant difference between the precisions of the two methods?

Fexptl

s^2 existing

⁄s

2

new

5.08

The two-tailed tabular value for Fwith 7 degrees of freedom for both the

numerator and the denominator is

F7,7=5.00 at the 95% probability level

As Fexptlis greaterthan Ftab, the null hypothesis is rejected; the two methods are

giving significantly different precisions.

This test is used to compare the experimental means of two sets of data or to

compare the experimental mean of one set of data with a known or reference

value. A statistic, t, is defined, depending on the circumstances, by one of three

alternative equations.

Comparison of two experimental means, x

_

Aand x

_

B

t = ×

(^1) ⁄ 2
(1)
where spooledis the pooled estimated standard deviation (Topic B2) for sets A and
B, and Nand Mare the numbers of values in sets A and B respectively. If N =M,
then the second term reduces to (N/ 2 )
(^1) ⁄ 2

. A simplified version of equation (4),
Topic B2, can be used to calculate spooledas there are only two sets of data.

spooled= (N−1)s^2 A+(M−1)s^2 B / N+M− (^2) 

(^1) ⁄ 2
(2)
In some circumstances, the use of equation (1) may not be appropriate for the
comparison of two experimental means. Examples of when this may be the case
are if
● the amount of sample is so restricted as to allow only one determination by
each of the two methods;
● the methods are to be compared for a series of samples containing different
levels of analyte rather than replicating the analysis at one level only;
● samples are to be analyzed over a long period of time when the same experi-
mental conditions cannot be guaranteed.
It may therefore be essential or convenient to pair the results (one from each
method) and use a paired t-testwhere t is defined by

NM
N+ M
(x

A−x


B)
spooled
t-test

(3.38)^2

(1.50)^2

B3 – Significance testing 37