9597

• UCREFRP Catalog Number: 9597
• Author: U.S. Environmental Protection Agency.
• Title: Biological Field and Laboratory Methods for Measuring the Quality of Surface Waters and Effluents.
• USFW Year: 1973.
• USFW - Doc Type: Cincinnati, OH.
• Copyright Material: NO

BIOLOGICAL METHODS

correct probability level requires a discussion of
two types of hypotheses testable using the
t-statistic.
The null hypothesis is a hypothesis of no
difference between a population parameter and
another value. Suppose the hypothesis to be
tested is that the mean, J.l, of some population
equals 10. Then we would write the null
hypothesis (symbolized Ho) as

Ho : /1. = 10

Here 10 is the value of e in the general form for
the t-statistic. An alternative to the null
hypothesis is now required. The investigator,
viewing the experimental situation, determines
the way in which this is stated. If the investi-
gator merely wants to answer whether the
sample indicates that J.l = 10 or not, then the
alternate hypothesis, Ha, is

Ha : p.=~ 10

If it is known, for example, that J.l cannot be less
than 10, then Ha is

Ha: /1.>10

and by similar reasoning the other possible Ha is
Ha: /1.<10

Hence, there are two types of alternate hy-
potheses: one where the alternative is simply
that the null hypothesis is false (Ha : J.l =1= 10);
the other, that the null hypothesis is false and,
in addition, that the population parameter lies
to one side or the other of the hypothesized
value [Ha: J.l (> or <) 10]. In the case of Ha : J.l
=1= 10, the test is called a two-tailed test; in the
case of either of the second types of alternate
hypotheses, the t-test is called a one-tailed test.

To use a t-table, it must be determined
whether the column headings (probability of a
larger value, or percentage points, or other
means of expressing a) are set for one-tailed or
two-tailed tests. Some tables are presented with
both headings, and the terms "sign ignored" and
"sign considered" are used. "Sign ignored"
implies a two-tailed test, and "sign considered"
impiies a one-tailed test. Where tables are given
for one-tailed tests, the column for any
probability (or percentage) is the column
appropriate to twice the probability for a two-
tailed test. Hence, if a column heading is .025

.

and the table is for one-tailed tests, use this same
column for .05 in a two-tailed test (double any
one-tailed test heading to get the proper two-
tailed test heading; or conversely, halve the two-
tailed test heading to obtain proper headings for
one-tailed tests).
Testing Ho : J.l = M (the population mean
equals some value M):
X-M
t = ---s:- (27)
x

where X is given by equation (11) or other
appropriate equation; M = the hypothesized
population mean; and Sx is given by equation
(15). The t-table is entered at the chosen proba-
bility level (often .05) and n- 1 degrees of free-
dom, where n is the number of observations in
the sample.
When the computed t-statistic exceeds the
tabular value there is said to be a 1 - a proba-
bility that Ho is false.
Testing Ho : J.ll = J.l2 (the mean of the popula-
tion from which sample 1 was taken equals the
mean of the population from which sample 2
was taken):

XI - X2
t=
s- -
XI - X2

(28)

where sX1 _ X2 = the pooled standard error

obtained by adding the corrected sums of
squares for sample 1 to the corrected sums of
squares for sample 2, and dividing by the sum of
the degrees of freedom for each times the sum
of the numbers of observations, i.e.,

(~XI)2 (~X2)2
XI2 - - + ~X22 -- (29)*
nl n2
sXI-X2 = (nl + n2) [(nl - 1) + (n2 - 1)]

An alternative and frequently useful form is

_ ~)SI2 + (n2 -1)s2~
sXI - X2 =V "'"'(ii0 n2) (nl + n2 - 2)

(30)

where S12 and S22 are each computed according
to equation (12).
For all conditions to be met where the t-test is
applicable, the sample should have been selected

*~ sign, when unsubscripted, will indicate summation for all
observations, hence ~I means sum of all observations in
sample 1.

I

12