30 INTRODUCTION TO BASIC CONCEPTS IN EXPERIMENTAL DESIGN

Sampling Sampling
distribution distribution
— under Hb under H,

100

101 102 | 103
-4——— — Do not reject H Reject H, — ——

Figure 1.5-3 Regions corresponding to prob-
abilities of making type | error (x) and type LU
error (8). The region corresponding to a type 1
error is determined by the experimenter when he
specifies x and /,. |, for a given 4, and true
alternative, x is made smaller, the probability of
making a type li error is increased.

the parameter is egual to or less than 100, a correct decision has ben made.
On the other hand, if the parameter is egual to 103, a type TI error has
been made.

POWER OF A TEST

If the parameter is egual to 100, the probability of making a correct
decision is 1 — a. If, on the other hand, the parameter is egual to 103,
the probability of making a correct decision is 1 — B. This latter prob-
ability is called the power of the test. It is simply the probability of deciding
that H, is wrong, given a decision rule and the true value under H,. The
possible decision outcomeS can be categorized as shown in Table 1.5-1.

TABLE 1.5-1 Decision Outcomes Categorized

True Situation

u < 103

u <100
correct decision type II error
zl1-4a <B
type ] error correct decision
z4 z l1-— B

An experimenter attempts to select an experimental design and
set of decision rules that will result in the highest power for a given type l

Decision