Anon

Inferential Statistics 375

the statistic t(x) either leads to rejection of the null hypothesis (i.e., δ(x) = d 0 ),
or not (i.e., δ(x) = d 1 ).
To determine when we have to make a decision d 0 or, alternatively, d 1 ,
we split the state space Δ of t(X) into two disjoint sets that we denote by
ΔA and ΔC. The set ΔA is referred to as the acceptance region while ΔC is the
critical region or rejection region.
When the outcome of the sample x is in ΔA, we do not reject the null
hypothesis (i.e., the result of the test is δ(x) = d 0 ). If, on the other hand, x
should be some value in ΔC, the result of the test is now the contrary (i.e.,
δ(x) = d 1 ), such that we decide in favor of the alternative hypothesis.

e rror Types

We have to be aware that no matter how we design our test, we are at risk of
committing an error by making the wrong decision. Given the two hypoth-
eses, H 0 and H 1 , and the two possible decisions, d 0 and d 1 , we can commit
two possible errors. These errors are discussed next.

Type i and Type ii error The two possible errors we can incur are charac-
terized by unintentionally deciding against the true hypothesis. Each error
related to a particular hypothesis is referred to using the following standard
terminology.

Type I error. The error resulting from rejection of the null hypothesis

(H 0 ) (i.e., δ(X) = d 1 ) given that it is actually true (i.e., θ ∈ Θ 0 ) is

referred to as a type I error.

Type II error. The error resulting from not rejecting the null hypothesis

(H 0 ) (i.e., δ(X) = d 0 ) even though the alternative hypothesis (H 1 )

holds (i.e., θ ∈ Θ 1 ) is referred to as a type II error.

In the following table, we show all four possible outcomes from a
hypothesis test depending on the respective hypothesis:

H 0 : θ in Θ 0 H 1 : θ in Θ 1

Decision

d 0 Correct Type II error

d 1 Type I error Correct

So, we see that in two cases, we make the correct decision. The first case
occurs if we do not reject the null hypothesis (i.e., δ(X) = d 0 ) when it actually
holds. The second case occurs if we correctly decide against the null hypoth-
esis (i.e., δ(X) = d 1 ), when it is not true and, consequently, the alternative