Probability and Statistics for Engineering and the Sciences (with CD-ROM and InfoTrac )

Stat 312: Lecture 12
Errors in hypothesis testing
Moo K. Chung
[email protected]
Oct 20, 2004
Type I error : you reject H
0
when H
0
is true.
P ( Type I error ) = P ( reject H
0
| H
0
true ) = α. The
resulting α is called the significance level of the test
and the corresponding test is called a level α test .
We will use test procedures that give α less than a
specified level (0.05 or 0.01 usually).
I believe that dogs are as smart as people. Assume
IQ of a dog follows X
i
∼ N ( μ, 10
2
) . IQ of 10 dogs
are measured: 30, 25, 70, 110, 40, 80, 50, 60, 100,
60. We want to test if dogs are as smart as people by
testing
H
0
: μ = 100 vs. H
1
: μ < 100 .
(1) Let
¯
X be a test statistic and R = ( -∞ , 90) to
be a rejection region. Let’s compute the probabil-
ity of making Type I error based on this testing
procedure. Under the assumption H
0
is true,
X
i
∼ N (100 , 10
2
) .
Under this condition,
¯
X ∼ N (100 , 10) and
α = P (
¯
X < 90) .
>help(pnorm)
...
Usage:
... pnorm(q, mean=0, sd=1,...)
...
> pnorm(90,100,sqrt(10))
[1] 0.0007827011
By using this test procedure, it is highly unlikely
to make Type I error.
(2) Let’s see what happens when we change the test
precedure. When R = ( -∞ , 95) with the same
test statistic
¯
X ,
α = P (
¯
X < 95) .
> pnorm(95,100,sqrt(10))
[1] 0.05692315
70 90 110
0.0
0.4
0.8
a
pnorm(a, 100, sqrt(10))
70 90 110
0.0
0.4
0.8