fundamentals of engineering supplied-reference handbook - Ventech!

HYPOTHESIS TESTING
Consider an unknown parameter θ of a statistical distribution. Let the null hypothesis be
H 0: θ=θ0
and let the alternative hypothesis be
H1: θ=θ 1
H : µ=µ ; H : µ≠µ
0 0 1 0
⎛µ 0 −µ ⎞ ⎛µ 0 −µ ⎞
β=Φ ⎜ +Ζα 2 −Φ −Ζα
2
⎝ σ n
⎟
⎠
⎜
⎝ σ n
⎟
⎠
H : µ=µ ; H : µ>µ
0 0 1 0
⎛µ 0 −µ ⎞
β=Φ ⎜ +Ζα
⎝ σ n
⎟
⎠
n =
( α β)
( µ −µ )
1 0
2 2
Ζ +Ζ σ
2
18
ENGINEERING PROBABILITY AND STATISTICS (continued)
Rejecting 0 H when it is true is known as a type I error, while accepting H 1 when it is wrong is known as a type II error.
Furthermore, the probabilities of type I and type II errors are usually represented by the symbols α and β , respectively:
α=probability (type I error)
β=probability (type II error)
The probability of a type I error is known as the level of significance of the test.
Assume that the values of α and β are given. The sample size can be obtained from the following relationships. In (A) and
(B), µ 1 is the value assumed to be the true mean.
(A)
An approximate result is
(B)
n �
( α 2 β)
( µ −µ )
1 0
2 2
Ζ +Ζ σ
2
Refer to the Hypothesis Testing table in the INDUSTRIAL ENGINEERING section of this handbook.