Hypothesis Testing The z-test for Proportions

The Data
• Let x1 , x2 , x3 , … , xn denote a sample from a
normal population with mean μ and standard
deviation σ.
• Let
n
∑ xi
i=
1 x = = the sample mean
n
• we want to test if the mean, μ, is equal to some
given value μ 0 .
• Obviously if the sample mean is close to μ 0 the
Null Hypothesis should be accepted otherwise the
null Hypothesis should be rejected.
The Test Statistic
• To decide to accept or reject the Null Hypothesis
(H 0 ) we will use the test statistic
x − μ0 x − μ0
z = = =
σ σ
x
x − μ0
n ≈
σ
x − μ0
n
s
n
• If H0 is true we should expect the test statistic z to
be close to zero.
• If H 0 is true we should expect the test statistic z to
have a standard normal distribution.
• If H A is true we should expect the test statistic z to
be different from zero.
The sampling distribution of z when H 0 is true:
The Standard Normal distribution
Reject H 0
0 z
Accept H 0
Reject H 0
P
P
Reject H 0
α/2
The Acceptance region:
[ Accept H 0 when true]
= P[
− zα
/ 2 ≤ z ≤ zα
/ 2]
= 1−
α
[ Reject H when true]
P[
z < −z
or z > z ] = α
0
α/2
0 z
− zα
/ 2 zα
/ 2
Accept H 0
• Acceptance Region
– Accept H0 if: − z ≤ z ≤ z
Reject H 0
= α / 2
α / 2
α / 2
α / 2
• Critical Region
– Reject H0 if: z < −z
z > z
• With this Choice
α / 2 or
α / 2
[ I Error]
[ Reject when true]
H P
P =
Type 0
[ < − > ] = α z z z z
= α / 2
α / 2 or
P
Summary
To test mean of a Normal population
H0 : μ = μ0 (some specified value of μ)
Against
HA : μ ≠ μ0
1. Decide on α = P[Type I Error] = the
significance level of the test (usual choices
0.05 or 0.01)
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