Hypothesis Testing The z-test for Proportions

Performing the Test
1. Decide on α = P[Type I Error] = the
significance level of the test
Choose (α = 0.05)
2. Collect the data
• The number of successful operations in the
sample of 200 cases is x = 187
x 187
pˆ
= = =
n 200
3. Compute the test statistic
0.
935 (93.5%)
pˆ
− p0
z = =
σ pˆ pˆ
− p0
p0(
1−
p0
)
n
=
0.
935−
0.
91
= 1.
235
0.
91(
1−
0.
91)
200
4. Make the Decision zα = z
• Accept H0 if: z ≤1.
645
• Reject H 0 if:
z > 1.
645
0 . 05
=
1.
645
Since the test statistic is in the Acceptance
region we decide to Accept H 0
Conclude that H0 : p ≤ 0.
91(
91%)
is true
There is a no significant (α = 5%) increase
in the success rate of the new procedure
over the older procedure
Comments
• When the decision is made to accept H 0 is
made it should not be conclude that we have
proven H 0 .
• This is because when setting up the test we
have not controlled β = P[type II error] =
P[accepting H 0 when H 0 is FALSE]
• Whenever H 0 is accepted there is a
possibility that a type II error has been
made.
In the last example
The conclusion that there is a no significant
(α = 5%) increase in the success rate of the
new procedure over the older procedure
should be interpreted:
We have been unable to proof that the new
procedure is better than the old procedure
An analogy – a jury trial
The two possible decisions are
– Declare the accused innocent.
– Declare the accused guilty.
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