12.2 Hypothesis Tests for One Population Proportion

12.2 Hypothesis Tests for One Population Proportion
1. A one-sample hypothesis test is a special case of the one-sample z-test for a
population mean.
2. One-Sample z-Test for a Population Proportion:
Purpose: To perform a hypothesis test for a population proportion, p.
Assumptions: 1. We have a simple random sample. 2. Both
n(1 − p 0
) are greater than 5.
np 0
and
a. Set the hypotheses for an assumed value p 0
: Null hypothesis: H 0
: p = p 0
Alternate hypothesis:
Two Tailed: H 0
: p ≠ p 0
Left Tailed: H 0
: p < p 0
Right Tailed: H 0
: p > p 0
b. Significance level: α
c. Test statistic, where ˆ p is the sample proportion: z 0
=
ˆ p − p 0
p 0
(1 − p 0
)
n
d. Critical Value Method: Calculate the critical values using Table II.
These values divide the rejection region and the non-rejection region:
Two Tailed Test: ±z α /2
Left Tailed Test: −z α
Right Tailed Test: z α
If the values of the test statistic,
Otherwise don't reject H 0
.
z 0
falls in the reject region, reject H 0
.
e. P-Value Method: Use Table II to find the area for z
0
and more extreme
than z 0
. This area is the P-value. (Two Tailed: Area left of – z 0
added
to the area to the right of z 0
. Left Tailed: Area to the left of – z 0
. Right
Tailed: Area to the right of z 0
.) Reject the null hypothesis, H 0
, if the
P-value ≤ α. Otherwise, don't reject H 0
.
f. Interpret the results.

3. A calculator can also be used to perform this test. Use #5: 1-Prop ZTest under
STAT TESTS.
12.3 Inferences for two Population Proportions, Using Independent
Samples
1. We will now study a method for comparing proportions computed for two
populations with one attribute. Are these proportions equal
2. We will use independent random samples from each population to perform a
hypothesis test and to find a confidence interval to determine whether the
proportion of one population with the attribute is equal to the proportion of the
other population with the same attribute.
3. To do this we need to know the distribution of the difference between two sample
proportions.
4. Notation:
Population 1 Population 2
Population Proportion p 1
p 2
Sample Size n 1
n 2
Number of Successes x 1
x 2
Sample Proportion
ˆ p 1
= x 1
n 1
ˆ p 2
= x 2
n 2
5. The Sampling Distribution of the Difference Between Two Sample
Proportions for Independent Samples
For independent samples of sizes and from the two populations:
n 1
n 2
µ ˆ p 1 − ˆ p 2
= p 1
− p 2
σ
p ˆ 1 − p ˆ 2
=
p 1
(1 − p 1
)
n 1
+ p 2 (1− p 2 )
n 2
ˆ p 1
− p ˆ 2
is approximately normally distributed for large n 1
and n 2
.
6. From the above we see that the standardized variable is
z =
z =
( p ˆ 1
− p ˆ 2
) − ( p 1
− p 2
)
p 1
(1 − p 1
)
+ p 2(1 − p 2
)
n 1
n 2
p ˆ 1
− p ˆ 2
1
p(1 − p) + 1 n 1
n 2
, which if p 1
= p 2
= p becomes

However, since p is not known, we will replace it with the pooled sample
x
proportion ˆ p p
= 1
+ x 2
.
n 1
+ n 2
7. Two-Sample z-Test for Two Population Proportions:
Purpose: To perform a hypothesis test to compare two population proportions, p 1
and p 2
.
Assumptions: 1. We have a simple random samples that are independent. 2. x 1
,
n 1
− x 1
, x 2
, and n 2
− x 2
, are all greater than 5.
a. Set the hypotheses assuming the population proportions are equal.
Null hypothesis: H 0
: p 1
= p 2
Alternate hypothesis:
Two Tailed: H 0
: p 1
≠ p 2
Left Tailed: H 0
: p 1
< p 2
Right Tailed: H 0
: p 1
> p 2
b. Significance level: α
c. Test statistic: z 0
=
p ˆ 1
− p ˆ 2
ˆ p p
(1 − p ˆ p
)
1
n 1
+ 1 n 2
d. Critical Value Method: Calculate the critical values using Table II.
These values divide the rejection region and the non-rejection region:
Two Tailed Test: ±z α /2
Left Tailed Test: −z α
Right Tailed Test: z α
If the values of the test statistic,
Otherwise don't reject H 0
.
z 0
falls in the reject region, reject H 0
.
e. P-Value Method: Use Table II to find the area for z 0
and more extreme
than z 0
. This area is the P-value. (Two Tailed: Area left of – z 0
added
to the area to the right of z 0
. Left Tailed: Area to the left of – z 0
. Right
Tailed: Area to the right of z 0
.) Reject the null hypothesis, H 0
, if the
P-value ≤ α. Otherwise, don't reject H 0
.
f. Interpret the results.

8. Two-Sample z-Interval Procedure: To find a confidence interval for the
difference between two population proportions, p 1
and p 2
.
Assumptions: 1. We have a simple random samples that are independent. 2. x 1
,
n 1
− x 1
, x 2
, and n 2
− x 2
, are all greater than 5.
For the confidence level of 1 – α, use Table II to find z α /2
The end points of the confidence interval for p 1
– p 2
are
( p ˆ 1
− p ˆ 2
) ± z α /2
⋅
p ˆ 1
(1− p ˆ 1
) p
+ ˆ 2
(1 − p ˆ 2
)
.
n 2
n 1
Interpret the confidence interval.
9. The margin of error for the estimate of p 1
– p 2
is
E = z α /2
⋅
p ˆ 1
(1 − p ˆ 1
) p
+ ˆ 2
(1 − p ˆ 2
)
.
n 2
n 1
This is half the length of the confidence interval and it represents the precision
with which ˆ p 1
– ˆ p 2
estimates p 1
– p 2
.
10. Sample Size for Estimating p 1
– p 2
: A (1−α)-level confidence interval for the
difference between two population proportions that has a margin of error of at
most E can be obtained by choosing
⎛
n 1
= n 2
= 0.5 z α /2 ⎞
⎝ E ⎠
2
rounded up to the nearest whole number.
If you can make educated guesses, ˆ p 1g
and p ˆ 2g
for ˆ p 1
and ˆ p 2
, you should
instead choose
n 1
= n 2
=
(
p ˆ 1g
(1 − p ˆ 1g
) + p ˆ 2g
(1− p ˆ 2g
)) ⎛
⎝
z α /2
E
⎞
⎠
2
rounded up to the nearest
whole number. (If a range an estimate for ˆ p 1
and ˆ p 2
are given as a range of
values are pick the values in the range that are closest to 0.5.)
11. Hypothesis test for two population proportions can be conducted with a calculator
using 2-PropZTest under STAT TEST. Confidence intervals can be found in
the calculator using 2-PropZInt under STAT TEST.