Inferences about Two Population Proportions
Page 1
Testing a Claim about a Population Proportion
Step 0: Verify Assumptions
The hypothesis test of a population proportion has two assumptions.
1. The samples are obtained using simple random sampling.
2. np0 q 0 ≥ 10 and n ≤ 0.05N (i.e., that sample size n is no more than 5% or the population
size N).
Step 1: State the Hypothesis
A claim is made regarding the population proportion. This claim is used to determine the null
and alternative hypotheses. The hypotheses can be structured in one of the following ways:
Two-Tailed
H0: p = p0
H1: p ≠ p0
Left-Tailed
H 0: p ≥ p0
H1: p < p0
Right-Tailed
H0: p ≤ p0
H1: p > p0
Note: p0 is the assumed value of the population proportions.
Step 2: Select a Level of Significance
The selection of the level of significance α is done based on the seriousness of making a Type I
error. (The typical value of α is 0.05.)
Step 3: Calculate the Test Statistic
The test statistic represents the number of standard deviations the proportion p̂ is from the
claimed population proportion, p0, based on the standard error of the sampling distribution of p0.
The test statistic approximately follows the z-distribution.
z0 =
pˆ − p 0
p0 q0
n
Step 4: Determine the Decision Criterion
The Classical Approach: Find the Critical Value
The level of significance is used to determine the critical value, represented by the z-values in
the figures below. The critical region includes the values of the shaded region. The shaded
region is α.
Two-Tailed
−zα / 2
Robert A. Powers
zα / 2
Left-Tailed
−zα
Right-Tailed
zα
University of Northern Colorado
Inferences about Two Population Proportions
Page 2
The Modern Approach: Find the p-Value
Based on the critical value z0, determine the probability that the difference between two
sample proportions is further from the expected difference than is hypothesized. This is
represented by the shaded region in the figures below.
Two-Tailed
P(Z < -|z0| or Z > |z0|)
− | z0 |
| z0 |
Left-Tailed
P(Z < -z0)
− z0
Right-Tailed
P(Z > z0)
z0
Step 5: Make a Decision
Reject the null hypothesis if the test statistics lies in the critical region or the probability
associated with the test statistic is less than the level of significance.
Do not reject the null hypothesis if the test statistic does not lie in the critical region or the
probability associated with the test statistic is greater than or equal to the level of significance.
Step 6: State the Conclusion
State the conclusion of the hypothesis test based on the decision made and with respect to the
original claim.
Reject H0
Do Not
Reject H0
Original Claim is H0
There is sufficient evidence (at the α
level) to reject the claim that … .
There is not sufficient evidence (at the
α level) to reject the claim that … .
Robert A. Powers
Original Claim is H1
There is sufficient evidence (at the α
level) to support the claim that … .
There is not sufficient evidence (at the
α level) to support the claim that … .
University of Northern Colorado