Hypothesis Test for a Proportion
This lesson explains how to conduct a hypothesis test of a proportion,
when the following conditions are met:
- The population size is at least 20 times as big as the sample
size.
This approach consists of four steps: (1) state the hypotheses,
(2) formulate an analysis plan, (3) analyze sample data, and
(4) interpret results.
State the Hypotheses
Every hypothesis test requires the analyst
to state a
null hypothesis
and an
alternative hypothesis. The hypotheses are stated in such
a way that they are mutually exclusive. That is, if one is
true, the other must be false; and vice versa.
Formulate an Analysis Plan
The analysis plan describes
how to use sample data to accept or reject the null
hypothesis. It should specify the following elements.
- Significance level. Often, researchers choose
significance levels
equal to
0.01, 0.05, or 0.10; but any value between 0 and
1 can be used.
- Test method. Use the
one-sample z-test
to determine whether the hypothesized population proportion differs
significantly from the observed sample proportion.
Analyze Sample Data
Using sample data, find the test statistic and its associated
P-Value.
Interpret Results
If the sample findings are unlikely, given
the null hypothesis, the researcher rejects the null hypothesis.
Typically, this involves comparing the P-value to the
significance level,
and rejecting the null hypothesis when the P-value is less than
the significance level.
Test Your Understanding
In this section, two hypothesis testing examples illustrate how to conduct a
hypothesis test of a proportion. The first problem involves a a
two-tailed test; the second problem, a one-tailed test.
Sample Size Calculator
As you probably noticed, the process of testing a hypothesis about a proportion
can be complex. Stat Trek's Sample Size Calculator can do the
same job quickly and easily. When you need to test a hypothesis, consider using the
Sample Size Calculator. The
calculator is free. It can found in the Stat Trek
main menu under the Stat Tools tab. Or you can tap the button below.
Sample Size Calculator
Problem 1: Two-Tailed Test
The CEO of a large electric utility claims that 80 percent of his 1,000,000
customers are very satisfied with the service they receive. To test this claim,
the local newspaper surveyed 100 customers, using simple random sampling.
Among the sampled
customers, 73 percent say they are very satisified. Based on these findings,
can we reject the CEO's hypothesis that 80% of the customers are very
satisfied? Use a 0.05 level of significance.
Solution: The solution to this problem takes four steps:
(1) state the hypotheses, (2) formulate an analysis plan,
(3) analyze sample data, and (4) interpret results.
We work through those steps below:
State the hypotheses. The first step is to
state the null hypothesis and an alternative hypothesis.
Null hypothesis: P = 0.80
Alternative hypothesis: P ≠ 0.80
Note that these hypotheses constitute a two-tailed test.
The null hypothesis will be rejected if the sample proportion
is too big or if it is too small.- Formulate an analysis plan. For this analysis,
the significance level is 0.05. The test method,
shown in the next section, is a
one-sample z-test.
- Analyze sample data. Using sample data, we
calculate the standard deviation (σ) and compute the z-score
test statistic (z).
σ = sqrt[ P * ( 1 - P ) / n ]
σ = sqrt [(0.8 * 0.2) / 100]
σ = sqrt(0.0016) = 0.04
z = (p - P) / σ = (.73 - .80)/0.04 = -1.75
where P is the hypothesized value of population proportion in
the null hypothesis, p is the sample proportion,
and n is the sample size.
Since we have a
two-tailed test, the P-value is the probability that the
z-score is less than -1.75 or greater than 1.75.
We use the
Normal Distribution Calculator
to find P(z < -1.75) = 0.04, and
P(z > 1.75) = 0.04. Thus, the P-value = 0.04 + 0.04 = 0.08.
- Interpret results. Since the P-value (0.08) is
greater than the significance level (0.05), we cannot reject the
null hypothesis.
Note: If you use this approach on an exam, you may also want to mention
why this approach is appropriate. Specifically, the approach is
appropriate because the sampling method was simple random sampling,
the sample included at least 10 successes and 10 failures, and
the population size was at least 10 times the sample size.
Problem 2: One-Tailed Test
Suppose the previous example is stated a little bit differently. Suppose
the CEO claims that at least 80 percent of the company's
1,000,000 customers are very satisfied. Again, 100 customers are
surveyed using simple random sampling. The result: 73
percent are very satisfied. Based on these results, should we
accept or reject the CEO's hypothesis? Assume a significance
level of 0.05.
Solution: The solution to this problem takes four steps:
(1) state the hypotheses, (2) formulate an analysis plan,
(3) analyze sample data, and (4) interpret results.
We work through those steps below:
- State the hypotheses. The first step is to
state the null hypothesis and an alternative hypothesis.
Null hypothesis: P >= 0.80
Alternative hypothesis: P < 0.80
Note that these hypotheses constitute a one-tailed test.
The null hypothesis will be rejected only if the sample
proportion is too small.
- Formulate an analysis plan. For this analysis,
the significance level is 0.05. The test method,
shown in the next section, is a
one-sample z-test.
- Analyze sample data. Using sample data, we
calculate the standard deviation (σ) and compute the z-score
test statistic (z).
σ = sqrt[ P * ( 1 - P ) / n ] = sqrt [(0.8 * 0.2) / 100]
σ = sqrt(0.0016) = 0.04
z = (p - P) / σ = (.73 - .80)/0.04 = -1.75
where P is the hypothesized value of population proportion in
the null hypothesis, p is the sample proportion,
and n is the sample size.
Since we have a
one-tailed test, the P-value is the probability that the
z-score is less than -1.75.
We use the
Normal Distribution Calculator
to find P(z < -1.75) = 0.04. Thus, the P-value = 0.04.
- Interpret results. Since the P-value (0.04) is
less than the significance level (0.05), we cannot accept the
null hypothesis.
Note: If you use this approach on an exam, you may also want to mention
why this approach is appropriate. Specifically, the approach is
appropriate because the sampling method was simple random sampling,
the sample included at least 10 successes and 10 failures, and
the population size was at least 10 times the sample size.
