A z-test for proportions helps decide whether sample data provide strong evidence about a population proportion. It is used when outcomes fall into two categories, such as success or failure, yes or no, or defective or not defective. The test compares an observed sample proportion to a hypothesized value, or compares two sample proportions to each other.
It matters because many real decisions in science, medicine, business, and polling are based on proportions.
Key Facts
- One-proportion hypotheses: H0: p = p0 and Ha: p < p0, p > p0, or p != p0.
- One-proportion test statistic: z = (p-hat - p0) / sqrt(p0(1 - p0) / n).
- Two-proportion hypotheses: H0: p1 = p2 and Ha: p1 < p2, p1 > p2, or p1 != p2.
- Pooled proportion for a two-proportion z-test: p-hat pooled = (x1 + x2) / (n1 + n2).
- Two-proportion test statistic: z = (p1-hat - p2-hat) / sqrt(p-hat pooled(1 - p-hat pooled)(1/n1 + 1/n2)).
- Decision rule: reject H0 when the p-value is less than or equal to alpha, and fail to reject H0 when the p-value is greater than alpha.
Vocabulary
- Population proportion
- The true fraction p of an entire population that has a certain characteristic.
- Sample proportion
- The observed fraction p-hat = x/n of a sample that has a certain characteristic.
- Null hypothesis
- The default claim H0 that the population proportion equals a stated value or that two population proportions are equal.
- P-value
- The probability, assuming H0 is true, of getting a test statistic as extreme as or more extreme than the observed one.
- Significance level
- The cutoff alpha for deciding when evidence is strong enough to reject the null hypothesis.
Common Mistakes to Avoid
- Using p-hat instead of p0 in the one-proportion standard error is wrong because the null distribution must be built using the proportion claimed by H0.
- Forgetting to pool in a two-proportion z-test is wrong when H0 says p1 = p2 because the test assumes one common proportion under the null hypothesis.
- Using a two-tailed p-value for a one-tailed alternative is wrong because the shaded rejection area must match the direction stated in Ha.
- Running a z-test when expected counts are too small is wrong because the normal approximation may be inaccurate unless counts such as np0 and n(1 - p0) are large enough.
Practice Questions
- 1 A company claims that 60% of customers prefer its new design. In a random sample of 150 customers, 78 prefer the new design. Test H0: p = 0.60 against Ha: p < 0.60 at alpha = 0.05. Find z, estimate the p-value, and state the conclusion.
- 2 Group A has 64 successes out of 200 trials, and Group B has 90 successes out of 250 trials. Test H0: p1 = p2 against Ha: p1 != p2 at alpha = 0.05 using a two-proportion z-test. Compute the pooled proportion, z-statistic, and conclusion.
- 3 A poll about support for a policy uses a random sample, but only 6 people in the sample oppose the policy. Explain why a z-test for a proportion may not be appropriate and what condition is being violated.