3.6 Setting Up a Test for a Population Proportion

State hypotheses first

A significance test starts with a null hypothesis (H0), a statement of 'no difference' such as p = p0, and an alternative hypothesis (Ha) stating what you suspect is true (p > p0, p < p0, or p ≠ p0). Hypotheses are always about the population parameter p, never the sample statistic.

Check conditions before testing

Before calculating, verify three conditions: Random (data from random sample/assignment), Independence (10% condition: sample ≤10% of population, or independent trials), and Large Counts (both np0 ≥ 10 and n(1-p0) ≥ 10, using the hypothesized proportion). These ensure the sampling distribution of p̂ is approximately Normal.

Define parameters and pick significance level

Clearly define p in context (e.g., 'p = true proportion of all adults who...'). State the significance level α (often 0.05) before collecting data — this is the threshold for deciding whether evidence is strong enough to reject H0. Setting this up correctly is essential before ever computing a test statistic.

Key terms

Null hypothesis (H0)

A statement that the population proportion equals a specific value, p0; assumed true until evidence suggests otherwise.

Alternative hypothesis (Ha)

A statement claiming p > p0, p < p0, or p ≠ p0; what the researcher suspects is true.

Random condition

Data must come from a random sample or random assignment to support cause-and-effect or generalization to the population.

Large Counts condition (test)

np0 ≥ 10 and n(1-p0) ≥ 10, using the hypothesized proportion, to ensure the sampling distribution of p̂ is approximately Normal.

Significance level (α)

The threshold probability (often 0.05) set before the test to decide whether results are statistically significant.

Try an AP-style question

A researcher wants evidence that MORE than 30% of students skip breakfast. Which hypotheses fit?

1. H₀: p = 0.3; Hₐ: p > 0.3
2. H₀: p = 0.3; Hₐ: p ≠ 0.3
3. H₀: p > 0.3; Hₐ: p = 0.3
4. H₀: p̂ = 0.3; Hₐ: p̂ > 0.3