A national survey claims that 80% of all Australians like Vegemite. You collect a random sample of 100 Australians, and 72 of them say they like Vegemite. You want to test whether the true proportion of Australians who like Vegemite is different from 80%.

Which statistical test should you use out of the following?

Question

A national survey claims that 80% of all Australians like Vegemite. You collect a random sample of 100 Australians, and 72 of them say they like Vegemite. You want to test whether the true proportion of Australians who like Vegemite is different from 80%.

Which statistical test should you use out of the following?

BlackTom AI · Accepted Answer

Since we're comparing a sample proportion to a hypothesized population proportion, the appropriate method is a one-sample z-test for a proportion.

1. Relevant knowledge points and concepts
- Hypothesis testing for a population proportion uses the sampling distribution of the sample proportion p̂, which is approximately normal for large n by the Central Limit Theorem.
- Test statistic: z = (p̂ − p0) / sqrt[p0(1 − p0)/n], where p0 is the claimed proportion under H0.
- Validity condition for the normal approximation: n p0 ≥ 10 and n(1 − p0) ≥ 10.
- Two-tailed test if we test “different from” (≠).

2. Step-by-step reasoning and calculations
- Sample size n = 100, observed successes x = 72, so p̂ = 72/100 = 0.72.
- Null hypothesis H0: p = p0 = 0.80. Alternative Ha: p ≠ 0.80 (two-tailed).
- Check approximation conditions: n p0 = 100 × 0.80 = 80 ≥ 10 and n(1 − p0) = 100 × 0.20 = 20 ≥ 10. Conditions satisfied.
- Compute standard error under H0: SE = sqrt[p0(1 − p0)/n] = sqrt[0.8 × 0.2 / 100] = sqrt[0.16 / 100] = sqrt[0.0016] = 0.04.
- Compute z-statistic: z = (0.72 − 0.80) / 0.04 = (−0.08) / 0.04 = −2.00.
- Two-tailed p-value = 2 × P(Z ≤ −2.00) ≈ 2 × 0.0228 = 0.0456.

3. Why the one-sample z-test is correct
- The question concerns a single sample proportion compared to a claimed population proportion. The sample size is large enough for the normal approximation, so the one-sample z-test for a proportion is the standard, straightforward test. The computed z and p-value show how the test would be carried out.

4. Why other tests are incorrect
- One-sample t-test: used for testing a population mean when σ is unknown, not for proportions.
- Two-sample z-test: used when comparing two independent proportions, not one proportion to a hypothesized value.
- Chi-square tests: used for categorical contingency tables or goodness-of-fit across multiple categories; overkill here for a single binary proportion (though an exact binomial test is an alternative).
- Exact binomial test: a valid alternative for small n or when normal approximation fails, but unnecessary here because n is large and conditions are satisfied.

Conclusion: The appropriate choice is the one-sample z-test for a proportion, which matches the provided answer.

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