Step 12: Inference for One Proportion

The idea: from a sample to the whole population

Suppose a college has 6,000 students and you want to know what proportion use a revision app. Asking everyone is slow, so you ask a random sample of 400 students, and 248 say yes. Your sample proportion is \( \hat{p} = 248/400 = 0.62 \).

But a different random sample of 400 would give a slightly different answer, perhaps 0.59 or 0.64. So 0.62 is a good guess at the true proportion \( p \), but almost certainly not exactly right. Inference gives two tools for handling this uncertainty:

• A confidence interval answers “what values of \( p \) are plausible?”
• A significance test answers “is a claimed value of \( p \) believable, given my data?”

The conditions you must check

Both procedures rely on the sampling distribution of \( \hat{p} \) being approximately Normal with a standard deviation we can calculate. That is only true if three conditions hold.

1. Random. The data come from a random sample from the population (or a randomised experiment). Without this, you cannot generalise at all.
2. 10% condition. When sampling without replacement, the sample size must be no more than 10% of the population: \( n \le 0.1N \). This makes individual observations close enough to independent for the standard deviation formula to work.
3. Large Counts. The expected numbers of successes and failures must both be at least 10, so the sampling distribution is approximately Normal.
   • For a confidence interval, use the sample: \( n\hat{p} \ge 10 \) and \( n(1-\hat{p}) \ge 10 \).
   • For a significance test, use the claimed value: \( np_0 \ge 10 \) and \( n(1-p_0) \ge 10 \).

One-sample z-interval for a proportion

A confidence interval has the shape point estimate ± margin of error. The margin of error is a critical value times the standard error of \( \hat{p} \).

The critical value \( z^* \) depends on the confidence level:

Higher confidence means a larger \( z^* \) and so a wider interval. A larger sample size means a smaller standard error and so a narrower interval.

What the interval and the confidence level mean

These two interpretations are tested every year, so learn the wording.

• Interpreting the interval: “We are 95% confident that the interval from 0.572 to 0.668 captures the true proportion of all students at this college who use a revision app.”
• Interpreting the confidence level: “If we took many random samples of the same size and built a 95% confidence interval from each, about 95% of those intervals would capture the true proportion.”

The confidence level describes the method, not one particular interval. Once you have calculated an interval, it either contains \( p \) or it does not; you just do not know which.

Worked example: State, Plan, Do, Conclude

Choosing a sample size

Before collecting data, you can choose \( n \) so the margin of error is no bigger than a target \( ME \). Solve

Here \( p^* \) is a guess for the proportion from a previous study. If you have no guess, use \( p^* = 0.5 \): it gives the largest possible value of \( p^*(1-p^*) = 0.25 \), so the sample is guaranteed to be big enough. Always round up to the next whole number.

One-sample z-test for a proportion

Now suppose someone makes a claim: “25% of students at our school walk to school.” You doubt it and think the true proportion is higher. A significance test checks whether your data give convincing evidence against the claim.

• The null hypothesis \( H_0: p = p_0 \) is the claim being tested (here \( p_0 = 0.25 \)).
• The alternative hypothesis \( H_a \) is what you suspect: \( p > p_0 \), \( p < p_0 \) (one-sided) or \( p \ne p_0 \) (two-sided). Decide it before looking at the data.

The test statistic measures how many standard deviations \( \hat{p} \) is from \( p_0 \), assuming \( H_0 \) is true:

The p-value is the probability, assuming \( H_0 \) is true, of getting a sample proportion at least as extreme as the one observed (in the direction of \( H_a \)). For a two-sided test, double the one-tail area.

Compare the p-value with the significance level \( \alpha \) (usually 0.05). If p-value < \( \alpha \), reject \( H_0 \): you have convincing evidence for \( H_a \). If p-value ≥ \( \alpha \), fail to reject \( H_0 \): you do not have convincing evidence for \( H_a \). Never say you “accept \( H_0 \)” or “prove” anything.

Confidence intervals and two-sided tests agree: a two-sided test at \( \alpha = 0.05 \) rejects \( H_0: p = p_0 \) roughly when \( p_0 \) lies outside the 95% confidence interval (the match is not exact for proportions, because the interval uses \( \hat{p} \) and the test uses \( p_0 \) in the standard deviation).

Practise: 6 questions

Try each one on paper first, then check the worked answer.

Check yourself

The short version

• Check Random, 10% (\( n \le 0.1N \)) and Large Counts before any inference: use \( \hat{p} \) for intervals and \( p_0 \) for tests.
• Interval: \( \hat{p} \pm z^*\sqrt{\hat{p}(1-\hat{p})/n} \), with \( z^* \) = 1.645, 1.960 or 2.576 for 90%, 95% or 99%.
• “95% confident” describes the method: about 95% of such intervals capture \( p \).
• Sample size: \( n \ge (z^*/ME)^2 p^*(1-p^*) \), using \( p^* = 0.5 \) if unsure, and always round up.
• Test: \( z = (\hat{p}-p_0)/\sqrt{p_0(1-p_0)/n} \); compare the p-value with \( \alpha \) and conclude in context with State, Plan, Do, Conclude.