Topic 6.6 Concluding a Test for a Population Proportion: compute the standardized z test statistic and P-value for a one-sample proportion test, compare to the significance level, and state a conclusion in context.

What this topic is asking

The College Board (Topic 6.6) wants you to carry out and conclude a one-sample proportion test: compute the standardized z statistic (using the null value $p_0$), find the P-value, compare it to $\alpha$, and state a conclusion in context, completing the test set up in Topic 6.4.

The standardized test statistic

The standard deviation in the denominator uses $p_0$, the hypothesized proportion, because the whole test assumes $H_0$ is true. This is the key contrast with the confidence interval, whose standard error used $\hat{p}$. The numerator is the gap between observed and claimed; dividing by the null standard deviation converts that gap into a z-score on the standard normal scale.

From z to the P-value

The P-value is the area beyond the observed z in the alternative's direction. A two-sided test doubles the one-tail area because "at least as extreme" means equally far in either direction. Match the tail to $H_a$ exactly; using the wrong tail (or forgetting to double) is a frequent and costly slip.

The decision rule and the conclusion

Compare the P-value to the preset $\alpha$. If P-value $\le \alpha$, the data are too surprising to credit $H_0$: reject $H_0$, and there is convincing evidence for $H_a$. If P-value $> \alpha$, the data are consistent with $H_0$: fail to reject $H_0$ (never "accept $H_0$"), and there is not convincing evidence for $H_a$. The conclusion sentence must do three things: state the decision (reject or fail to reject), tie it to the comparison ($P$ versus $\alpha$), and translate it into context ("there is convincing evidence that the proportion of ... is less than ..."). A bare "reject $H_0$" without context loses the final mark.

Tests and intervals agree

A two-sided test at level $\alpha$ and a $C = 1 - \alpha$ confidence interval reach the same verdict: $H_0$ is rejected exactly when $p_0$ falls outside the interval. This duality, foreshadowed in Topic 6.3, is a useful check and a common exam theme. (The two procedures use slightly different standard deviations, $p_0$ for the test versus $\hat{p}$ for the interval, so they can disagree in rare borderline cases, but conceptually they tell the same story about whether $p_0$ is plausible.)

Try this

Q1. Write the test statistic formula and say which proportion goes in the standard deviation. [1 point]

• Cue. $z = \dfrac{\hat{p} - p_0}{\sqrt{p_0(1-p_0)/n}}$; the standard deviation uses the null value $p_0$.

Q2. A one-sided upper test gives $z = 1.20$. Find the P-value. [1 point]

• Cue. $\text{P-value} = P(Z > 1.20) \approx 0.1151$.