AP Statistics Unit 6: Inference for Proportions — Free Review Games.

95% confidence means the method produces intervals that capture the truth 95% of the time.

ME = \(z^*\) times the standard error of \(\hat{p}\).

H0 usually claims no effect, no difference, or a specific parameter value.

A small p-value means the observed result is unlikely under H0, providing evidence against it.

The z* for 95% confidence is 1.96.

CI uses p-hat; hypothesis tests use the null value p0.

When the p-value is below the significance level, we reject the null hypothesis.

Type I error is a false positive: concluding there is an effect when there isn't.

Type II error is a false negative: missing a real effect.

Power = 1 - beta = probability of correctly rejecting a false H0.

Under H0: p1=p2, so we estimate the common proportion with the pooled p-hat.

Larger n reduces standard error, which narrows the confidence interval.

Higher confidence requires a wider interval to maintain the coverage guarantee.

Randomness, large counts, and the independence (10%) condition must all be checked.

p-hat is the midpoint of the interval: (0.42+0.58)/2 = 0.50.

\(\hat{p}\) is the point estimate calculated from sample data, specifically the number of successes divided by the sample size \(n\). The choice "The population proportion of successes" is wrong because that parameter, denoted \(p\), is unknown and is exactly what we are trying to estimate. A confidence interval always centers on a sample statistic to estimate an unknown population parameter.

A higher confidence level requires a larger critical value \(z^*\), which increases the margin of error and therefore widens the interval. The option "The interval becomes narrower" is incorrect because greater confidence requires capturing a broader range of plausible values, not a smaller one. Students should remember the tradeoff: higher confidence always costs precision when sample size is fixed.

Random sampling minimizes systematic bias, making it reasonable to generalize results from the sample to the broader population. The option claiming it "guarantees the sample proportion equals the population proportion" is false since sampling always involves variability, no matter how random the selection. The core principle is that randomness supports generalizability, not certainty about exact values.

The 10% condition checks that the sample is no more than 10% of the population, which keeps trials approximately independent when sampling without replacement. The option "To ensure the sample size is large enough for a Normal approximation" describes the success/failure condition instead, a separate requirement. Students must keep these two conditions distinct: 10% condition addresses independence, success/failure addresses Normality.

For a confidence interval, since there is no hypothesized value, the sample proportion \(\hat{p}\) is used in place of \(p\) to check that both expected successes and failures are at least 10. The choice using \(np_0\) is incorrect for intervals because \(p_0\) only exists in hypothesis testing where a claimed value is being tested. Remembering to swap \(\hat{p}\) for \(p_0\) depending on interval versus test is a common exam distinction.

A two-sided alternative states that the true proportion differs from the hypothesized value in either direction, symbolized with \(\neq\). The option \(H_a: p > p_0\) represents a one-sided test looking only for an increase, which is a different hypothesis structure. Correctly identifying one-sided versus two-sided alternatives is essential because it affects how the p-value is calculated.

The significance level \(\alpha\) is a threshold set before the test that defines the probability of committing a Type I error, rejecting a true null hypothesis. The option "The probability that \(H_0\) is true" is incorrect because \(\alpha\) is a decision criterion chosen by the researcher, not a probability calculated from the data about \(H_0\) itself. Students should treat \(\alpha\) as a pre-set risk tolerance for false rejections.

The p-value quantifies how surprising the observed sample result would be under the assumption that the null hypothesis is correct, based on the sampling distribution. The option "The probability that \(H_0\) is false" is wrong because p-values are calculated assuming \(H_0\) is true and never directly measure the probability of any hypothesis being true or false. Keeping this conditional interpretation in mind prevents a very common AP scoring error.

Margin of error depends on \(SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\), and since \(n\) is in the denominator, a smaller \(n\) produces a larger standard error and thus a larger margin of error. The option "It stays the same" ignores the inverse relationship between sample size and precision built into the formula. A key exam takeaway is that smaller samples yield less precise, wider intervals.

The z-statistic standardizes the distance between \(\hat{p}\) and \(p_0\) by dividing by the standard error, expressing that distance in standard error units. The option describing "the exact difference" is incorrect because it omits the standardization step that makes the statistic comparable across different sample sizes and proportions. Recognizing the z-statistic as a standardized distance helps students correctly interpret its magnitude.

The null hypothesis for comparing two proportions states there is no difference between the true population proportions, written as \(p_1 = p_2\) or equivalently \(p_1 - p_2 = 0\). The option using \(\hat{p}_1 = \hat{p}_2\) is incorrect because hypotheses are always statements about population parameters, not sample statistics. Students must remember that hypotheses reference unknown parameters, never the observed data itself.

A confidence interval is constructed to capture a range of reasonable values for the unknown population parameter \(p\), based on the observed sample. The option "A range of plausible values for the sample proportion" is wrong because \(\hat{p}\) is already known exactly from the data, so there is nothing uncertain about it to estimate. The central purpose of any confidence interval is inference about an unknown parameter, not the already-known statistic.

For a two-proportion confidence interval, each sample's own proportion is used separately because there is no assumption of equal population proportions, unlike in hypothesis testing. The option with a pooled \(\hat{p}\) under a single square root is the formula used for the hypothesis test statistic, not the interval, because the test assumes \(p_1 = p_2\) under \(H_0\). Distinguishing pooled versus unpooled standard errors based on interval versus test context is a frequently tested nuance.

Because the null hypothesis in a two-proportion test assumes the two population proportions are equal, pooling the data to estimate that single common proportion is statistically justified under \(H_0\). The option about equal sample sizes is incorrect because pooling works with unequal \(n_1\) and \(n_2\) as long as the equality assumption from \(H_0\) holds. Students should link the use of pooling directly to the assumption embedded in the null hypothesis, not to sample size equality.

Solving the margin of error formula \(ME = z^*\sqrt{\frac{p(1-p)}{n}}\) for \(n\) gives \(n = \left(\frac{z^*}{ME}\right)^2 p(1-p)\), which matches the first option exactly. The option with the fraction inverted, \(\left(\frac{0.03}{1.96}\right)^2\), would produce a sample size far too small because it incorrectly reverses the relationship between margin of error and required sample size. Students should practice algebraically solving the ME formula for \(n\) since sample size determination is a common free-response task.

Since the p-value of \(0.032\) is less than \(\alpha = 0.05\), the observed result is unlikely enough under \(H_0\) to justify rejecting it in favor of \(H_a\). The option "Accept \(H_0\) as true" is incorrect because hypothesis tests never prove a null hypothesis true, they only fail to find sufficient evidence against it. A key exam rule is that small p-values relative to \(\alpha\) lead to rejecting \(H_0\), never to proving \(H_a\) with certainty.

The test statistic is \(z = \frac{\hat{p}-p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} = \frac{0.6-0.5}{\sqrt{\frac{0.5(0.5)}{150}}} \approx \frac{0.1}{0.0408} \approx 2.45\), matching the first option. The option \(z \approx 0.60\) mistakenly treats the numerator alone as the final answer without dividing by the standard error. Always remember that the z-statistic standardizes the observed difference by the standard error under \(H_0\), using \(p_0\) rather than \(\hat{p}\) in that denominator.

Since standard error involves \(\sqrt{n}\) in the denominator, quadrupling \(n\) divides the standard error, and therefore the margin of error, by \(\sqrt{4}=2\). The option stating it is "cut to one-fourth" incorrectly applies the scaling factor directly to \(n\) rather than to its square root. Students should remember that precision improves with the square root of sample size, not proportionally with sample size itself.

In this scenario \(np_0 = 40 \times 0.02 = 0.8\), which is far below the required threshold of 10, so the Normal approximation used for the sampling distribution is not valid. The option describing a random sample of 200 voters testing a 45% claim is fine because \(np_0\) and \(n(1-p_0)\) would both comfortably exceed 10. Always verify the success/failure condition numerically before trusting the z-test results, since violating it invalidates the Normal-based p-value.

Because the claim specifically states the proportion is greater than 70%, the alternative hypothesis should be one-sided and directional, \(p > 0.70\), while the null retains the equality value being tested against. The option with \(H_a: p \neq 0.70\) is incorrect because it tests for a difference in either direction rather than specifically testing the claim of an increase. Students should always translate directional claims like "more than" or "less than" into one-sided alternative hypotheses.

Since the value 0, which represents no difference between proportions, falls outside the interval \((0.02, 0.18)\), this corresponds to rejecting \(H_0\) at the equivalent significance level. The option "Fail to reject \(H_0\), because 0 is not contained in the interval" contradicts the correct logical link between confidence intervals and two-sided tests, since excluding the null value should lead to rejection, not failure to reject. This CI-to-test correspondence is a powerful shortcut: whenever a confidence interval excludes the null value, the corresponding two-sided test at the matching significance level rejects \(H_0\).

A Type I error occurs when a true null hypothesis is incorrectly rejected, which here means concluding the patient is sick despite actually being healthy. The option describing missing an actual disease is a Type II error, since it involves failing to reject a false null hypothesis. Framing errors within a real context like this helps clarify which error is more costly in a given situation, a skill tested on free-response items.

Increasing the sample size reduces the standard error, which sharpens the sampling distribution and makes it easier to detect a true effect, thereby increasing power. The option about moving the true parameter closer to \(p_0\) actually decreases power, because a smaller true difference from the null value is harder to distinguish statistically. Students should recognize the primary levers of power: larger sample size, larger effect size, and higher \(\alpha\) all tend to increase power.

A two-proportion z-test is specifically designed to determine whether an observed difference between two sample proportions reflects a genuine difference in the underlying population proportions, or could plausibly be due to chance. The option about sample proportions being "numerically identical" misunderstands the test's purpose, since sample proportions almost never match exactly even when the populations are equal. The core goal of any two-sample test is comparing population parameters using evidence from independent samples.

The correct interpretation states confidence about the fixed but unknown population parameter, based on the method's long-run capture rate across repeated samples. The option claiming "90% probability that \(\hat{p}\) falls between 0.45 and 0.55" is wrong because \(\hat{p}\) is a known, already-calculated statistic from this specific sample, not a random variable requiring a probability statement. Correct CI interpretation always references the unknown parameter, never the already-observed sample statistic or individual population members.

Hypothesis testing operates under the assumption that \(H_0\) is true until evidence suggests otherwise, so the standard error uses the hypothesized value \(p_0\) to model the sampling distribution consistent with that assumption. The option claiming \(p_0\) "always produces a smaller standard error" is false because the relative size depends on the specific values of \(p_0\) and \(\hat{p}\) in each problem. This distinction between using \(p_0\) for tests and \(\hat{p}\) for intervals reflects the different purposes of testing an assumed value versus estimating an unknown one.

With very large sample sizes, even small differences from a hypothesized value can produce statistically significant results, so significance alone does not confirm the effect is practically meaningful. The option claiming a small p-value "guarantees the effect is large" is incorrect because p-values reflect the strength of evidence against \(H_0\), not the magnitude of the effect itself. Exam-takers should evaluate both the p-value and the size of the estimated effect, such as a confidence interval, when judging real-world importance.

The success/failure condition for a two-proportion test requires checking that expected successes and failures in each group, computed using the pooled proportion under \(H_0\), are all at least 10 to justify the Normal approximation. The option requiring \(n_1 = n_2\) is incorrect because the two-proportion z-test formula is fully valid with unequal sample sizes, as long as other conditions are met. Independence between the two samples, along with adequate expected counts, are the key structural requirements for this test.

A larger sample size reduces the standard error, which directly shrinks the margin of error and narrows the confidence interval while confidence level remains fixed. The option stating the interval "becomes wider" reverses the true relationship, since more data generally increases precision rather than uncertainty. This tradeoff between sample size and interval width is central to designing efficient studies in statistics.

Since \(0.55\) falls outside the interval \((0.61, 0.79)\), a two-sided test of \(H_0: p=0.55\) at \(\alpha=0.05\) would be rejected, consistent with the CI-to-test correspondence rule. The option \(p_0=0.65\) lies within the interval, meaning a test of that hypothesized value would fail to reject \(H_0\) rather than reject it. This linkage lets students quickly evaluate multiple hypothesized values just by checking whether each falls inside or outside a previously calculated confidence interval.

Because margin of error is proportional to \(\frac{1}{\sqrt{n}}\), halving the margin of error requires increasing \(n\) by a factor of \(2^2=4\) to compensate for the square root relationship. The option suggesting the sample size doubles mistakenly applies a linear rather than a square relationship between \(n\) and margin of error. This square-root tradeoff between precision and sample size frequently appears in both multiple-choice and free-response sample-size problems.

Failing to reject \(H_0\) despite a sizable observed difference often indicates that the sample sizes were too small, producing large standard errors that limit the test's power to detect real effects. The option stating the null hypothesis "was proven true" is incorrect because failing to reject never proves a null hypothesis, it only reflects insufficient evidence against it given the data collected. Recognizing that small samples can mask real differences reinforces why researchers must carefully plan adequate sample sizes before testing.

The pooled proportion combines total successes over total sample size: \(\hat{p}_c = \frac{80(0.30)+100(0.45)}{80+100} = \frac{24+45}{180} = \frac{69}{180} \approx 0.383\), matching the first option. The option \(\hat{p}_c \approx 0.375\) results from simply averaging the two sample proportions equally rather than weighting by their respective sample sizes. Weighting by sample size is essential because larger samples should contribute proportionally more to the combined estimate under \(H_0\).

A two-sided p-value accounts for extreme values in both tails of the sampling distribution, so it is calculated as twice the one-sided tail probability corresponding to the same z-statistic. The option "remain identical" ignores that the two-sided alternative considers deviations in either direction, not just the one specified in the one-sided test. Students should remember this doubling relationship when converting between one-sided and two-sided p-values for the same test statistic.

Surveying only rally attendees creates a non-random, biased sample that likely overrepresents supporters of the rally's cause, undermining the validity of generalizing results to the entire city regardless of sample size. The option claiming large sample size "always eliminates concerns" is a common misconception, since a large biased sample simply produces a very precise but inaccurate estimate. This illustrates that random sampling, not just sample size, is essential for valid statistical inference.

Power increases as the true parameter value moves farther from the hypothesized value, because larger true differences are easier to detect against the natural variability of the sampling distribution. The option stating power is "unaffected by the distance" contradicts the fundamental relationship between effect size and the ability to detect a real difference. Effect size, along with sample size and \(\alpha\), is one of the three main factors that determine the power of a test.

The sample proportion is the midpoint of the interval, \(\hat{p} = \frac{0.40+0.50}{2} = 0.45\), and checking \(n\hat{p} = 400(0.45) = 180\) and \(n(1-\hat{p}) = 220\) confirms both comfortably exceed 10, satisfying the success/failure condition. The option claiming the 10% condition is "automatically satisfied" is incorrect because that condition depends on knowing the population size, which is not given by the confidence interval alone. Extracting \(\hat{p}\) from the interval midpoint is a useful technique for verifying conditions retroactively on exam problems.

Because the standard error of the difference combines variability from both samples, a very small \(n_2\) contributes a disproportionately large term to the standard error, limiting the precision and thus the power of the test even though \(n_1\) is large. The option claiming power is "determined solely by the larger sample size" ignores that the smaller sample's variability still weighs heavily in the combined standard error formula. Balanced sample sizes generally produce more powerful two-sample tests than highly unequal designs with the same total observations.

Since \(z^*=1.645\) corresponds to 90% confidence rather than the correct 95% value of approximately 1.96, using the smaller critical value produces a smaller margin of error, resulting in an interval that is too narrow for the stated confidence level. The option claiming the interval would be "too wide" reverses the actual effect, since a smaller \(z^*\) always shrinks the margin of error, not expands it. Recognizing the correct critical value for a stated confidence level is essential, since using the wrong one systematically misrepresents the true precision of the estimate.

Standard error decreases as \(n\) increases, so Survey B, with its much larger sample size, will have a smaller standard error and therefore a narrower confidence interval despite sharing the same \(\hat{p}\) as Survey A. The option claiming smaller samples yield "more precise estimates" reverses the true relationship, since smaller samples produce larger standard errors and wider, less precise intervals. This example highlights that identical point estimates can still lead to very different levels of precision depending on sample size.

With \(np_0 = 20(0.05)=1\), which is far below the required threshold of 10, the sampling distribution of \(\hat{p}\) is not well approximated by a Normal distribution, making the calculated p-value untrustworthy. The option claiming a sample size of 20 is "always too small for any hypothesis test" is incorrect because adequacy depends on the specific value of \(p_0\), not on sample size in isolation. This reinforces that checking the success/failure condition numerically is necessary before trusting any z-test result, regardless of how straightforward the calculation appears.

Because a 95% confidence level uses a larger critical value than a 90% level, the resulting interval is wider, but this wider range corresponds to a higher long-run success rate at capturing the true population proportion. The option claiming the 90% interval is "wider but less likely to capture" the parameter reverses both relationships, since lower confidence levels actually produce narrower, less reliable intervals. This tradeoff between width and reliability is fundamental to understanding what confidence level actually controls in interval estimation.

A test statistic close to zero, such as \(z=-0.45\), indicates that the observed difference between sample proportions is small in standard error units, producing a large p-value and weak evidence against \(H_0\). The option stating the negative value means "\(p_1\) must be less than \(p_2\) in the population" incorrectly generalizes a sample-based statistic to a definitive claim about population parameters, which the test cannot establish with such weak evidence. Students should interpret the magnitude of \(z\), not just its sign, since values near zero always indicate results consistent with the null hypothesis.

Because the researcher specifically wants to test whether the new design increases the rate compared to the old design, a one-sided two-proportion z-test with a directional alternative $p_{new} > p_{old}$ directly matches the research question. The option using a two-sided alternative would be appropriate only if the researcher were open to detecting a difference in either direction, which does not match the stated goal of testing for an increase specifically. Matching the alternative hypothesis direction to the specific wording of the research question is critical for correctly setting up two-sample proportion tests.

Although the tiny p-value indicates strong statistical evidence that a true difference exists, the confidence interval reveals the actual difference is quite small, between 0.001 and 0.004, which may have limited real-world impact depending on context. The option claiming the small p-value "guarantees the difference is large" repeats a common misconception, since statistical significance reflects confidence that an effect exists, not its practical size. Distinguishing statistical significance from practical significance is essential, especially with large samples that can detect trivially small true differences.