Inference for proportions practice questions (AP Statistics)

Sample questions

All 2 free AP Statistics questions on this unit, from the free AP Statistics Practice Exam. Each answer is behind its own toggle; the whole paper, its PDF and the full answer key are on the exam page.

1. Question 12 marks · Confidence interval interpretation
   A 95 percent confidence interval for a population mean is computed as $(12.1,\ 15.3)$. Which interpretation is correct?
   • A.
     There is a 95 percent probability that the population mean is between 12.1 and 15.3
   • B.
     95 percent of the data values lie between 12.1 and 15.3
   • C.
     If we repeated the sampling many times, about 95 percent of the intervals built this way would capture the population mean
   • D.
     95 percent of sample means will fall between 12.1 and 15.3
   • E.
     The sample mean has a 95 percent chance of being in the interval
   Show answerFrom the free AP Statistics Practice Exam →
2. Question 28 marks · One-proportion z-test
   Significance test. A city official claims that 60 percent of voters support a new transit measure. A polling firm takes a random sample of 600 voters from the city's 90,000 registered voters and finds 390 who support the measure. Does the sample provide convincing evidence, at the 5 percent significance level, that MORE than 60 percent of all the city's voters support the measure?
   (a)

   Define the parameter of interest and state the hypotheses.

   [2]
   (b)

   Verify that the conditions for a one-proportion z-test are met.

   [2]
   (c)

   Compute the test statistic and the p-value.

   [2]
   (d)

   State a conclusion in the context of the problem.

   [2]
   Show answerFrom the free AP Statistics Practice Exam →

Key terms

The definitions and formulas AP Statistics expects for inference for proportions, typeset as they appear on the paper.

Confidence interval structure

Point estimate plus or minus a margin of error, where the margin is a critical value times the standard error. Interpret the interval ("we are 95 percent confident the interval captures the true proportion of...") in context.

$$\text{statistic} \pm (\text{critical value}) \times (\text{standard error})$$

One-proportion z-interval

Conditions: random sample, 10 percent condition, and at least 10 successes and 10 failures in the sample.

$$\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$$

Confidence level interpretation

The confidence level is the long-run capture rate of the method: if many samples were taken and an interval built from each, about 95 percent of the intervals would contain the parameter. It is not the probability that this interval contains it.

Sample size for a given margin of error

Solve the margin-of-error formula for $n$, using a guessed proportion $p^*$ or the conservative $p^* = 0.5$, and round up.

$$n = \left(\frac{z^*}{ME}\right)^2 p^*(1 - p^*)$$

One-proportion z-test

State hypotheses about $p$, check conditions using the hypothesised $p_0$ for the large counts condition, compute the test statistic and p-value, and conclude in context.

$$z = \frac{\hat{p} - p_0}{\sqrt{\dfrac{p_0(1 - p_0)}{n}}}$$

p-value and conclusion

The probability, assuming $H_0$ is true, of a statistic at least as extreme as the one observed. Compare it with the significance level: small p-value, reject $H_0$; otherwise fail to reject. Never "accept" $H_0$.

Type I and Type II errors; power

Type I: rejecting a true $H_0$ (probability $\alpha$). Type II: failing to reject a false $H_0$ (probability $\beta$). Power is $1 - \beta$ and rises with sample size, a larger $\alpha$, or a true value further from $H_0$.

Two-proportion z-test

Compares proportions from two independent samples using a pooled estimate of the common proportion under $H_0$.

$$z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\hat{p}_c(1 - \hat{p}_c)\left(\dfrac{1}{n_1} + \dfrac{1}{n_2}\right)}}, \qquad \hat{p}_c = \frac{X_1 + X_2}{n_1 + n_2}$$

Frequently asked questions

Other AP Statistics topics

• Exploring one-variable data
• Exploring two-variable data
• Collecting data
• Probability, random variables and distributions
• Sampling distributions
• Inference for means
• Chi-square tests
• Inference for slopes

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