Inference for means practice questions (AP Statistics)

Sample questions

The one free AP Statistics question on this unit, from the free AP Statistics Free-Response Practice Questions (FRQ) with Answers. Each answer is behind its own toggle; the whole paper, its PDF and the full answer key are on the exam page.

1. Question 14 marks · Two-sample t-test
   Inference. A teacher wants to know whether spacing study sessions out over several days produces different test scores from cramming the same amount of study into one session. She randomly assigns 24 volunteer students to two groups of 12. The spaced group studies a unit for 30 minutes on each of four days; the massed group studies for two hours on a single day. All 24 students then take the same test. The spaced group has a mean score of 78.5 with a standard deviation of 6.2; the massed group has a mean score of 72.1 with a standard deviation of 7.4. Dotplots of the scores in each group show no strong skewness and no outliers. Do the data provide convincing evidence, at the 5 percent significance level, that the mean test score differs between the two study methods?
   (a)

   State the hypotheses, defining any parameters you use.

   [1]
   (b)

   Identify the appropriate test and verify that the conditions for it are met.

   [1]
   (c)

   Calculate the test statistic and the p-value.

   [1]
   (d)

   State your conclusion in the context of the question.

   [1]
   Show answerFrom the free AP Statistics Free-Response Practice Questions (FRQ) with Answers →

Key terms

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

t-distribution

Used when the population standard deviation is unknown and estimated by $s$. It is bell-shaped with heavier tails than the Normal, indexed by degrees of freedom; with more degrees of freedom it approaches the Normal.

One-sample t-interval

Conditions: random sample, 10 percent condition, and Normal population or large sample ($n \ge 30$) or no strong skew and outliers in a small sample's graph. Degrees of freedom $n - 1$.

$$\bar{x} \pm t^* \frac{s}{\sqrt{n}}, \qquad \text{df} = n - 1$$

One-sample t-test

Tests a claim about a population mean. Report the test statistic, degrees of freedom and p-value, then conclude in context.

$$t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$$

Paired t procedures

When each observation in one sample is matched with one in the other (before and after, twins), analyse the differences as a single sample. Identify pairing from the design before choosing the procedure.

Two-sample t procedures

For two independent samples. The calculator's degrees of freedom are not an integer; the conservative choice is the smaller of $n_1 - 1$ and $n_2 - 1$.

$$t = \frac{(\bar{x}_1 - \bar{x}_2) - 0}{\sqrt{\dfrac{s_1^2}{n_1} + \dfrac{s_2^2}{n_2}}}$$

Robustness

t procedures are robust against non-Normality except with outliers or strong skew in small samples, which is why the Normal condition is checked with a graph when $n$ is small.

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 proportions
• Chi-square tests
• Inference for slopes

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