Inference for Quantitative Data: Means

What Unit 4 covers

Unit 4 is where you do inference for quantitative data, meaning numerical measurements whose center is a mean. It builds directly on the ideas you used for proportions in Unit 3, but the parameter is now a population mean, written $\mu$ (the Greek letter mu), instead of a proportion.

The College Board weights Unit 4 at 10 to 20 percent of the multiple-choice section and allots about 18 class periods. There are 10 topics, split into two halves: one-sample and matched-pairs procedures (topics 4.1 to 4.5), then two-sample procedures (topics 4.6 to 4.10).

The big change from Unit 3 is that you almost never know the population standard deviation $\sigma$ (sigma) for a quantitative variable. You estimate it with the sample standard deviation $s$, and that forces you to use the t-distribution instead of the normal z-distribution. See t-test vs z-test for the full contrast.

The 10 topics in Unit 4

• 4.1 Sampling distributions for sample means. Find the mean and standard deviation of the sampling distribution of the sample mean $\bar{x}$ (x-bar), and check when it is approximately normal.
• 4.2 Constructing a confidence interval for a population mean or mean difference. Describe t-distributions and build a one-sample t-interval, including for matched pairs.
• 4.3 Justifying a claim based on that interval. Interpret the interval and the confidence level, and relate sample size, width, and margin of error.
• 4.4 Setting up a test for a population mean or mean difference. Choose the one-sample t-test, state hypotheses about $\mu$ or the population mean difference $\mu_d$, and check conditions.
• 4.5 Carrying out that test. Compute the t-statistic and p-value, then decide by comparing the p-value to the significance level $\alpha$ (alpha).
• 4.6 Sampling distributions for the difference between two sample means. Find the mean and standard deviation of $\bar{x}_1 - \bar{x}_2$.
• 4.7 Constructing a confidence interval for the difference between two population means. Build a two-sample t-interval.
• 4.8 Justifying a claim based on that interval. Interpret it, and use whether the interval contains 0.
• 4.9 Setting up a test for the difference between two population means. Choose the two-sample t-test and state the hypotheses.
• 4.10 Carrying out that test. Compute the two-sample t-statistic and p-value and conclude in context.

Sampling distributions for sample means

For a population with mean $\mu$ and standard deviation $\sigma$, when the sampled values are independent, the sampling distribution of the sample mean $\bar{x}$ has mean $\mu_{\bar{x}} = \mu$ and standard deviation $\sigma_{\bar{x}} = \sigma/\sqrt{n}$, where $n$ is the sample size.

That standard deviation shrinks as $n$ grows, which is why larger samples give more precise estimates. The central limit theorem tells you the shape. If the population is normal, $\bar{x}$ is normal for any $n$. If the population is not normal, $\bar{x}$ is approximately normal once $n \ge 30$, and a strongly skewed population may need a sample much larger than 30.

You can watch this happen in the sampling distribution interactive.

Why means use the t-distribution

Because you rarely know $\sigma$, you replace it with the sample standard deviation $s$. The quantity $s/\sqrt{n}$ is the standard error of the mean, an estimate of the true standard deviation $\sigma/\sqrt{n}$. Those two ideas trip up many students, so read standard error vs standard deviation.

Using $s$ adds extra uncertainty, so the standardized statistic follows a t-distribution rather than the standard normal. t-distributions are symmetric and bell-shaped with heavier tails than the normal curve. Each one is set by its degrees of freedom (df); for one sample, $df = n - 1$. As df grows, the t-distribution approaches the standard normal, and you look up critical values on the t-table.

One-sample and matched-pairs t procedures

A one-sample t-interval estimates $\mu$ as $\bar{x} \pm t^* \dfrac{s}{\sqrt{n}}$, where $t^*$ is the critical value for the central C percent of a t-distribution with $n - 1$ degrees of freedom. The one-sample t-test uses the statistic

where $\mu_0$ is the value of the mean claimed by the null hypothesis.

Matched pairs, for example a pre-test and post-test on the same students, are handled by taking the difference within each pair to get one sample of differences. You then run the same one-sample procedures on those differences, estimating the population mean difference $\mu_d$. The one-sample t-test calculator runs both the interval and the test.

Two-sample t procedures

When you compare two independent groups, the parameter is the difference between population means $\mu_1 - \mu_2$. The sampling distribution of $\bar{x}_1 - \bar{x}_2$ has standard deviation $\sqrt{\dfrac{\sigma_1^2}{n_1} + \dfrac{\sigma_2^2}{n_2}}$, and its standard error replaces each $\sigma$ with the matching sample standard deviation $s$.

The two-sample t-interval is $(\bar{x}_1 - \bar{x}_2) \pm t^* \sqrt{\dfrac{s_1^2}{n_1} + \dfrac{s_2^2}{n_2}}$, and the two-sample t-test uses

The degrees of freedom fall between $n_1 + n_2 - 2$ and the smaller of $n_1 - 1$ and $n_2 - 1$, and on the exam you find them with technology. If a two-sample interval contains 0, you do not have convincing evidence of a difference. The two-sample t-test calculator handles the arithmetic.

Conditions to check before any t procedure

Every t procedure in Unit 4 asks you to verify conditions before you trust the result, and you should state them explicitly on the free-response section.

• Randomization: the data come from a random sample or a randomized experiment, using two independent random samples for two-sample procedures.
• 10 percent: when sampling without replacement, each sample is at most 10 percent of its population, written $n \le 0.10N$, where $N$ is the population size.
• Sample data (normality): the population is stated to be approximately normal, or $n \ge 30$. If $n < 30$, the sample should be free of strong skewness and outliers. For matched pairs, apply this to the differences.

State your hypotheses in terms of population parameters, not sample statistics, and write the conclusion in context by comparing the p-value to $\alpha$.

How Unit 4 fits the exam

Unit 4 mirrors Unit 3 step for step, so the College Board expects you to keep the procedures straight and pick the right one from the wording of a problem. A question asking whether data give convincing evidence of something is asking for a significance test, not a description.

Free-response question 3 is always a full inference problem, a hypothesis test or a confidence interval, and Unit 4 procedures are fair game there. A graphing calculator with statistical capabilities is expected, and a formula sheet and tables are provided. For the wider picture see the AP Statistics hub and the official AP Statistics course page.