Sampling Distributions for Sample Means
Calculate mu_x-bar = mu and sigma_x-bar = sigma / sqrt(n), verify conditions, and apply the Central Limit Theorem to determine the shape of the sampling distribution.
AP Statistics Unit 4 Review: Means
- 10–20% of AP multiple choice
- 10 topics
- AP exam review verified for 2027
Unit 4 teaches you to use sample means and the t-distribution to estimate and test claims about population means. The three scenarios are: one population mean, a matched-pairs mean difference, and the difference between two independent population means. Each scenario requires checking the same three conditions (random, 10%, normal/large sample) and then applying the appropriate t-interval or t-test formula.
Fiveable's AP Statistics Unit 4 review covers 10 topics, with guided practice questions for the unit.
AP Statistics unit 4 topics
Constructing a Confidence Interval for a Population Mean or Population Mean Difference
Build a one-sample t-interval using x-bar plus or minus t*(s / sqrt(n)) with df = n - 1; apply the same procedure to matched-pairs differences.
Justifying a Claim Based on a Confidence Interval for a Population Mean or Population Mean Difference
Interpret a t-interval in context, identify the relationships among confidence level, sample size, and margin of error, and use the interval to support or refute a claim.
Setting Up a Test for a Population Mean or Population Mean Difference
State H_0 and H_a in terms of mu or mu_d, identify the one-sample t-test or matched-pairs t-test, and verify the three conditions before calculating anything.
Carrying Out a Test for a Population Mean or Population Mean Difference
Calculate t = (x-bar - mu_0) / (s / sqrt(n)) with df = n - 1, find the p-value, compare to alpha, and write a conclusion in context referencing the population.
Sampling Distributions for the Difference Between Two Sample Means
Find the mean and standard deviation of the sampling distribution of x-bar_1 minus x-bar_2 using the addition rule for variances, and verify normality conditions for each group.
Constructing a Confidence Interval for the Difference Between Two Population Means
Apply the two-sample t-interval formula (x-bar_1 - x-bar_2) plus or minus t* times sqrt(s_1^2/n_1 + s_2^2/n_2), with df from technology, to estimate mu_1 - mu_2.
Justifying a Claim Based on a Confidence Interval for the Difference Between Two Population Means
Interpret the two-sample t-interval in context; determine whether 0 is inside or outside the interval to decide if there is evidence of a difference between the two population means.
Setting Up a Test for the Difference Between Two Population Means
State H_0: mu_1 = mu_2 and the appropriate H_a, identify the two-sample t-test, and verify randomization, 10%, and normal/large sample conditions for both groups.
Carrying Out a Test for the Difference Between Two Population Means
Calculate t = (x-bar_1 - x-bar_2) / sqrt(s_1^2/n_1 + s_2^2/n_2) with df from technology, find the p-value, and state a conclusion in context referencing both populations.
Unit 4 review notes
4.1
Sampling Distribution of the Sample Mean
The sampling distribution of x-bar describes how sample means vary across repeated random samples. Its center equals the population mean and its spread shrinks as sample size grows. Knowing its shape lets you calculate probabilities about x-bar before doing formal inference.
- Mean of x-bar: mu_x-bar = mu; the sampling distribution is centered at the population mean.
- Standard deviation of x-bar: sigma_x-bar = sigma / sqrt(n); also called the standard error when sigma is estimated by s.
- Normal population: If the population is normal, x-bar is exactly normal for any sample size.
- Central Limit Theorem: If the population is not normal, x-bar is approximately normal when n is at least 30 (more if the population is extremely skewed).
- 10% condition: Require n to be at most 10% of the population so that sampled values are approximately independent.
Given mu = 50, sigma = 10, and n = 25, what are the mean and standard deviation of the sampling distribution of x-bar, and what shape does it have if the population is normal?
4.2
One-Sample t-Interval for a Population Mean or Mean Difference
When sigma is unknown, estimate mu with a one-sample t-interval. For matched pairs, compute differences within each pair first, then treat those differences as a single sample. Interpret the interval in context and use it to justify or refute a claim about the population mean.
- t-interval formula: x-bar plus or minus t* times (s / sqrt(n)), where df = n - 1 and t* is the critical value for the chosen confidence level.
- Standard error: SE_x-bar = s / sqrt(n); replaces sigma / sqrt(n) because sigma is unknown.
- Matched pairs: Subtract within each pair to get one sample of differences d; apply the one-sample t-interval to d-bar with df = n - 1.
- Interpreting the interval: We are C% confident the interval (a, b) captures the true population mean (or mean difference) of [variable] for [population].
- Width and sample size: Increasing n decreases the margin of error; width is approximately proportional to 1 / sqrt(n). Increasing the confidence level increases the critical value and widens the interval.
A researcher records the difference in reaction times (before minus after) for 20 subjects. Conditions are met. The mean difference is 15 ms and s = 8 ms. Construct and interpret a 95% t-interval for the population mean difference.
4.4
One-Sample t-Test for a Population Mean or Mean Difference
A one-sample t-test evaluates whether sample data provide convincing evidence against a null hypothesis about mu. For matched pairs, the test runs on the sample of differences. After calculating the test statistic and p-value, compare the p-value to alpha and state a conclusion in context.
- Hypotheses: H_0: mu = mu_0; H_a: mu < mu_0, mu > mu_0, or mu not equal to mu_0. For matched pairs: H_0: mu_d = 0.
- Test statistic: t = (x-bar - mu_0) / (s / sqrt(n)), with df = n - 1.
- p-value interpretation: Assuming H_0 is true, the p-value is the probability of observing a test statistic as extreme as or more extreme than the one calculated, in the direction of H_a.
- Conclusion language: If p-value is less than or equal to alpha, reject H_0 and state there is sufficient evidence that [H_a in context]. If p-value is greater than alpha, fail to reject H_0 and state there is insufficient evidence.
- Conditions: Random, 10% (for observational data), and normal/large sample; for matched pairs, check conditions on the differences.
A sample of 36 students has x-bar = 72 and s = 12. Test H_0: mu = 75 vs. H_a: mu < 75 at alpha = 0.05. Calculate t, find the p-value, and write a conclusion in context.
4.6
Sampling Distribution of the Difference Between Two Sample Means
When comparing two independent groups, the relevant statistic is x-bar_1 minus x-bar_2. Its sampling distribution has a predictable center and spread, and it is approximately normal under the same conditions used for one-sample means applied to each group.
- Mean of the difference: mu_(x-bar_1 - x-bar_2) = mu_1 - mu_2.
- Standard deviation of the difference: sigma_(x-bar_1 - x-bar_2) = sqrt(sigma_1^2 / n_1 + sigma_2^2 / n_2); variances add for independent groups.
- Normality condition: The difference is approximately normal if both populations are normal, or if both n_1 and n_2 are at least 30.
- Randomized experiments: If data come from a randomized experiment, only the randomization condition is required; the 10% condition is not needed.
Two independent populations have mu_1 = 80, sigma_1 = 10, n_1 = 25 and mu_2 = 75, sigma_2 = 8, n_2 = 36. Find the mean and standard deviation of the sampling distribution of x-bar_1 minus x-bar_2.
4.7
Two-Sample t-Interval for the Difference Between Two Population Means
Use a two-sample t-interval when you have two independent samples and want to estimate mu_1 minus mu_2. The point estimate is x-bar_1 minus x-bar_2, and the margin of error uses a pooled standard error. Whether 0 is inside or outside the interval determines whether you have evidence of a difference.
- Two-sample t-interval formula: (x-bar_1 - x-bar_2) plus or minus t* times sqrt(s_1^2 / n_1 + s_2^2 / n_2); df found using technology.
- Standard error for two means: SE = sqrt(s_1^2 / n_1 + s_2^2 / n_2).
- Degrees of freedom: Use technology; df falls between the smaller of (n_1 - 1, n_2 - 1) and (n_1 + n_2 - 2).
- Interpreting 0 in the interval: If 0 is inside the interval, there is insufficient evidence of a difference between the two population means. If 0 is outside, there is evidence of a difference in the direction of the subtraction order.
- Conditions: Two independent random samples (or random assignment), 10% condition for each group (unless randomized experiment), and normal/large sample for each group.
A 95% two-sample t-interval for mu_A minus mu_B is (-3.2, 1.8). Does this provide convincing evidence that the two population means differ? Explain.
4.9
Two-Sample t-Test for the Difference Between Two Population Means
A two-sample t-test assesses whether two independent groups have different population means. The null hypothesis is that the two means are equal (difference equals 0). The test statistic divides the observed difference in sample means by the standard error of that difference.
- Hypotheses: H_0: mu_1 = mu_2 (or mu_1 - mu_2 = 0); H_a: mu_1 < mu_2, mu_1 > mu_2, or mu_1 not equal to mu_2.
- Test statistic: t = (x-bar_1 - x-bar_2) / sqrt(s_1^2 / n_1 + s_2^2 / n_2); df found using technology.
- p-value: Found using the t-distribution with the technology-calculated df; represents the probability of observing a difference as extreme as the one found if the population means are equal.
- Conclusion: Compare p-value to alpha; reject or fail to reject H_0. State the conclusion in context referencing both populations and using non-definitive language.
- Conditions: Same as for the two-sample t-interval: random, 10% (unless randomized experiment), and normal/large sample for each group.
Two groups have x-bar_1 = 48, s_1 = 6, n_1 = 30 and x-bar_2 = 44, s_2 = 8, n_2 = 35. Set up the hypotheses for a two-sided test, calculate the test statistic, and describe what the p-value represents.
4.2
Choosing the Right t-Procedure
Selecting the correct procedure is a required skill on the AP exam. The key distinctions are: how many groups, whether the groups are paired or independent, and whether you are estimating (interval) or testing (significance test).
- One-sample t-interval: One quantitative sample; estimate mu with x-bar plus or minus t*(s / sqrt(n)).
- One-sample t-test: One quantitative sample; test a claim about mu using t = (x-bar - mu_0) / (s / sqrt(n)).
- Matched pairs t-interval or t-test: Two dependent samples; compute differences within pairs, then apply one-sample procedures to the differences.
- Two-sample t-interval: Two independent samples; estimate mu_1 - mu_2.
- Two-sample t-test: Two independent samples; test whether mu_1 - mu_2 = 0.
A study measures blood pressure before and after a treatment for the same 25 patients. Which procedure is appropriate and why?
| Scenario | Procedure | Parameter | df |
|---|---|---|---|
| One sample | One-sample t-interval or t-test | mu | n - 1 |
| Matched pairs | One-sample t-interval or t-test on differences | mu_d | n - 1 |
| Two independent samples (interval) | Two-sample t-interval | mu_1 - mu_2 | Technology |
| Two independent samples (test) | Two-sample t-test | mu_1 - mu_2 | Technology |
Key terms
| Term | Definition |
|---|---|
| t-Distribution | A family of symmetric, bell-shaped distributions with heavier tails than the standard normal, used when sigma is unknown. Identified by degrees of freedom; approaches the standard normal as df increases. |
| One Sample T-test | A significance test that uses x-bar and s to test a claim about one population mean mu when sigma is unknown; test statistic is t = (x-bar - mu_0) / (s / sqrt(n)) with df = n - 1. |
| Two-sample t-test | A significance test that uses two independent samples to test whether mu_1 - mu_2 = 0; test statistic is t = (x-bar_1 - x-bar_2) / sqrt(s_1^2/n_1 + s_2^2/n_2) with df from technology. |
| Matched Pairs Design | A design in which two measurements are taken on the same subject or on deliberately paired subjects; inference is performed on the sample of within-pair differences using one-sample t-procedures. |
| dependent samples | Samples where observations are paired or related, such as before-and-after measurements on the same subjects; require matched pairs analysis rather than two-sample procedures. |
| Independent groups | Two groups whose observations are unrelated to each other; required for two-sample t-procedures. Contrast with matched pairs, where observations are linked. |
| 10% Condition | Requires that the sample size is no more than 10% of the population when sampling without replacement, supporting approximate independence between observations. |
| randomness condition | Requires that data come from a random sample or randomized experiment before any t-interval or t-test is applied. |
| Critical Value | The t* multiplier determined by the confidence level and degrees of freedom; used to compute the margin of error in a t-interval. |
| Confidence Level | The long-run percentage of t-intervals from repeated random samples that would capture the true population mean or mean difference. |
| Significance Level | The threshold alpha used to decide whether to reject H_0; if the p-value is less than or equal to alpha, the result is statistically significant. |
| Statistically Significant | A result is statistically significant when its p-value is less than or equal to alpha, providing evidence against the null hypothesis about mu or mu_1 - mu_2. |
| One-tailed test | A hypothesis test with a directional alternative hypothesis (H_a: mu < mu_0 or H_a: mu > mu_0); the p-value is the area in one tail of the t-distribution. |
| Independence Condition | Requires that observations within a sample are independent of each other, supported by the 10% condition for random samples or by random assignment in experiments. |
How to review unit 4
Common unit 4 mistakes
Confusing matched pairs with two independent samples
If the same subjects are measured twice or subjects are deliberately paired, use a one-sample t-procedure on the differences. Running a two-sample t-test on paired data ignores the pairing and inflates variability, leading to incorrect results.
Misinterpreting the confidence level
The confidence level describes the long-run capture rate across repeated samples, not the probability that this specific interval contains the parameter. Avoid saying 'there is a 95% probability that mu is in this interval' after the interval is computed.
Forgetting to state the order of subtraction for differences
For matched pairs and two-sample procedures, the sign of the interval or test statistic depends on which group is subtracted from which. Always define the order of subtraction when stating the parameter and interpret results accordingly.
Skipping or incompletely verifying conditions
On free-response questions, all three conditions (random, 10%, normal/large sample) must be checked explicitly with reference to the context. Simply listing condition names without connecting them to the problem earns no credit.
Using z-procedures instead of t-procedures
Because sigma is unknown for quantitative variables, t-procedures are required in Unit 4. Using z* as the critical value or the standard normal to find p-values is incorrect unless the problem explicitly states that sigma is known.
How this unit shows up on the AP exam
Full four-step inference in free-response questions
AP Statistics free-response questions on inference for means typically require all four steps: state the parameter and procedure, verify conditions with context-specific justification, calculate the test statistic or interval, and write a conclusion in context. Partial credit is awarded at each step, so skipping condition checks or writing a conclusion without referencing the population costs points.
Procedure identification and justification
Multiple-choice and free-response questions frequently present a scenario and ask which t-procedure is appropriate. You must distinguish between one-sample, matched pairs, and two-sample situations based on whether observations are paired or independent, and justify your choice by referencing the study design.
Interpreting intervals and p-values in context
The AP exam consistently tests whether students can write correct, context-specific interpretations of confidence intervals and p-values. For intervals, the interpretation must reference the population parameter and use 'confident' language. For p-values, the interpretation must state the probability is computed assuming H_0 is true and describe the event in terms of the variable and population.
More ways to review
Topic study guides
Open the individual guides for Unit 4 when you want a closer review of one topic.
Practice questions
Use AP-style practice after you review the notes so you can check what you understand.
FRQ practice
Practice free-response reasoning and compare your answer with scoring guidance.
Cram archive videos
Watch past review streams filtered to Unit 4 when you want a video walkthrough.
Official unit cheatsheet
Open the Fiveable one-page unit review, then explore visual cheatsheets for a quick refresher.
Score calculator
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Unit 4 printables
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get printablesFrequently Asked Questions
What should I be able to do after AP Stats Unit 4?
By the end of Unit 4, you should be able to decide whether a one-sample, matched-pairs, or two-sample t procedure applies, identify the parameter, verify randomization and distribution conditions, use the correct standard error, and interpret results in context. The exam rewards the method and the interpretation, so practice writing what a result means in the problem's context instead of stopping at a calculator output.
What are common mistakes in AP Stats Unit 4?
Common Unit 4 mistakes include treating paired data as independent samples, using a z procedure when the population standard deviation is unknown, failing to check for strong skew or outliers in small samples, or interpreting a confidence interval as a probability about a fixed parameter. Slow down long enough to identify the variables, population, and requested conclusion before calculating.
How should I study AP Stats Unit 4?
Start with the vocabulary and conditions, then mix short calculations with full-sentence interpretations. For each missed question, label whether the problem was choosing a method, doing the calculation, or interpreting the result. Finish with timed mixed practice so you have to recognize the method without a topic label.
How does AP Stats Unit 4 connect to the rest of the course?
Unit 4 uses the same inference logic as Unit 3 but applies it to quantitative responses and t distributions rather than proportions and z procedures.