Fall 2026 Rehearsal — Section II: Free Response

Four complete investigations

This section contains four original generated free-response questions, each worth 10 points, for the course effective Fall 2026 and the revised May 2027 exam format. They are not official College Board items. Allow 90 minutes for the section. Use a graphing calculator with statistical capabilities and appropriate formulas and tables.

Question 1 emphasizes formulating a statistical question and planning data collection. Question 2 emphasizes constructing, calculating, describing, and comparing data representations. Question 3 requires a complete inference procedure. Question 4 brings together method selection, probability, summary statistics, inference, and regression. The four questions receive equal weight; the two rehearsal sections each contribute 50% to the optional practice composite.

Show the statistical argument

A useful response connects what you did to why it answers the question. Give the parameter a name and state what population and response it describes. In an experiment, distinguish random selection of participants from random assignment of treatments. Explain how your design deals with a known source of variation, and describe an operational chance procedure rather than only saying “randomize.” In a graph, provide the variable, units, scale, and all values needed to reconstruct the display.

For inference, choose the procedure before entering calculator values. Conditions need supporting evidence from the actual study: a random mechanism, the applicable sampling fraction, and a distribution or count check. In a paired analysis, examine the differences. In independent-sample inference, check each sample and population separately. A condition name without the study-specific reason leaves the reader unable to evaluate it.

Show the calculation that identifies the standard error, the test statistic or interval, and the reference distribution. Include a tail direction for a p-value. A calculator may produce the result, but a bare screen value does not tell the reader whether you used the correct input and model. For a one-sample t statistic, the structure is $t=(\bar x-\mu_0)/(s/\sqrt n)$ . Interpret the final result for the population quantity, with units and an appropriate scope of generalization.

Use words such as “convincing statistical evidence” or “we are confident,” rather than “proved.” A confidence interval for a mean difference does not contain a stated percentage of individual observations. A positive residual says the observed response lies above the prediction; it does not make the fitted model causal. Wherever the prompt defines an order such as old minus new, keep that order consistent through the sign, tail, and conclusion.

Keyboard entry and rounding

Plain-text notation is acceptable here: write mu_d for a population mean difference, sqrt(n) for a square root, and p_A - p_B for a difference in proportions. Define the notation once and keep it consistent. If a task asks you to construct a plot, complete coordinates or labeled plotting instructions are an accessible alternative to a drawing; merely naming a graph does not construct it. Keep intermediate digits and report enough precision to show the intended calculation. Sensible rounding and equivalent parameter definitions are addressed in the separate analytical rubrics.

Work all parts before opening the separate worked answers and rubrics. For a timed attempt, about 22 minutes per question is a useful starting allocation, with the remaining time for checks. No timer is enforced by this page. Read each rubric after your attempt and locate the actual evidence in your response; a feeling of familiarity is not the same as having carried out the action.

Original generated practice for the Fall 2026 course.

1. Mulch, sunlight, and water use

A community garden wants to investigate whether a new mulch reduces the mean weekly volume of irrigation water needed by tomato plants. Owners of 120 garden plots volunteer. Each plot contains the same tomato cultivar. Sixty plots receive full sun, and sixty receive partial shade. Sun exposure is expected to affect water use.

The two treatments are new mulch and standard mulch. Both are permitted for food gardening. Over four weeks, technicians will follow the same soil-moisture target and record the liters of irrigation water supplied to each plot each week. The response for each plot will be its mean weekly irrigation volume, in liters. The owners permit either mulch to be used.

Part a

Write a valid investigative question for this experiment. State the response, the direction of the treatment comparison, and the population or scope for which a causal conclusion would be appropriate.

Your reasoning

Part b

Describe a randomized block design using all 120 plots. Explain why the blocks are appropriate. Give an operational random assignment procedure, the number of plots receiving each treatment in each block, and at least two concrete controls for the way water use will be measured or the plots maintained.

Your reasoning

Part c

A gardener proposes assigning new mulch to every full-sun plot and standard mulch to every partial-shade plot. Explain the specific problem with this plan and why even a large observed treatment-group difference would not isolate the effect of mulch.

Your reasoning

Part d

Suppose your randomized design finds convincing evidence of lower mean water use with the new mulch. State the strongest appropriate conclusion and its generalization limit. Also describe one ethical safeguard for carrying out the experiment on volunteer-owned plots.

Your reasoning

Worked answer and rubric

2. Museum puzzle times

A museum records completion times for all 40 attendees in workshop A. Times are recorded to the nearest whole minute. The full frequency table is below; there are no other completion times.

Workshop A completion times
Time (minutes)	4	5	6	7	8	12
Number of attendees	4	8	12	8	4	4

A separate workshop B also has 40 attendees. Its five-number summary, in minutes, is minimum 3, Q1 = 5, median = 7, Q3 = 9, maximum 11. For all quartiles in this question, use the median of the lower half and the median of the upper half of the ordered data.

Part a

Construct a labeled dotplot for workshop A, keeping the horizontal axis on an evenly spaced time scale. You may provide a drawing or complete plotting instructions giving the axis label, time locations, and number of dots at every occupied location. Do not merely name the type of graph.

Your reasoning

Part b

Calculate the mean and median completion time for workshop A. Show how the frequencies or ordered positions produce each result.

Your reasoning

Part c

Calculate Q1, Q3, the IQR, and both 1.5×IQR outlier fences for workshop A. Identify all observed outlier values, if any. Values exactly on a fence are not outliers.

Your reasoning

Part d

Compare typical completion time and the variability of the middle half of attendees in workshops A and B, using the median and IQR. Explain why those two statistics are useful for workshop A.

Your reasoning

Part e

One of workshop A’s 40 attendees is chosen uniformly at random. Calculate the probability that the attendee’s completion time is at least 8 minutes. Also calculate the percentile rank of a 6-minute completion time, using the percentage of times less than or equal to 6.

Your reasoning

Worked answer and rubric

3. Comparing two route-planning methods

A delivery depot wants to determine whether a new route-planning method lowers mean driving time relative to its old method. A simple random sample of 20 drivers is selected without replacement from the depot’s 6,000 drivers. Each selected driver completes one comparable route using each method; the order of the methods is randomized independently for each driver. Routes have the same distance and stop schedule within each driver’s comparison, and the study conditions are otherwise held constant. Assume results from different drivers can be treated as independent.

For each driver, define d = old-method driving time − new-method driving time, in minutes. For the 20 differences, the sample mean is 4.6 minutes and the sample standard deviation is 7.2 minutes. The distribution of differences is roughly symmetric with no outliers. The population SD of differences is unknown. Use α = 0.05.

Optional t-table reference for df = 19 (rounded)
Positive t cutoff	Upper-tail probability
1.729	0.05
2.539	0.01
2.861	0.005
3.174	0.0025

You may use technology for a p-value or give a correct bound using this table.

Part a

Define the population parameter and state the null and alternative hypotheses for whether the new method lowers mean driving time.

Your reasoning

Part b

Identify the appropriate inference procedure and its degrees of freedom. Justify it by verifying the randomization, sampling-fraction, and sample-shape conditions using the information given.

Your reasoning

Part c

Calculate the standard error and test statistic. Obtain the appropriately directed p-value or a correct bound, showing enough work to identify the reference distribution and tail.

Your reasoning

Part d

Make the decision at α = 0.05 and state a conclusion about the population in context. Avoid wording that proves a hypothesis or promises a decrease for every driver.

Your reasoning

Worked answer and rubric

4. Parcel times and dispatch delays

Two parcel depots, A and B, each served more than 20,000 different customers last month. For this question, every customer placed exactly one order during that month. A researcher independently takes a simple random sample of 100 customers from each depot’s customer list, without replacement, and records elapsed time from order to delivery, in days. Assume sampled customer responses can be treated as independent within each sample, and the two depot samples are independent.

Complete delivery-time frequencies for depot A
Delivery time (days)	1	2	3	4	5
Number of sampled customers	10	20	40	20	10

For A, the sample standard deviation is 1.101 days (rounded). For B, the sample mean is 3.6 days and the sample standard deviation is 1.4 days (rounded). The B sample has no strong skewness or outliers. Neither population SD is known. A technology calculation for a 90% two-sample t-interval gives df≈187.6 and t*≈1.653; use that critical value.

Separately, for 40 parcels from depot A, a researcher records x = hours between order and dispatch and y = days between order and delivery. The observed x values range from 4 through 40 hours. A scatterplot supports a linear model, with no influential outliers. The fitted line is $\hat y=1+0.08x$ , and r=0.75. One parcel has x=25 hours and observed y=4 days.

Part a

Identify the appropriate procedure to estimate μA−μB, the difference in population mean delivery time for A minus B. Explain why these samples are independent rather than paired, and verify the conditions for the procedure using the given design, population sizes, and sample information.

Your reasoning

Part b

A sampled depot-A customer is selected uniformly at random. Calculate the probability that this customer’s order took at most 2 days. Calculate the sample mean delivery time for A from the complete frequency table.

Your reasoning

Part c

Calculate a 90% confidence interval for μA−μB using the supplied critical value. Interpret it in context and state whether it gives convincing evidence of a difference in population mean delivery times.

Your reasoning

Part d

For the separately modeled parcel with x=25 hours, calculate the predicted delivery time and its residual, then explain what the residual says about the prediction. Calculate r² and interpret it for this parcel regression.

Your reasoning

Worked answer and rubric

Glossary and deeper reading · 8 refreshers

These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.

Population parameter
A population parameter is a numerical feature of the population or probability model being studied. It is fixed for that defined population, although its value may be unknown. For example, the true proportion of all enrolled students who walk to school is a parameter.
Read more: Population parameter in the captured Statistics reading →
Population
A population is the full set of individuals or measurements a study aims to describe. It must be defined clearly before deciding whether a sample supports generalization. For example, all students enrolled in a particular school this year form a population.
Read more: Population in the captured Statistics reading →
Statistical inference
Statistical inference uses data and a study design to draw conclusions beyond the observed values. Random sampling supports population generalization, while well-conducted random assignment supports treatment-effect conclusions. For example, a random sample can estimate a school-wide preference without surveying every student.
Read more: Statistical inference in the captured Statistics reading →
Statistical independence
Statistical independence means that learning one outcome does not change the probability distribution of another. For two events, it implies that the probability of both occurring equals the product of their probabilities. For example, independent draws with replacement keep the second draw’s color probabilities unchanged by the first.
Read more: Statistical independence in the captured Statistics reading →
Standard error
A standard error describes the standard deviation of a statistic’s sampling distribution, usually estimated from data. It measures uncertainty in an estimate rather than the spread of individual observations. For example, the estimated standard error of a sample mean is the sample standard deviation divided by the square root of the sample size.
Read more: Standard error in the captured Statistics reading →
Test statistic
A test statistic summarizes how the sample departs from the null model. Its reference distribution under the null supplies the probabilities used in testing. For example, a z-statistic expresses a sample proportion’s difference from a hypothesized proportion in null-model standard-error units.
Read more: Test statistic in the captured Statistics reading →
Confidence interval
A confidence interval is a data-based range produced by a method with a specified long-run coverage rate for a population parameter. For a fixed population, the calculated interval either contains the parameter or it does not; the confidence level describes the method over repeated samples. For example, a 95% method captures the true parameter in about 95% of repeated intervals when its assumptions hold.
Read more: Confidence interval in the captured Statistics reading →
Mean
The mean is a measure of center often called an average. For a finite data set, the arithmetic mean is the sum of the values divided by their number; for a random variable, the mean is its probability-weighted expected value when that value exists. Means can be pulled toward extreme values and are not generally resistant. For example, the arithmetic mean of 2, 4, and 9 is 5.
Read more: Mean in the captured Statistics reading →

Four complete investigations

Show the statistical argument

Keyboard entry and rounding

Part a

Part b

Part c

Part d

Part a

Part b

Part c

Part d

Part e

Part a

Part b

Part c

Part d

Part a

Part b

Part c

Part d

Population parameter

Population

Statistical inference

Statistical independence

Standard error

Test statistic

Confidence interval

Mean

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