FRQs 1-5 – Free Response

Overview

• These five questions form the first part of Section II
• 65 minutes total for FRQ 1-5 (about 13 minutes per question)
• Each question worth 4 points, totaling 20 points
• Makes up 37.5% of your total exam score
• Calculator use is allowed and encouraged throughout
• Each question focuses on a specific area:
  • FRQ 1: Focus on Exploring Data
  • FRQ 2: Focus on Sampling and Experimental Design
  • FRQ 3: Focus on Probability and Sampling Distributions
  • FRQ 4: Focus on Inference
  • FRQ 5: Multi-Focus (combines multiple areas)

The standard FRQs test your ability to apply statistical methods in context. Unlike multiple-choice where you select from given options, here you must show complete statistical reasoning. Each question typically has multiple parts (a, b, c, sometimes d) that build on each other. The scoring emphasizes communication - you need to explain your reasoning, not just show calculations.

Strategy Deep Dive

FRQs demand comprehensive statistical communication beyond problem-solving. show conceptual understanding through clear reasoning, not just mechanical procedures.

The Communication Framework

Think of each FRQ response as a mini statistical report. Your reader (the grader) needs to follow your logic from problem to solution. This means stating what you're doing, why you're doing it, and what your results mean in context. A calculation without explanation rarely earns full credit.

For exploring data questions, don't just calculate statistics - describe what they reveal about the data. When you find that the mean is 45.7 and the median is 52.3, explain that this left skew suggests a few unusually low values pulling the mean down. The grader wants to see that you understand what these numbers tell us about the real-world context.

For inference procedures, the structure is even more critical. State your hypotheses in both symbols and words. Verify conditions before proceeding - this shows you understand when procedures are valid. After calculating, always return to the context with your conclusion. A p-value of 0.023 isn't the end - explain what this means about your original question.

Managing Multi-Part Questions

FRQs build logically, with later parts often depending on earlier work. Essential strategy: Attempt all parts regardless of earlier difficulties. The grading often allows you to earn points on later parts even if earlier parts are incorrect. Make a reasonable assumption about what part (a) should have given you and proceed.

This building structure also provides hints. If part (a) asks you to create a scatterplot and part (b) asks about the correlation, that's telling you the variables are likely correlated. If part (c) then asks about prediction, regression is probably appropriate. The question structure guides your thinking if you let it.

Condition Checking and Assumptions

One of the biggest differentiators between scores is proper condition checking. For inference procedures, always explicitly verify:

For confidence intervals and hypothesis tests about means: Random sampling (or random assignment for experiments), the 10% condition if sampling without replacement, and normality (via sample size or data shape). Don't just list these - explain why each is met. "The problem states that students were randomly selected" shows you identified randomness in the problem context.

For proportions: Random sampling, the 10% condition, and the success/failure condition (np ≥ 10 and n(1-p) ≥ 10). Calculate these values - don't just state the condition. Show that 50(0.3) = 15 ≥ 10 and 50(0.7) = 35 ≥ 10, so the condition is satisfied.

Calculator Integration

Your calculator should handle computations, but you must show what you're computing and why. Write something like: "Using 2-PropZTest on the calculator with x₁ = 45, n₁ = 200, x₂ = 38, n₂ = 180..." This shows the grader you know which procedure to use and what values to input. Then report the key results: "This gives z = 2.34 with p-value = 0.0192."

But don't let the calculator do your thinking. You still need to set up the problem, check conditions, and interpret results. The calculator is a tool for computation, not a replacement for statistical reasoning.

Rubric Breakdown

Understanding how FRQs are scored transforms how you write responses. Each part is scored as essentially correct (E), partially correct (P), or incorrect (I). The key point is earns each score level:

Essentially Correct (E)

• Complete and correct statistical reasoning
• Appropriate procedure selected and executed properly
• Conditions checked when required
• Clear conclusion in context
• Minor arithmetic errors don't prevent an E if work is shown

Partially Correct (P)

• Right idea but incomplete execution
• Correct procedure but missing condition check
• Correct work but conclusion not in context
• Major components present but some confusion evident

Incorrect (I)

• Wrong procedure selected
• Major conceptual errors
• Missing most required components
• Work that doesn't address the question asked

The overall question score (0-4) depends on how many parts earn E or P. Generally:

• 4: All parts essentially correct
• 3: Three parts essentially correct OR two E and one or two P
• 2: Two parts essentially correct OR one E and one or two P OR three P
• 1: One part essentially correct OR two P
• 0: No parts essentially or partially correct

Common Patterns by Question Type

FRQ 1: Exploring Data

These questions typically involve describing distributions, comparing groups, or analyzing relationships. You'll often need to create or interpret graphs, calculate summary statistics, and explain what they reveal.

Pattern recognition: Look for whether you're dealing with one variable (histogram, boxplot, mean/median) or two variables (scatterplot, correlation, regression). For comparing groups, parallel boxplots or back-to-back stemplots are common. Always comment on shape, center, variability, and unusual features.

Common tasks include:

• Creating appropriate graphical displays
• Calculating and interpreting summary statistics
• Comparing distributions using comparative language
• Identifying and discussing outliers in context

Key insight: These questions reward thorough description. Don't just say "skewed right" - explain what this means about the data. Don't just identify an outlier - discuss whether it's plausible in context and how it affects your analysis.

FRQ 2: Sampling and Experimental Design

These questions test whether you understand how data is collected and what conclusions are valid. You might design a study, identify flaws in a given design, or explain what conclusions can be drawn.

Pattern recognition: Immediately identify whether you're dealing with an observational study or an experiment. For experiments, look for random assignment, control groups, and replication. For sampling, identify the population, sampling method, and potential biases.

Common tasks include:

• Describing how to put in place random sampling or random assignment
• Identifying and explaining sources of bias
• Explaining why random assignment allows causal conclusions
• Distinguishing between parameters and statistics

Key insight: Precision in language matters enormously here. "Bias" has a specific meaning (systematic error), different from "variability" (random error). "Random sampling" allows generalization to the population; "random assignment" allows causal conclusions. Using these terms correctly shows deep understanding.

FRQ 3: Probability and Sampling Distributions

These questions often involve calculating probabilities, working with distributions, or understanding sampling variability. You might work with normal distributions, binomial situations, or sampling distributions of statistics.

Pattern recognition: Identify what type of random variable you have (discrete or continuous) and what distribution applies. For sampling distributions, the statistic being studied (mean, proportion, difference) determines your approach.

Common tasks include:

• Calculating probabilities using normal or binomial distributions
• Finding values that correspond to given probabilities
• Describing sampling distributions (shape, center, spread)
• Using the Central Limit Theorem appropriately

Key insight: Always define your random variable clearly. Instead of just calculating P(X > 50), write "Let X = the weight of a randomly selected apple in grams. We need P(X > 50)." This clarity prevents errors and shows the grader you understand what you're calculating.

FRQ 4: Inference

The inference question is often the most procedural, requiring a complete hypothesis test or confidence interval. Structure and completeness are crucial for earning full credit.

Pattern recognition: Identify whether you need a hypothesis test (making a decision about a claim) or confidence interval (estimating a parameter). Then determine what parameter you're working with (mean, proportion, difference of means, etc.) to select the appropriate procedure.

For hypothesis tests, always include:

1. Parameter definition and hypotheses (in symbols and words)
2. Conditions verification (with work shown)
3. Test statistic calculation and p-value
4. Decision about H₀ using α comparison
5. Conclusion in context

For confidence intervals, always include:

1. Parameter definition
2. Conditions verification
3. Calculation showing critical value and standard error
4. The interval
5. Interpretation in context

Key insight: The conclusion is where many students lose points. For a hypothesis test, connect your statistical decision back to the original question. For a confidence interval, explain what the interval tells us about the parameter, not about individual values or sample statistics.

FRQ 5: Multi-Focus

This question combines elements from multiple areas, often telling a story through data that requires various statistical approaches. You might explore data, then conduct inference, or use probability to inform a study design decision.

Pattern recognition: Read the entire question first to see the big picture. Part (a) might involve exploring data that sets up an inference procedure in part (c). Understanding this connection helps you know what to emphasize in earlier parts.

Common combinations:

• Explore data → identify need for inference → conduct test
• Calculate probability → design study → analyze results
• Compare groups descriptively → test for significant difference

Key insight: These questions test whether you can integrate different statistical ideas. Show how each part connects to the overall investigation. If part (a) reveals an outlier and part (c) asks for inference, discuss how that outlier affects your choice of procedure.

Time Management Reality

With 65 minutes for 5 questions, you have about 13 minutes per question. But not all questions require equal time. Here's a realistic pacing strategy:

Minutes 0-5: Survey and Plan Read all five questions quickly. Identify which ones seem most straightforward for you. Note which require graphs (these take time to draw properly) and which are more computational. This initial survey prevents surprises and helps you allocate time wisely.

Minutes 5-50: Work Through Questions Start with questions where you feel most confident. A solid answer on an "easier" question is worth the same points as struggling through a harder one. For each question:

• Read carefully and identify all parts
• Plan your approach before writing
• Show work clearly
• Leave space to add thoughts if time permits

Exploring data questions often take longer because of graphing. Budget 15-16 minutes for these. Pure inference questions might only need 10-12 minutes if you're efficient with calculator use. Multi-focus questions vary widely - gauge based on the specific tasks required.

Minutes 50-60: Strategic Review Return to any parts you skipped. Even attempting a reasonable approach can earn partial credit. Add context to conclusions if you rushed. Check that you've answered what was asked - it's frustrating to lose points for solving the wrong problem.

Minutes 60-65: Final Points Grab If completely stuck on a part, write what you would do if you had more time. "I would check the normality condition by examining a normal probability plot" shows understanding even without execution. Define parameters, state what test you'd use, or explain what a result would mean. Partial credit adds up.

Common Statistical Misconceptions to Avoid

The FRQs often test whether you truly understand concepts or just memorize procedures. Avoiding these misconceptions separates strong responses from mediocre ones:

Exploring Data Misconceptions

Don't confuse the mean being greater than the median with right skew - it indicates left skew. The few low values pull the mean down more than they affect the median. When comparing groups, discuss variability, not just centers. Two groups can have identical means but very different spreads, leading to different practical implications.

Experimental Design Misconceptions

Random assignment is not the same as random sampling. You can have one without the other. An experiment with volunteers using random assignment can establish causation but can't generalize beyond those volunteers. Blocking is not the same as stratifying - blocking happens in experiments to reduce variability, while stratifying is a sampling technique.

Probability Misconceptions

Independence and mutual exclusivity are different concepts entirely. Events can be independent but not mutually exclusive (rolling a 3 on one die and a 4 on another). Don't confuse P(A|B) with P(B|A) - the order matters enormously in conditional probability.

Inference Misconceptions

A confidence interval contains plausible values for the parameter, not predictions for individual observations. If a 95% CI for mean height is (68, 72) inches, this doesn't mean 95% of people have heights in this range. The interval is about the population mean, not individual values.

Statistical significance doesn't mean practical importance. With large samples, tiny differences can be statistically significant but meaningless in practice. Always consider the context and magnitude of differences, not just p-values.

Final Thoughts

FRQ success requires technical accuracy combined with accessible explanation. Write for a statistically literate reader unfamiliar with your specific problem context.

The students who excel don't just perform procedures - they tell statistical stories. They explain why they're checking conditions, what their calculations reveal, and what conclusions are reasonable given the data. They use context throughout their responses, not just in a final sentence.

Practice with released FRQs is essential, but don't just check if your answer is correct. Study the scoring guidelines to understand what earners full credit. Often, it's not about getting the exact right number but about demonstrating statistical thinking. Pay attention to:

• How complete responses are structured
• What level of detail is expected for conditions
• How conclusions should be stated
• When calculator output should be referenced

Remember that partial credit is your friend. Even if you can't complete a problem fully, showing that you know which procedure to use, why conditions matter, or what a conclusion should address earns points. Never leave a part blank - educated attempts based on statistical reasoning often earn partial credit.

Your calculator proficiency matters, but it's not everything. Know how to perform procedures efficiently, but spend your mental energy on interpretation and communication. The calculator handles computation; you handle the thinking.

The five standard FRQs follow consistent patterns: data exploration, study design critique, probability applications, inference procedures, and integrated analysis. Systematic preparation for these predictable formats, combined with practiced communication skills, positions you to earn the substantial 37.5% of total points available from this section.