Choose a criterion from your original response or a reviewer's feedback. Follow its practice link, then attempt the changed case before opening its solution.
Keep an evidence record
Keep your original answer and name the specific missing criterion. Choosing a category is your judgment, not an automated diagnosis.
Use the linked lesson task to repair the reasoning. Record what changed and why. Browser Back returns here after the guided practice.
Write the changed-case response on paper or in your own document. Record the date, tools used and whether you had already seen its solution.
Compare with the rubric or ask a reviewer for feedback, retain your first attempt, and revise. A matching number alone does not establish sound statistical reasoning.
This page does not collect or save responses or solution exposure. The linked lesson practice has its own local draft records. These are original public practice tasks; a changed case is new to you only if you have not seen it before. Repeating an exposed solution is practice, not an independent assessment.
These four routes add no points to STAT-1, create no required lesson completion, and do not predict an AP score. They cover selected inference criteria from one question; other items and skills still need review.
Browser printing gives all four prompts and blank review records. Solutions and rubrics are excluded even when open. Choose the relevant section; suggested writing lines are not answer-length requirements.
Free response 3 · part a
Design and dependence
In stat-fr-03 the archives contain distinct teams with no matching relation, so independent-group Welch inference is appropriate. The repair is to recover that design argument, not to choose a method from the two reported SDs. In this changed case the same units appear twice. Separate column summaries do not determine the variability of their differences.
A simple random sample of 18 employees is selected without replacement from 900 eligible employees. Each completes the same timed task before and after a training session, with no randomized control group. Every employee has both measurements. The surviving summaries are before: mean 42 minutes, SD 6 minutes; after: mean 39 minutes, SD 8 minutes. The pair identifiers and SD of the differences were lost. A colleague proposes an independent-group Welch test of a decrease at the prespecified 5% level. Define the parameter using d=after−before. Decide whether that test or a paired t test can be calculated legitimately from the surviving summaries. Use the covariance identity to demonstrate two possible difference SDs consistent with these margins. State what must be recovered and what a valid negative mean-change result would still fail to establish. Assume a normal population of differences would be a reasonable model if the paired data were recovered.
Record ID: stat-fr-03-retry-independent-design
Date / reviewer / tools / prior solution exposure: __________________________
Original criterion and evidence of the error: ______________________________
Repair explanation and first changed-case response:
Revision and next evidence needed: _____________________________________
Review this changed case: reasoning and criteria
Opening this public solution changes the evidence available for subsequent attempts. Keep the earlier response separately and record exposure yourself.
The parameter is the mean within-employee change μd=E(after−before) for the eligible population under this before/after protocol, with H₀:μd=0 and Hₐ:μd<0. The independent units are the 18 employees, not 36 separate measurements. The sampling fraction is 18/900=2%, supporting the usual small-fraction approximation across sampled employees; it does not make the two readings within an employee independent. The sample mean difference is recoverable as 39−42=−3 minutes. But s_d²=s_after²+s_before²−2s_before,after=100−2s_before,after is not determined by the marginal SDs. Covariance +48 gives s_d=2, whereas covariance −48 gives s_d=14. These are achievable: choose any 18 standardized numbers z with sample mean 0 and sample SD 1; before values 42+6z and after values 39+8z or 39−8z preserve the supplied margins and yield those respective difference SDs. Thus the potential t statistics are −3√18/2≈−6.364 and −3√18/14≈−0.909 with df 17. Under the stated normal-difference model their lower-tail probabilities would be about 0.00000353 and 0.187994, leading to different 5% decisions. These are counterexample calculations, not a valid p-value for the missing original pairs. Refuse both an independent-group substitution and an invented paired test. Recover the actual differences, their SD, or the within-pair covariance, and check the sampling/difference-model assumptions. Even a valid negative mean change in this uncontrolled before/after design would not isolate a causal training effect: practice, time or other changes could explain it.
Review criteria
Defines the oriented population mean change, equality null and lower alternative; identifies employees as the independent units.
Checks the 2% sampling fraction without treating within-employee readings as independent.
Recovers the mean −3 but proves the variance is unidentified using covariance and the two feasible SDs 2 and 14; if supplied, accept t values within 0.01 and counterexample p-values within 0.0001.
Explicitly refuses an unsupported original-sample test and requests differences, difference SD or covariance.
Separates before/after evidence from a causal training claim and identifies a plausible alternative explanation.
These criteria support manual review. They do not award an automatic score or certify independent understanding.
A Welch interval uses separate estimated variance contributions and a potentially fractional reference degrees of freedom. Its confidence level and the alternative of a test are separate choices. This changed case deliberately makes a two-sided 95% interval exclude zero while the prespecified one-sided 1% test does not reject.
Two independent simple random samples of distinct repair jobs come from separate archives of 1800 and 2600 jobs. No technician or job appears twice or in both samples. Archive A: n=18, mean=26.4 minutes, sample SD=4.8 minutes. Archive B: n=26, mean=23.6 minutes, sample SD=3.1 minutes. Treat normal population shapes as a reasonable working model; workflows were observed, not randomly assigned. Define delta=mu_A minus mu_B. Without assuming equal population variances, calculate the two variance contributions, Welch SE and degrees of freedom, then obtain a two-sided 95% interval using technology. Independently test H0:delta=0 against the prespecified Ha:delta>0 at alpha=0.01. A colleague says that zero exclusion by the interval forces rejection at that level. Evaluate this claim and interpret both results. Show calculator queries or formulas and retain unrounded intermediate values. Report SE and t to at least four decimals, df to at least two decimals, p to at least five decimals and interval endpoints to at least three decimals.
Record ID: stat-fr-03-retry-welch-uncertainty
Date / reviewer / tools / prior solution exposure: __________________________
Original criterion and evidence of the error: ______________________________
Repair explanation and first changed-case response:
Revision and next evidence needed: _____________________________________
Review this changed case: reasoning and criteria
Opening this public solution changes the evidence available for subsequent attempts. Keep the earlier response separately and record exposure yourself.
The independent sampling units and separate archives support an independent-group procedure; the fractions are 18/1800=1% and 26/2600=1%. With the stated normal-shape model and independence assumptions, use the approximate Welch t reference, not a pooled equal-variance or paired procedure. The estimate is 2.8 minutes. The variance contributions are v=4.8^2/18=1.28 and w=3.1^2/26=0.369615384615, in squared minutes. SE=sqrt(v+w)=1.284373537806 minutes. The degrees of freedom are (v+w)^2/[v^2/17+w^2/25]=26.720362740206; do not round them to 42 or use n_A+n_B-2. Query the t quantile at cumulative probability 0.975 with these df: t*=2.052835731423. The interval is 2.8 plus or minus t*SE, giving [0.163392109098,5.436607890902] minutes. This estimates the population mean difference, not a range containing 95% of individual job times; nominal confidence refers to the repeated interval procedure. For the test, t=2.8/SE=2.180051143676 and p=P(T_df>=t)=0.019120568511. Do not reject at 0.01. There is insufficient evidence at that prespecified level for a larger A population mean; this does not prove equal means. The 95% two-sided interval corresponds to a two-sided 5% reference rule, or a directional 2.5% rule when its sign agrees with a prespecified direction. It is not the one-sided 1% rule. Thus the two results are consistent. Observational workflow membership also prevents a causal conclusion from this comparison. Report values as numerical approximations under the stated design/model assumptions.
Review criteria
Defines delta as the A population mean minus the B population mean, justifies independent groups, checks both 1% sampling fractions and retains the supplied shape assumptions. Does not infer random assignment.
Uses v=1.28 and w=0.3696153846, SE approximately 1.284374 (absolute tolerance 0.005) and Welch df approximately 26.72036 (absolute tolerance 0.05). Separate variance contributions are required; a correct number from unjustified pooling does not repair the method.
Obtains t* approximately 2.052836 (absolute tolerance 0.005) and the 95% interval [0.163392,5.436608] minutes (each endpoint within 0.02), with a population-mean/repeated-procedure interpretation.
Uses signed t approximately 2.180051 (absolute tolerance 0.005), the upper-tail p approximately 0.0191206 (absolute tolerance 0.0001), and does not reject at 0.01 without claiming equality.
Explains the confidence-level/alternative mismatch rather than letting interval zero exclusion override the specified one-sided 1% test. Numerical tolerances are answer-review allowances, not additional sampling uncertainty or permission to change the decision rule.
These criteria support manual review. They do not award an automatic score or certify independent understanding.
Choose the tail from the parameter and prespecified alternative, not from the size of an unsigned statistic. Reversing the order of the groups is legitimate only if the hypotheses and tail reverse too. Retain enough p-value precision to make the requested decision.
A service compares two observational archives of response times. Independent simple random samples contain 24 distinct incidents from the 2400-incident A archive and 40 from the 4000-incident B archive. Each incident has a distinct independently operating response team; no team appears in both samples. A has mean 18.2 minutes and sample SD 5.4 minutes; B has mean 21.5 minutes and sample SD 4.8 minutes. Assume normal population shapes are a reasonable working model and do not assume equal variances. Before inspecting the data, the service specified delta=mu_A minus mu_B, H0:delta=0, Ha:delta<0 and alpha=0.01. Calculate the Welch statistic, df and the correctly oriented p-value, then conclude. Audit three proposed shortcuts: (i) replace t by |t| but still request the lower tail; (ii) double the smaller tail; (iii) round p to two decimal places before deciding. Finally rewrite the entire test using mu_B minus mu_A and verify that the decision is unchanged. Show the numerical tail queries; report t to at least four decimals, df to at least two decimals and p to at least five decimals.
Record ID: stat-fr-03-retry-tail-direction
Date / reviewer / tools / prior solution exposure: __________________________
Original criterion and evidence of the error: ______________________________
Repair explanation and first changed-case response:
Revision and next evidence needed: _____________________________________
Review this changed case: reasoning and criteria
Opening this public solution changes the evidence available for subsequent attempts. Keep the earlier response separately and record exposure yourself.
The parameter is the A archive mean minus the B archive mean, estimated by -3.3 minutes. The sampling fractions are 1% in each archive, groups are independent under the stated design, and the working normal-shape assumption supports the approximate Welch reference. Use v=5.4^2/24=1.215 and w=4.8^2/40=0.576. Then SE=sqrt(1.791)=1.338282481392 and df=(1.791)^2/[1.215^2/23+0.576^2/39]=44.127760462022. The signed statistic is -3.3/SE=-2.465847118142. Query the lower t tail: P(T_df<=-2.465847118142)=0.008813185714. Since this is below 0.01, reject H0 at the prespecified level. Under the sampling/model assumptions there is evidence of a smaller mean response time in archive A. The evidence does not by itself establish a causal workflow effect or the probability that H0 is true. Shortcut (i) requests P(T_df<=+2.465847118142)=0.991186814286, which is the wrong tail event. Shortcut (ii) gives 0.017626371428, the two-sided p-value for a different alternative; it must not replace this prespecified one-sided test. Shortcut (iii) displays 0.01 and loses the information that the unrounded value is strictly below the threshold. Compare the retained p-value with alpha, then round for reporting. If eta=mu_B minus mu_A, the estimate and statistic become +3.3 and +2.465847118142, while Ha becomes eta>0 and H0 remains eta=0. Its upper-tail probability is again 0.008813185714 by symmetry. This is an algebraic reexpression of the same prespecified claim, not a post-data switch to whichever scientific alternative looks favorable.
Review criteria
States the oriented parameter, equality null and lower alternative before interpreting the data. Checks independent units, both 1% sampling fractions and the stated normal-shape model without assuming equal variances.
Calculates SE approximately 1.338282 (absolute tolerance 0.005), Welch df approximately 44.12776 (absolute tolerance 0.05) and signed t approximately -2.465847 (absolute tolerance 0.005).
Uses the lower-tail query and p approximately 0.00881319 (absolute tolerance 0.00005), rejects at 0.01 and gives a conditional population-mean conclusion without causal or posterior-null claims.
Identifies the wrong-event value approximately 0.99118681 and different two-sided value approximately 0.01762637 (absolute tolerance 0.0001 each). Explains why rounding to 0.01 before comparison discards decision-relevant precision.
Reverses the parameter, statistic and alternative together, giving the same upper-tail p and conclusion. Tolerances concern displayed numerical answers; the signed hypotheses, tail and decision must still be correct.
These criteria support manual review. They do not award an automatic score or certify independent understanding.
Part d of stat-fr-03 uses observational archives, so random sampling supports a population comparison without establishing causation. This retry changes the design to random assignment among volunteers. That changes the causal argument, but does not supply random population sampling. Confidence coverage, individual outcomes and the prespecified test level remain separate claims.
Forty-eight adult volunteers at one training center are randomly assigned, 24 each, to task protocols A and B. Assignment is implemented as planned, there is no interference between participants, and the task and measurement conditions are otherwise controlled. The volunteers are not a random sample of adults. The response is task-completion time in minutes. Under the stated working conditions for Welch inference, sample means are A=31 and B=34, with SD 5 in each group. A correctly computed 95% interval for the oriented mean effect A−B is approximately [−5.9054,−0.0946] minutes; the prespecified lower-tail test of zero effect at α=0.01 has p≈0.021636. Audit the statement: “A causes every adult to finish between 0.095 and 5.905 minutes faster with 95% probability, and zero is outside the interval, so the 1% test is significant.” Give a corrected causal and population-scope statement, explain the interval, and make the specified decision. Identify what further evidence would support extending the result to a wider population.
Record ID: stat-fr-03-retry-interpretation-scope
Date / reviewer / tools / prior solution exposure: __________________________
Original criterion and evidence of the error: ______________________________
Repair explanation and first changed-case response:
Revision and next evidence needed: _____________________________________
Review this changed case: reasoning and criteria
Opening this public solution changes the evidence available for subsequent attempts. Keep the earlier response separately and record exposure yourself.
Random assignment, implemented treatment and the no-interference/controlled-setting assumptions support a causal comparison of the protocols for these experimental participants in this setting. The estimated mean effect A−B is −3 minutes: A is faster on average in the sample. Random assignment does not make the volunteers a representative random sample of all adults, so a general adult-population claim requires additional sampling or justified transport evidence. Under the inference assumptions, the 95% interval estimates a mean effect; its nominal coverage belongs to the repeated interval-producing procedure. It neither contains 95% of individual effects nor assigns a 95% probability that every adult has an improvement in that range. Individual effects may vary and are not identified by these two group summaries. The lower-tail p-value 0.021636 exceeds 0.01, so fail to reject the zero-mean-effect null at the prespecified 1% level; this is not proof of no effect. A two-sided 95% interval corresponds to a different threshold from a one-sided 1% test, so its exclusion of zero does not implement the requested rule. Here a 95% interval is evidence at its own confidence level while the stricter requested test does not reject. Broader sampling across the intended population and settings, with relevant participation and treatment-effect differences examined, would support a more defensible extension; the present randomization alone does not provide it.
Review criteria
Recognizes that randomized assignment changes the causal justification relative to stat-fr-03, and qualifies it by the experimental participants and setting.
Distinguishes random assignment from random sampling and refuses the unsupported all-adults generalization.
Interprets the interval for a mean effect and the repeated procedure, not individual ranges or a probability for every adult.
Uses p>0.01 to fail to reject, avoids proving the null, and explains why the 95% two-sided interval cannot replace the stated one-sided test.
Names meaningful additional population/setting evidence rather than claiming randomization automatically establishes representativeness.
These criteria support manual review. They do not award an automatic score or certify independent understanding.
