EXD.2:4 - Solution

EXD.2:4.1 - Select the relation and a discriminating instance

Recover the current question through EXD.1 when necessary. Obtain the definition, derivation, causal account or rationale through the supplied-account entry.

State what the recipient must connect. In a pooled mean, each group mean multiplied by its count reconstructs the group sum; the group sums and counts combine separately. Select small values that preserve this relation while exposing a likely competing rule. Unequal counts and unequal group means distinguish pooling individuals from giving groups equal influence.

Use a real case when its evidence and complexity fit. A constructed case is useful when its assumptions and answer criterion can be made explicit. Identify which it is.

EXD.2:4.2 - Work the correspondence, not only the answer

Show the intermediate quantities, relations and conditions needed to obtain the result. Connect each significant step to the claim: which quantity instantiates which part, why this operation is legitimate, and what condition makes the inference usable.

For a worked calculation, name the units and what each numerator or denominator counts. For a causal example, distinguish observations from an inferred mechanism. For an analogy, identify the relation that corresponds and a feature that does not transfer. A picture of two similar objects cannot supply that correspondence by itself.

If a recipient contribution would help locate the difficulty, first invite a small unassisted mapping: “Which quantity here is the total represented by the group mean?” Retain the response. Then provide targeted component guidance, such as identifying the count and asking how to recover the sum. A recipient with little relevant preparation may instead need the worked relation first.

EXD.2:4.3 - Vary the condition that matters

Choose a changed case that distinguishes the intended relation from the plausible alternative. Ask what should stay the same, what should change and why. Keep irrelevant complexity low enough for the relation to remain visible.

Inspect the rule as well as the number. Equal group counts allow an unweighted average of the group means to agree with the pooled mean. Equal group means can also make the two numerical answers coincide despite unequal counts. The governing target still determines which weighting rule the explanation should express.

Use EXD.5 when a recipient’s own account and supported correction serve a learning purpose. Use EXD.4 when the response changes the question, premise or next explanation.

EXD.2:4.4 - Bound the example and stop at its useful result

State the admissible correspondence and the nearest consequential limit. A branch can be a unit in one aggregation question and contain several equally weighted people in another. The same data then support different summaries.

Retain the smallest example or pair that supplies the needed distinction. Add another case only when it can reveal a remaining wrong generalization or support the intended next use. An example is evidence of the particular relation it demonstrates; claims about learning from it need the recipient evidence appropriate to that claim.