Items / Reference templates / BP011
REF-BP011-1
C05 expected value · C05.2 linearity of expectation · target easy, CD2 · split test · reference items carry their template's native demand and difficulty
Pending
The number of hours X for a repair job has mean E[X] = 15. The invoice in dollars is Y = 4X + 40. What is E[Y]?
Explanation (as written by the item's author)
E[Y] = 4(15) + 40 = 100.
| Option | Code | Catalogue meaning | Author's rationale |
|---|---|---|---|
| A | M-VAR-LINEAR | Treats variance as linear: Var(aX) = a*Var(X) instead of a^2*Var(X). | Squares the multiplier, as the variance rule would. |
| B (key) | KEY | Linearity of expectation: aE[X] + b. | |
| C | MISC-OTHER | Distractor is not attributable to any catalogued misconception (itself a quality signal). | Scales the mean but drops the additive constant. |
| D | MISC-OTHER | Distractor is not attributable to any catalogued misconception (itself a quality signal). | Adds the constant but does not scale the mean. |
Quality flags · 0 fired
- numeric middle key (info) firedOptions are numeric and the key is one of the two middle values. Informational; tested at set level against the 0.5 chance rate.
- duplicate options (integrity) not firedTwo options are identical after normalisation.
- numerically equal options (integrity) not firedTwo options denote the same number (e.g. 0.5 and 1/2 and 50%).
- missing options (integrity) not firedFewer than four non-empty options.
- key out of range (integrity) not firedThe keyed answer is a probability outside [0, 1] (or a percentage > 100).
- all none of above (cue) not firedAn option is 'all/none of the above' or 'both A and B'.
- key uniquely longest (cue) not firedThe key is the longest option by at least 20% (length cue).
- key uniquely shortest (cue) not firedThe key is the shortest option by at least 20%.
- distractor out of range (cue) not firedA distractor is a probability outside [0, 1]; it can be eliminated without the construct.
- absolute terms in distractors (cue) not firedDistractors (only) use absolute terms such as always/never/must/proves, a known cue that they are wrong.
- stem option overlap cue (cue) not firedThe key alone repeats a content word from the stem.
- grammatical mismatch (cue) not firedThe stem ends in 'a'/'an' and some options do not agree with it.
- key is hedged only (cue) not firedOnly the key contains hedging language (may/might/plausibly/likely).
- negative stem (style) not firedStem uses NOT/EXCEPT/LEAST phrasing.
- overlong stem (style) not firedStem exceeds 110 words.
- inconsistent numeric format (style) not firedNumeric options mix fractions, decimals and percentages, or use differing decimal precision.
- unit mismatch (style) not firedOptions carry different units.
- option length imbalance (style) not firedLongest option is more than 2.5x the shortest.
- rationale value mismatch (metadata) not firedA distractor's stated rationale computes a value ('= x') that is not the option it is attached to: the distractor does not actually execute the misconception it claims.
Validator output
PoT not yet run for this item.
Answering probes
Probes not yet run for this item.
Shortcut results
| Heuristic | P(key) |
|---|---|
| longest option | 0.50 |
| shortest option | 0.00 |
| convergence | 0.25 |
| stem overlap | 0.25 |
| numeric middle | 0.50 |
| hedged option | 0.25 |
| position prior | 0.00 |
Provenance and generation trace
- Method
- reference_template / —
- Model
- none
- Source
- parameterised template T-C05.2-linearity, written by the AssessAI build agent (Claude); no external item bank; not human-reviewed
- Prompt hash
—- LLM calls · cost
- 0 · $0.00000 · 0.0 s
- Parser notes
- none