"Show that" answer discipline

X ~ N(170, 625). Show that P(X > 196) = 0.15, correct to 2 significant figures. (3 marks)

Which response below is certain to score all 3 marks?

• Standardise to get $z = 1.04$, look up $\Phi(1.04) = 0.8508$ from the tables, subtract from 1 to get $0.1492$, then state $0.1492 = 0.15$ (2 s.f.)

  Correct. All three mark-scheme steps are visible in order: the standardisation (M1), the accurate table value with the 1 − Φ step (A1), and the connection between that accurate value and the printed target (A1 cso). Nothing here relies on the examiner inferring work that isn't shown.

• B"$P(X > 196) = 0.15$, as given in the question."

  This is the exact response a real examiner report records candidates giving on this question type, verbatim: some students "simply wrote down 0.1500… with no working at all." It scores zero, not partial credit, because there is no method mark to attach the dependent accuracy marks to — the answer being numerically correct is irrelevant when nothing demonstrates how it was reached.

• CStandardise to $z = 1.04$ correctly, then write "$P(Z > 1.04) = 0.15$" straight away with no table value shown

  This earns the method mark for the standardisation, but not the accuracy mark that follows it — the accurate table value (0.8508, or the resulting 0.1492) never appears, only the given target repeated. This is precisely the 'missing out the intermediate step and the accurate answer' failure the Oct 2021 report names, distinct from writing nothing at all.

• DUse a calculator to find $P(X > 196) \approx 0.149$ directly, and write only that decimal

  A calculator route can be entirely correct mathematically and still fail a 'show that' question, because the mark scheme is built around the standardisation and table-lookup steps specifically, not just around producing a numerically close decimal. Without the standardised value $z = 1.04$ shown, there is no method visible for the examiner to credit.

Traps tested: Answer only no intermediate step

Show that the interquartile range of the volunteers' ages (grouped data, n = 50) is 16.25.

A script states $Q_1 = 26$ and $Q_3 = 42.25$, giving IQR $= 16.25$, with no working shown for either quartile. What is the safest description of what this earns?

• Little or nothing — the subtraction is correct but nothing demonstrates the two quartile values were actually interpolated rather than chosen to be 16.25 apart

  Correct. This is the reverse-engineering trap a real examiner report names directly: "the main errors, since the answer was given, were making up two ages... which differed by 16 as their 'working'." Both quartiles here are also individually wrong (26.25 and 42.5 are the correct values) in a way genuine interpolation working would have caught — which is itself evidence the numbers were picked to match the gap, not derived from the data.

• BFull marks — the final answer matches the value given in the question

  This is exactly the assumption the whole lesson has been arguing against: on a 'show that' question, matching the given value is necessary but proves nothing on its own, because the value was never in doubt. The marks are attached to the interpolation method, which is entirely absent here.

• CFull marks for Q1 and Q3 individually, since the method for finding a quartile doesn't need to be shown, only the final IQR

  The interpolation method is exactly what the mark scheme is testing at this stage of the paper (2.3, 'simple interpolation may be required') — a quartile value with no cumulative frequency and no interpolation line behind it is an unsupported assertion, not a scored calculation.

• DIt cannot be assessed without knowing whether the student used the n or the n+1 convention for the quartile positions

  That convention choice matters for a genuinely worked interpolation (and the mark scheme tolerates either), but it is irrelevant here — there is no interpolation shown to apply either convention to. The problem with this script isn't which formula was used; it's that no formula was used at all.

Traps tested: Reverse engineered working · Answer only no intermediate step · Irrelevant technicality substituted for the real issue

Having correctly calculated the outlier boundaries as 8 and 52, a question asks you to "show that there are 3 outliers" in a dataset. What must your answer additionally include to be safe?

• The three specific data values that lie outside 8 and 52, named explicitly

  Correct. A real examiner report on exactly this question type records the gap directly: "having gained the correct limits some did not list the 3 outliers in this part in order to show there are 3 outliers." The boundaries alone show where an outlier WOULD be; they do not show that exactly three data points actually are.

• BNothing further — stating the correct boundaries already proves there are exactly 3 outliers

  The boundaries define the rule, not the count. Demonstrating that exactly three data points satisfy that rule requires checking the data against it, which is a separate, required step — and it's specifically the step the examiner report records candidates skipping.

• CA recalculation of the boundaries using a different outlier rule, to double-check the count

  The spec is explicit that any outlier rule used will be given in the question — inventing or substituting a second rule is not what's being asked for, and risks the separate, also-documented trap of applying a remembered generic rule instead of the one the question specified.

• DA box plot showing the outliers as individual points beyond the whiskers

  A box plot can illustrate outliers but this question asks you to 'show that there are 3' as a written justification, not to draw one. Drawing a diagram doesn't substitute for stating which three values were checked against the boundaries and found to lie outside them.

Traps tested: Shown result not fully stated · Unrequested second method substituted · Diagram substituted for required written justification

A real WST01 question gives $S_{xy} = 12105.12$ and $S_{xx} = 16769.78$, and asks you to show that the regression line is $g = -42.3 + 0.722d$ (3 s.f.) — the actual Jun 2024 Q4(c).

What is the safest working practice for how many decimal places to carry the gradient $b = S_{xy}/S_{xx}$ to, before rounding down to the printed line?

• Carry $b$ to at least one more decimal place than the 3 s.f. printed in the line — work with $b = 0.7218$, not $0.722$ — then round only at the very last step

  Correct, and it is Pearson's own stated advice on this real question: "Students should be encouraged in these types of questions to give answers to at least one more decimal place than the given value." Rounding the gradient to 3 s.f. before it feeds into the intercept calculation is exactly what cost marks here.

• BStop at $b = 12105.12 \div 16769.78 = 0.722$, matching the 3 s.f. already printed in the line

  This is the exact working the examiner report records as insufficient on this real question: "b = 12105.12/16769.78 = 0.722 was not accurate enough to gain the final mark due to it being a 'show that' question." The 0.722 itself is correct — the working simply doesn't carry enough precision to prove it was calculated rather than copied from the given line.

• CIt doesn't matter, since the mark scheme only checks the final printed line against the question

  It matters specifically because the final line IS being checked against a printed target — the whole reason a 'show that' question is harder to get marks on than an ordinary one is that the examiner can see exactly how close your working came, not just whether you wrote the right final digits.

• DAs few decimal places as possible, to keep the working short and reduce the chance of a transcription error

  This trades one risk for a larger one. A transcription error in a longer decimal is a real but recoverable risk; rounding too early on a 'show that' question is the documented, repeated way marks are actually lost on this question type.

Traps tested: Insufficient accuracy in a show that · Answer only no intermediate step