A-LEVEL MATHEMATICS / NORMAL DISTRIBUTION / INDEPENDENT SUPPLEMENT
METHOD SUMMARY + ORIGINAL PRACTICE
Normal distribution:
setup, tails and cutoffs
A-Level Maths
Four worked examples, four original shaded diagrams and 18 original practice tasks. Translate a verbal event
into bounds before using a calculator. Keep raw values and standardised values consistent.
Study route Pages
Right tail: two equivalent calculation routes 2
Intervals and calculator input checks 3
Inverse probabilities and upper cutoffs 4
Unknown parameters and model checks 5
Practice A / B / C: six tasks each 6-8
Separated solutions A / B / C 9-11
Repair checklist, scope and references 12
How to use this guide
First sketch the requested area and label the mean. Write the event as an inequality. Decide whether the
calculator inputs use the original scale or z values. Predict whether the answer should be above or below one
half, then calculate. Use separate paper for full solutions; ruled space is for your setup.
Scope and prerequisites
Selected AQA A-level Mathematics 7357 N2/N3 normal-probability and model-reasoning skills. You need
basic probability, inequalities, algebra and a calculator with normal cumulative/inverse functions. This is a
narrow supplement: no binomial approximation, hypothesis testing, sampling-distribution chapter or
complete-course coverage.
Created with AI assistance. Numerical answers were separately recomputed in code; every PDF page visually reviewed. No
independent expert review, achieved grade, course attendance or exam-board endorsement is claimed. All practice contexts
and diagrams are original and synthetic.
Version 1.0 | 28 September 2026 | Original methods and practice 1
, A-LEVEL MATHEMATICS / NORMAL DISTRIBUTION / INDEPENDENT SUPPLEMENT
01 / RIGHT TAILS
Same area, two valid routes
Notation: X ~ N(mu, s2) means mean mu and variance s squared. Here s is the positive standard deviation.
For Z ~ N(0, 1), write F(z) = P(Z <= z), the area to the LEFT. Standardise using z = (x - mu)/s.
Worked example A: a synthetic strip length
Let X have mean 120 mm and SD 8 mm. Find P(X > 132). The boundary is 12 mm above the mean: z = (132
- 120)/8 = 1.5.
0 (mean) 1.5
Standardised z axis; tails continue beyond plot
The shaded right tail is smaller than half of the entire area.
Route Use these inputs / expression
Original scale Normal CDF with mean 120, SD 8, lower bound 132 and upper bound +infinity.
Standardised scale Mean 0, SD 1, lower bound 1.5 and upper bound +infinity; equivalently 1 - F(1.5).
F(1.5) = 0.9331928, so P(X > 132) = 1 - 0.9331928 = 0.0668 (4 decimal places). This is about 6.68%,
consistent with a threshold above the mean.
Do not standardise the boundary to 1.5 and then leave mean 120 and SD 8 in the calculator. That mixes two scales. Direct
normal and standard-normal calculations agree when every input describes the same model.
Version 1.0 | 28 September 2026 | Original methods and practice 2
METHOD SUMMARY + ORIGINAL PRACTICE
Normal distribution:
setup, tails and cutoffs
A-Level Maths
Four worked examples, four original shaded diagrams and 18 original practice tasks. Translate a verbal event
into bounds before using a calculator. Keep raw values and standardised values consistent.
Study route Pages
Right tail: two equivalent calculation routes 2
Intervals and calculator input checks 3
Inverse probabilities and upper cutoffs 4
Unknown parameters and model checks 5
Practice A / B / C: six tasks each 6-8
Separated solutions A / B / C 9-11
Repair checklist, scope and references 12
How to use this guide
First sketch the requested area and label the mean. Write the event as an inequality. Decide whether the
calculator inputs use the original scale or z values. Predict whether the answer should be above or below one
half, then calculate. Use separate paper for full solutions; ruled space is for your setup.
Scope and prerequisites
Selected AQA A-level Mathematics 7357 N2/N3 normal-probability and model-reasoning skills. You need
basic probability, inequalities, algebra and a calculator with normal cumulative/inverse functions. This is a
narrow supplement: no binomial approximation, hypothesis testing, sampling-distribution chapter or
complete-course coverage.
Created with AI assistance. Numerical answers were separately recomputed in code; every PDF page visually reviewed. No
independent expert review, achieved grade, course attendance or exam-board endorsement is claimed. All practice contexts
and diagrams are original and synthetic.
Version 1.0 | 28 September 2026 | Original methods and practice 1
, A-LEVEL MATHEMATICS / NORMAL DISTRIBUTION / INDEPENDENT SUPPLEMENT
01 / RIGHT TAILS
Same area, two valid routes
Notation: X ~ N(mu, s2) means mean mu and variance s squared. Here s is the positive standard deviation.
For Z ~ N(0, 1), write F(z) = P(Z <= z), the area to the LEFT. Standardise using z = (x - mu)/s.
Worked example A: a synthetic strip length
Let X have mean 120 mm and SD 8 mm. Find P(X > 132). The boundary is 12 mm above the mean: z = (132
- 120)/8 = 1.5.
0 (mean) 1.5
Standardised z axis; tails continue beyond plot
The shaded right tail is smaller than half of the entire area.
Route Use these inputs / expression
Original scale Normal CDF with mean 120, SD 8, lower bound 132 and upper bound +infinity.
Standardised scale Mean 0, SD 1, lower bound 1.5 and upper bound +infinity; equivalently 1 - F(1.5).
F(1.5) = 0.9331928, so P(X > 132) = 1 - 0.9331928 = 0.0668 (4 decimal places). This is about 6.68%,
consistent with a threshold above the mean.
Do not standardise the boundary to 1.5 and then leave mean 120 and SD 8 in the calculator. That mixes two scales. Direct
normal and standard-normal calculations agree when every input describes the same model.
Version 1.0 | 28 September 2026 | Original methods and practice 2
