Chapter 12 — Probability and statistics
Lesson map
Lesson 12 published. Map, example, EX034.
flowchart TD
N0["Learning outcomes"]
N1["12.1 Proportions and intervals"]
N2["Lab — Wilson interval for a reporting proportion"]
N3["12.2 Comparing proportions"]
N4["12.3 Small samples"]
N5["Synthesis"]
N0 --> N1 --> N2 --> N3 --> N4 --> N5
N5 --> EX["Worked example"]
EX --> Q["Exercise EX034"]
Logical drawing
flowchart LR
N0["Learning outcomes"]
N1["12.1 Proportions and intervals"]
N2["Lab — Wilson interval for a reporting proportion"]
N3["12.2 Comparing proportions"]
N4["12.3 Small samples"]
N5["Synthesis"]
N0 --> N1 --> N2 --> N3 --> N4 --> N5 --> Q["EX034"]
[Learning outcomes]
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v
[12.1 Proportions and intervals]
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v
[Lab — Wilson interval for a reporting proportion]
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v
[12.2 Comparing proportions]
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v
[12.3 Small samples]
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v
[Synthesis]
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v
[Worked example] --> [Exercise EX034]
Learning outcomes
You will (1) estimate a proportion with a confidence interval; (2) compare two proportions cautiously; (3) explain why small samples mislead.
12.1 Proportions and intervals
Ashcombe ran an announced, staff-agreed simulation in which 120 staff received one practice email. Of these, 18 reported it within an hour. The proportion reporting is 15%. Nadia learns that a single number hides uncertainty, so she computes a 95% confidence interval using the Wilson method, which behaves well with small counts. The interval shows the range of rates consistent with the data. She reports the rate together with its interval so that managers do not treat a single estimate as exact. The interval is 9.7% to 22.5%.
Lab — Wilson interval for a reporting proportion
# Wilson 95% interval for reporting proportions (synthetic).
from math import sqrt
def wilson(k, n, z=1.96):
p = k / n; d = 1 + z * z / n; c = (p + z * z / (2 * n)) / d; h = z * sqrt(p * (1 - p) / n + z * z / (4 * n * n)) / d
return c - h, c + h
for name, k, n in (("all staff", 18, 120), ("Central", 6, 30), ("Riverside", 3, 24)):
lo, hi = wilson(k, n); print(f"{name}: {k}/{n} = {k / n:.1%}, 95% CI {lo:.1%} to {hi:.1%}")
Run with python3 lab_ci.py. Verified output:
all staff: 18/120 = 15.0%, 95% CI 9.7% to 22.5%
Central: 6/30 = 20.0%, 95% CI 9.5% to 37.3%
Riverside: 3/24 = 12.5%, 95% CI 4.3% to 31.0%
12.2 Comparing proportions
Two branches show different reporting rates: Central 6 of 30 (20%, interval 9.5% to 37.3%) and Riverside 3 of 24 (12.5%, interval 4.3% to 31.0%). Nadia compares them and finds that their intervals overlap heavily. The apparent difference could easily arise by chance. She resists the temptation to praise one branch and criticise the other. Comparing groups fairly requires attention to sample size, and in human factors work it also requires care, because public comparisons between teams can create pressure and resentment that damage the reporting culture the programme seeks to build.
12.3 Small samples
With small branches, one person's action changes the rate greatly. At Quay, with five staff, one extra report moves the rate by twenty points. Nadia explains this to managers using the lab output and suggests reporting branch results only when at least twenty staff are included, or combining small branches. This rule protects both the accuracy of the figures and the privacy of staff in small teams, where individual actions could otherwise be guessed from a group rate. Managers accept the rule.
Worked example
Nadia computes a Wilson interval for 18 of 120, compares two branches cautiously and sets a minimum group size of twenty.
Synthetic case. Data are synthetic.
Errors and limits
Intervals assume independent responses. One simulation is a single snapshot.
Exercises
- EX034 — Overall interval. What was the 95% interval for 18 of 120? Rubric: 3 = about 9.7% to 22.5%;; 2 = range;; 1 = 15% exactly;; 0 = none.
- EX035 — Branch comparison. Why not praise Central over Riverside? Rubric: 3 = their intervals overlap heavily;; 2 = chance;; 1 = vague;; 0 = none.
- EX036 — Minimum group. What minimum group size did Nadia suggest? Rubric: 3 = twenty staff;; 2 = 20;; 1 = five;; 0 = none.
Synthesis
Proportions need intervals, comparisons need care and small groups need protecting.
Continue studying — chapter 12 of 40
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