Statistics Lab / Appendix

The parity ledger

A port is only useful if it computes the same thing. Every number in this table was written down in the 2020 submission; the right-hand column is the TypeScript port recomputing it from the original inputs and printing it at the same precision. The comparison runs in the test suite on every push.

Printed R outputs reproduced exactly: 34 / 34
Hand calculations agreeing to their rounding: 6 / 6
Closed-form derivations evaluated: 2 / 2
Reference implementation: R 4.6.1

42 values in all: 34 printed R outputs, 6 hand calculations and 2 symbolic derivations. marks an identical string, ≈ agreement to the table rounding the hand working used, and = a closed form evaluated numerically.

How it was checked

Reference values. A script in the repository (scripts/r_reference.R) re-runs the original computations in R — the same functions, data and seeds as the 2020 notebooks — and writes the results to a test fixture: summaries, quantiles, intervals, test statistics, the full lm fit with its diagnostics, and a grid of distribution functions.

Two levels of comparison. Internally every quantity must agree with R to about nine significant figures. The ledger then checks the stricter, human-facing claim: formatted at the precision printed in 2020, the port’s value is the same string. Even the full summary(lm) printout is reproduced character for character.

Hand calculations used rounded table values (for example t₉ = 2.262), so they agree to the last digit or two, not exactly; the ledger shows both.

Reproducing R’s random numbers

The midrange-versus-mean simulation (A1 Q4) depends on set.seed(2529). To replay it, the port re-implements R’s default generator: the seed-scrambling LCG, the Mersenne-Twister state, R’s guard against 0 and 1, and normal draws by inversion from two uniforms (2²⁷u₁ + u₂ fed to Wichura’s AS241). The first three uniforms to 17 significant digits (the port’s column is computed when this page is built):

The uniform stream is bit-identical to R’s over 100,000 draws. Normal and exponential draws can differ from R 4.6 on Apple silicon in the last bit or two (at most 4 and 1 units in the last place), because R’s build fuses multiply-adds inside qnorm and JavaScript cannot. That is far below any printed precision, so every 2020 value still matches.

1R 0.9589315231423825TS0.9589315231423825
2R 0.75501735182479024TS0.75501735182479024
3R 0.017361181322485209TS0.017361181322485209

First three uniform draws after set.seed(2529) in R and in the port
Draw	R 4.6	TypeScript port
1	0.9589315231423825	0.9589315231423825
2	0.75501735182479024	0.75501735182479024
3	0.017361181322485209	0.017361181322485209

# R
> set.seed(2529); sprintf("%.17g", runif(3))

// TypeScript (src/lib/rng.ts)
new RRandom(2529).runif(3)

Assignment 1 — estimation

Median plant heightQ1(a)
2020
68.95
Port
68.95
R output
Mean plant heightQ1(a)
2020
98.17
Port
98.17
R output
Upper quartile (Type 7)Q1(a)
2020
133.57
Port
133.57
R output
Interquartile rangeQ1(a)
2020
113.55
Port
113.55
R output
Exponential MLE θ̂ = x̄Q1(b)
2020
98.16667
Port
98.16667
R output
Standard error of λ̂Q2(c)i
2020
0.788583
Port
0.788583
R output
95% CI for λQ2(c)ii
2020
(1.118327, 5.189367)
Port
(1.118327, 5.189367)
R output
Midrange bias (seed 2529)Q4
2020
0.043088
Port
0.043088
R output
Sample-mean bias (seed 2529)Q4
2020
-0.020101
Port
-0.020101
R output
Midrange variance (seed 2529)Q4
2020
0.158578
Port
0.158578
R output
Sample-mean variance (seed 2529)Q4
2020
0.069188
Port
0.069188
R output

Assignment 1 — estimation: values from the 2020 submission and the port
Question	Quantity	2020 submission	TypeScript port	Source
Q1(a)	Median plant height	68.95	68.95	R output
Q1(a)	Mean plant height	98.17	98.17	R output
Q1(a)	Upper quartile (Type 7)	133.57	133.57	R output
Q1(a)	Interquartile range	113.55	113.55	R output
Q1(b)	Exponential MLE θ̂ = x̄	98.16667	98.16667	R output
Q2(c)i	Standard error of λ̂	0.788583	0.788583	R output
Q2(c)ii	95% CI for λ	(1.118327, 5.189367)	(1.118327, 5.189367)	R output
Q4	Midrange bias (seed 2529)	0.043088	0.043088	R output
Q4	Sample-mean bias (seed 2529)	-0.020101	-0.020101	R output
Q4	Midrange variance (seed 2529)	0.158578	0.158578	R output
Q4	Sample-mean variance (seed 2529)	0.069188	0.069188	R output

Assignment 2 — intervals, regression and testing

X-squared statisticQ1(b)
2020
4.7269
Port
4.7269
R output
p-valueQ1(b)
2020
0.02969
Port
0.02969
R output
95% CI for p₁ − p₂Q1(c)
2020
(0.005118323, 0.094881677)
Port
(0.005118323, 0.094881677)
R output
95% t interval for μXQ2(a)
2020
(12.7897, 17.3303)
Port
(12.7895, 17.3305)agrees to table rounding
Hand calculation
Experiments for width 2 (σ = 3)Q2(b)
2020
35
Port
35
Hand calculation
95% CI for μX − μYQ2(c)
2020
(-7.71898, -2.81102)
Port
(-7.71884, -2.81116)agrees to table rounding
Hand calculation
F statisticQ2(e)
2020
15.28
Port
15.28
R output
F test p-valueQ2(e)
2020
0.00163
Port
0.00163
R output
95% CI for σ²X / σ²YQ2(e)
2020
(3.167976, 64.130185)
Port
(3.167976, 64.130185)
R output
Intercept estimateQ3(a)
2020
-32.3442
Port
-32.3442
R output
Slope (sales per customer)Q3(a)
2020
6.4005
Port
6.4005
R output
Slope standard errorQ3(a)
2020
0.6345
Port
0.6345
R output
Residual standard errorQ3(a)
2020
64.6
Port
64.6
R output
Multiple R-squaredQ3(a)
2020
0.8497
Port
0.8497
R output
F-statistic (1 and 18 DF)Q3(a)
2020
101.7
Port
101.7
R output
Overall p-valueQ3(a)
2020
7.816e-09
Port
7.816e-09
R output
95% CI for the interceptQ3(b)
2020
(-163.027250, 98.338933)
Port
(-163.027250, 98.338933)
R output
95% CI for the slopeQ3(b)
2020
(5.067368, 7.733558)
Port
(5.067368, 7.733558)
R output
Mean sales at 100 customers, 95% CIQ3(c)
2020
607.7022 (576.7281, 638.6762)
Port
607.7022 (576.7281, 638.6762)
R output
Sales at 100 customers, 95% PIQ3(d)
2020
607.7022 (468.4949, 746.9094)
Port
607.7022 (468.4949, 746.9094)
R output
Type I error e⁻²Q5(a)
2020
e^-2
Port
0.1353closed form evaluated numerically
Derivation
Power e^(−4/5)Q5(c)
2020
e^-0.8
Port
0.4493closed form evaluated numerically
Derivation
Size-0.05 critical value kQ5(d)
2020
5.991
Port
5.991
Hand calculation

Assignment 2 — intervals, regression and testing: values from the 2020 submission and the port
Question	Quantity	2020 submission	TypeScript port	Source
Q1(b)	X-squared statistic	4.7269	4.7269	R output
Q1(b)	p-value	0.02969	0.02969	R output
Q1(c)	95% CI for p₁ − p₂	(0.005118323, 0.094881677)	(0.005118323, 0.094881677)	R output
Q2(a)	95% t interval for μX	(12.7897, 17.3303)	agrees to table rounding(12.7895, 17.3305)	Hand calculation
Q2(b)	Experiments for width 2 (σ = 3)	35	35	Hand calculation
Q2(c)	95% CI for μX − μY	(-7.71898, -2.81102)	agrees to table rounding(-7.71884, -2.81116)	Hand calculation
Q2(e)	F statistic	15.28	15.28	R output
Q2(e)	F test p-value	0.00163	0.00163	R output
Q2(e)	95% CI for σ²X / σ²Y	(3.167976, 64.130185)	(3.167976, 64.130185)	R output
Q3(a)	Intercept estimate	-32.3442	-32.3442	R output
Q3(a)	Slope (sales per customer)	6.4005	6.4005	R output
Q3(a)	Slope standard error	0.6345	0.6345	R output
Q3(a)	Residual standard error	64.6	64.6	R output
Q3(a)	Multiple R-squared	0.8497	0.8497	R output
Q3(a)	F-statistic (1 and 18 DF)	101.7	101.7	R output
Q3(a)	Overall p-value	7.816e-09	7.816e-09	R output
Q3(b)	95% CI for the intercept	(-163.027250, 98.338933)	(-163.027250, 98.338933)	R output
Q3(b)	95% CI for the slope	(5.067368, 7.733558)	(5.067368, 7.733558)	R output
Q3(c)	Mean sales at 100 customers, 95% CI	607.7022 (576.7281, 638.6762)	607.7022 (576.7281, 638.6762)	R output
Q3(d)	Sales at 100 customers, 95% PI	607.7022 (468.4949, 746.9094)	607.7022 (468.4949, 746.9094)	R output
Q5(a)	Type I error e⁻²	e^-2	closed form evaluated numerically0.1353	Derivation
Q5(c)	Power e^(−4/5)	e^-0.8	closed form evaluated numerically0.4493	Derivation
Q5(d)	Size-0.05 critical value k	5.991	5.991	Hand calculation

Assignment 3 — nonparametric and likelihood methods

Days with cases ≥ 15Q1(a)
2020
14
Port
14
R output
Sign test p-valueQ1(a)
2020
0.9608231
Port
0.9608231
R output
Wilcoxon WQ1(b)
2020
38
Port
38
R output
Wilcoxon p-valueQ1(b)
2020
0.04798
Port
0.04798
R output
Type 7 sample quantile π̂₀.₂₅Q2(b)
2020
1.855
Port
1.855
Hand calculation
Standard error of π̂₀.₂₅Q2(d)
2020
0.706
Port
0.706
Hand calculation
Lower critical value of YQ5(c)
2020
37.11096
Port
37.11096
R output
Upper critical value of YQ5(c)
2020
64.7806
Port
64.7806
R output

Assignment 3 — nonparametric and likelihood methods: values from the 2020 submission and the port
Question	Quantity	2020 submission	TypeScript port	Source
Q1(a)	Days with cases ≥ 15	14	14	R output
Q1(a)	Sign test p-value	0.9608231	0.9608231	R output
Q1(b)	Wilcoxon W	38	38	R output
Q1(b)	Wilcoxon p-value	0.04798	0.04798	R output
Q2(b)	Type 7 sample quantile π̂₀.₂₅	1.855	1.855	Hand calculation
Q2(d)	Standard error of π̂₀.₂₅	0.706	0.706	Hand calculation
Q5(c)	Lower critical value of Y	37.11096	37.11096	R output
Q5(c)	Upper critical value of Y	64.7806	64.7806	R output

The ledger is generated by src/lib/__tests__/parity.test.ts in the web app; the original files are in coursework/ of the project repository (private for now).