MAST20005 Statistics Lab

§3.1Coursework A2 Q2(a), Q4

Ninety-five out of a hundred

With σ unknown, the interval for a normal mean replaces σ by the sample standard deviation and pays for it with a heavier-tailed critical value:

\bar{x} \pm t_{n-1,\;1-\alpha/2}\,\frac{s}{\sqrt{n}}

It works because $T=\frac{\bar X-\mu}{S/\sqrt n}\sim t_{n-1}$ whatever the values of μ and σ — the pivot identified in Q4. Draw many samples and the intervals scatter around the truth; about 95% of them catch it.

Random seed (R’s set.seed)

Sample size n10

Number of intervals100

Confidence level95%

Interval

Each sample is rnorm(10, 15, 3). With seed 1, n = 10 and 100 t intervals, R’s own loop over t.test() also gives 99 hits.

Intervals covering μ

99 / 100

Intervals missing μ

1

Observed coverage

99.0%

Wilson 95% 94.6% to 99.8%

Nominal coverage

95%

expect ≈ 5 misses

Interval covers μ = 15
Interval misses μ (marked ×)
True mean μ

Figure 3.1.100 independent 95% t intervals, each from a fresh sample of size 10. The procedure, not any single interval, carries the 95% guarantee: in the long run that share of intervals catches the fixed but unknown μ.

Show the numbers as a table

Simulated intervals
#	x̄	lower	upper	covers μ
1	15.397	13.721	17.072	yes
2	15.747	13.451	18.042	yes
3	14.599	12.548	16.650	yes
4	15.362	13.627	17.097	yes
5	15.402	14.124	16.681	yes
6	15.430	13.253	17.608	yes
7	16.354	13.588	19.119	yes
8	14.257	12.906	15.608	yes
9	15.382	13.662	17.102	yes
10	15.337	13.078	17.596	yes
11	16.042	14.031	18.054	yes
12	14.641	13.283	15.998	yes
13	14.981	13.695	16.268	yes
14	13.715	12.222	15.208	yes
15	13.333	10.521	16.146	yes
16	14.246	11.905	16.586	yes
17	15.631	13.471	17.790	yes
18	17.257	15.305	19.209	no
19	14.463	12.385	16.541	yes
20	14.557	12.640	16.474	yes
21	16.581	13.781	19.382	yes
22	15.009	13.036	16.982	yes
23	13.940	11.718	16.163	yes
24	13.909	11.465	16.353	yes
25	15.091	12.925	17.257	yes
26	15.056	12.974	17.138	yes
27	15.073	13.116	17.029	yes
28	15.382	12.392	18.373	yes
29	15.224	13.416	17.033	yes
30	15.624	13.769	17.480	yes
31	16.365	14.062	18.668	yes
32	14.898	13.293	16.504	yes
33	15.696	13.506	17.886	yes
34	14.870	13.047	16.694	yes
35	14.812	12.138	17.487	yes
36	15.321	12.985	17.657	yes
37	14.223	11.772	16.674	yes
38	14.821	13.157	16.486	yes
39	14.314	11.790	16.838	yes
40	16.227	14.416	18.039	yes
41	15.380	13.331	17.429	yes
42	15.503	13.387	17.620	yes
43	14.203	11.720	16.686	yes
44	14.451	12.375	16.527	yes
45	13.642	10.550	16.735	yes
46	13.378	11.182	15.575	yes
47	16.897	14.986	18.808	yes
48	14.909	12.804	17.013	yes
49	14.923	11.837	18.010	yes
50	15.539	11.974	19.104	yes
51	14.885	12.676	17.094	yes
52	13.866	11.736	15.996	yes
53	14.701	12.936	16.467	yes
54	15.707	13.693	17.721	yes
55	14.925	12.891	16.958	yes
56	14.110	11.805	16.416	yes
57	14.770	13.152	16.388	yes
58	15.450	12.804	18.097	yes
59	16.008	13.973	18.043	yes
60	14.242	12.002	16.481	yes
61	15.439	13.133	17.744	yes
62	14.514	11.960	17.067	yes
63	15.782	13.193	18.370	yes
64	13.812	12.105	15.519	yes
65	13.221	10.924	15.518	yes
66	14.125	11.276	16.974	yes
67	14.703	12.131	17.275	yes
68	13.381	11.448	15.314	yes
69	14.869	12.744	16.994	yes
70	14.168	11.732	16.605	yes
71	14.959	12.198	17.719	yes
72	15.371	12.551	18.191	yes
73	14.989	12.931	17.047	yes
74	15.237	13.470	17.004	yes
75	14.963	11.779	18.146	yes
76	14.091	11.236	16.947	yes
77	15.607	13.604	17.610	yes
78	15.378	13.211	17.545	yes
79	14.900	13.066	16.735	yes
80	14.512	11.925	17.099	yes
81	14.059	11.843	16.276	yes
82	14.695	11.710	17.680	yes
83	15.349	13.075	17.622	yes
84	16.076	13.584	18.568	yes
85	15.620	12.232	19.008	yes
86	13.203	11.381	15.024	yes
87	15.239	13.304	17.174	yes
88	15.534	13.947	17.121	yes
89	15.828	13.648	18.008	yes
90	14.699	12.457	16.941	yes
91	14.427	12.569	16.285	yes
92	15.573	13.323	17.824	yes
93	15.555	13.285	17.825	yes
94	14.857	12.043	17.670	yes
95	16.559	14.605	18.512	yes
96	15.155	13.118	17.193	yes
97	13.768	10.466	17.071	yes
98	14.607	11.688	17.525	yes
99	15.599	14.448	16.750	yes
100	14.021	12.466	15.577	yes

With 100 intervals the observed coverage is itself an estimate. The Wilson interval above shows how far it could plausibly sit from the true coverage. Appendix B runs this experiment 5,000 times per setting, for skewed data too.

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§3.2Coursework A2 Q2(b)

How many walks are enough?

When σ is known the half-width of a z interval is $E = z_{1-\alpha/2}\,\sigma/\sqrt n$ . Solving for n gives the planning formula

n = \left\lceil \left(\frac{z_{1-\alpha/2}\,\sigma}{E}\right)^{2} \right\rceil .

With σ = 3 minutes and a target of ±1 minute, (1.96 × 3)² ≈ 34.57, so 35 walks are needed.

Known σ (minutes)3.0

Half-width E (± minutes)1.00

Confidence level95%

Critical value z

1.960

(zσ / E)²

34.57

Walks needed

35

round up to a whole sample

Figure 3.2.Required sample size against the target half-width for σ = 3.0. Halving the width quadruples the number of observations — the 1/√n law behind every interval in this chapter.

§3.3Coursework A2 Q2(c)–(e)

Two routes, two samples

For independent normal samples with a common variance, pool the two sample variances,

s_p^2=\frac{(n_X-1)s_X^2+(n_Y-1)s_Y^2}{n_X+n_Y-2},

and the interval for the difference in means is

\bar x-\bar y \pm t_{n_X+n_Y-2,\;1-\alpha/2}\; s_p\sqrt{\tfrac{1}{n_X}+\tfrac{1}{n_Y}} .

The equal-variance assumption can itself be examined: $\frac{S_X^2/\sigma_X^2}{S_Y^2/\sigma_Y^2}\sim F_{n_X-1,\,n_Y-1}$ , which inverts to an interval for $\sigma_X^2/\sigma_Y^2$ — what var.test() reports.

Sample X

10 values

Sample Y

8 values

Confidence level95%

Defaults are simulated (X ~ N(15, 3²), Y ~ N(20.3, 0.8²)) to mimic the shape of the assignment data. Paste your own values to analyse them.

95% CI for μX − μY

(-6.84, -1.85)

pooled sp = 2.486, df = 16

F = s²X / s²Y

9.881

df = (9, 7)

95% CI for σ²X / σ²Y

(2.049, 41.47)

F test p-value

0.00638

variances differ

Sample X
Sample Y
Mean with its t interval

Figure 3.3.Each dot is one walk. The bar under each row is that sample’s own 95% t interval for its mean. The pooled interval for the difference assumes equal variances — the F test on the right checks how plausible that is, and here it is doubtful.

Show the numbers as a table

sample	n	mean	sd	t interval
X (route A)	10	16.280	3.191	(13.997, 18.563)
Y (route B)	8	20.625	1.015	(19.776, 21.474)

§3.4Coursework A2 Q1

Do two councils differ?

For large samples the difference of sample proportions is approximately normal. The test of $\;H_0: p_1 = p_2\;$ uses the pooled proportion in its standard error, and its square is the chi-squared statistic that prop.test() prints; the interval uses each group’s own proportion:

\hat p_1-\hat p_2 \pm z_{1-\alpha/2}\,\widehat{\mathrm{se}},

\widehat{\mathrm{se}}=\sqrt{\frac{\hat p_1(1-\hat p_1)}{n_1}+\frac{\hat p_2(1-\hat p_2)}{n_2}} .

Yes, group 1

Asked, group 1

Yes, group 2

Asked, group 2

Confidence level95%

Defaults are the survey counts from the assignment (support for more bicycle lanes in two inner-Melbourne councils). Equivalent to prop.test(…, correct = FALSE).

p̂₁ − p̂₂

0.0500

0.650 vs 0.600

X² (= z²)

4.7269

z = 2.174

p-value

0.02969

reject H₀ at α = 0.05

95% CI for p₁ − p₂

(0.0051, 0.0949)

Estimate and interval for p₁ − p₂
No difference (H₀)

Figure 3.4.Here the test and the interval tell the same story: the interval excludes zero and the two-sided test rejects at the matching level. (The test pools the proportions for its standard error; the interval does not, so the two can disagree near the boundary.)