Construct and interpret confidence intervals for a population mean

State the target and the sampling model

The target is a population mean μ. The observed sample mean x̄ estimates it; the standard error measures how that statistic varies under repeated sampling. This interval does not describe the range of individual observations. Keep the target population, sampling method and any dependence assumptions visible

Turn a central normal probability into an interval

For iid normal observations with known σ, Z=(X̄-μ)/(σ/√n) is standard normal. Choose z>0 so Pr(-z≤Z≤z)=C. Rearranging gives Pr(X̄-zσ/√n≤μ≤X̄+zσ/√n)=C. Before sampling the endpoints are random and μ is fixed

Split the excluded area between two tails

For confidence C, each tail has probability (1-C)/2, so the positive critical value uses less-than area (1+C)/2. For 95%, the two tails are 0.025 each and Φ(z)=0.975, giving z≈1.959964. For 90% and 99%, positive critical values are ≈1.644854 and 2.575829. Use calculator quantiles, not exact empirical-rule percentages

For the upper-tail statement below, which less-than area should be used to find z?

For a 90% equal-tail interval, what less-than area gives the positive critical value?

Construct a known-sigma mean interval

An iid normal sample has n=64, x̄=24.6 and known σ=4. The SE is 4/8=0.5. A 95% interval has margin z×0.5≈0.979982 and endpoints ≈23.620018, 25.579982. Report [23.62, 25.58] to two decimal places. Retain the full quantile until the final reporting step

For an iid normal sample with n=64 and known σ=4, give the standard error

For a 95% mean interval with x̄=24.6, known σ=4 and n=64, give the lower endpoint to two decimal places

Use estimated sigma for an approximate large-sample interval

If σ is unknown, use the observed sample SD s in the large-sample construction x̄±z s/√n, provided the sampling and normal-approximation conditions are adequate. For n=100, x̄=15.2, s=6, the estimated SE is 0.6. A nominal 95% interval has endpoints ≈14.024022, 16.375978, reported [14.02, 16.38]. Its coverage is approximate. Large n does not fix bias or dependence

For n=100 and observed sample SD s=6, give the estimated standard error

Separate centre, margin and width

For an unrounded interval [8.5, 11.5], the centre is 10, the margin is 1.5 and total width 3. These endpoints alone do not determine the confidence level; you also need the sampling model and standard error. The midpoint is the realised sample mean in this symmetric construction, not a known value of μ

Give the margin of the interval [8.5, 11.5]

Plan a sample size using an unrounded margin

For known σ=5, a 95% interval and target margin at most 1, require n≥(z×5/1)²≈96.0365. The minimum integer is 97. At n=96 the margin is slightly greater than 1; at 97 it is less. Round the required size upward, using the unrounded z before taking the ceiling. A pilot estimate substituted for σ would make this a planning approximation

With known σ=5, 95% confidence and a target margin of at most 1, give the minimum integer sample size

Compare widths while keeping other inputs fixed

At fixed confidence and known σ, width is 2 zσ/√n. Quadrupling n halves width. At fixed n and σ, higher confidence increases z and widens the interval. With unknown σ, replacing s by a different observed s also changes width; larger n alone need not make every realised estimated-sigma interval shorter

At fixed confidence and known σ, what is new width divided by old width when n is quadrupled?

Interpret confidence before and after sampling

In the valid known-sigma normal model, a C-level procedure covers fixed μ with probability C before the sample is drawn. After one interval is observed, its endpoints and μ are fixed: it either contains μ or does not. The frequentist confidence level describes the procedure; it does not itself assign a posterior probability to μ or assert that C percent of individual values lie inside

Which statement is supported by a 95% confidence method?

Count coverage with a fixed reference parameter

For a known simulation value μ=10, consider supplied intervals [8, 11], [10, 12], [6, 9], [9.5, 10.5]. Counting equality as containment, three cover and one misses, so the fixture coverage is 3/4. This checks the counting rule; four supplied intervals are not a claim about a nominal confidence level

For fixed μ=10, give the coverage fraction of [8, 11], [10, 12], [6, 9] and [9.5, 10.5], counting endpoint equality as containment

Technology: simulate complete confidence intervals

The Python program takes 1000 fresh samples in each of three predeclared runs: normal μ=10, σ=2 at n=4 and 49, and a skew population [0, 0, 0, 0, 5] at n=49 with μ=1, σ=2. It retains sample means, unbiased sample SDs, and intervals at 80%, 90%, 95%, 99%. Known-sigma intervals are computed in every run; estimated-sigma z intervals are included only for n≥30 and labelled approximate

Technology: runnable Python source

import random, math, statistics, json LEVELS = [0.80, 0.90, 0.95, 0.99] def interval_run(model, n, R, seed): if model not in ['normal', 'skew'] or type(n) is not int or n < 2: raise ValueError('Use a listed model and integer n >= 2.') if type(R) is not int or R < 1: raise ValueError('Use a positive integer repeat count.') mu, sigma = (10, 2) if model == 'normal' else (1, 2) rng = random.Random(seed) critical = [statistics.NormalDist().inv_cdf((1+C)/2) for C in LEVELS] records = [] for repetition in range(R): sample = [rng.gauss(mu, sigma) if model == 'normal' else rng.choice([0, 0, 0, 0, 5]) for _ in range(n)] centre = sum(sample)/n s = math.sqrt(sum((x-centre)**2 for x in sample)/(n-1)) # Do not drop a sample just because its observed spread is small known = [[centre-z*sigma/math.sqrt(n), centre+z*sigma/math.sqrt(n)] for z in critical] estimated = ([[centre-z*s/math.sqrt(n), centre+z*s/math.sqrt(n)] for z in critical] if n >= 30 else []) records.append(dict(mean=centre, s=s, known=known, estimated=estimated)) coverage = lambda kind: [sum(row[kind][j][0] <= mu <= row[kind][j][1] for row in records)/R for j in range(4)] return dict(model=model, n=n, R=R, seed=seed, mu=mu, sigma=sigma, levels=LEVELS, critical=critical, known_coverage=coverage('known'), estimated_coverage=coverage('estimated') if n >= 30 else [], records=records) if __name__ == '__main__': print(json.dumps([interval_run('normal', 4, 1000, 8400), interval_run('normal', 49, 1000, 8401), interval_run('skew', 49, 1000, 8402)]))

Retain the full output and its provenance

The displayed reference runs were generated with this exact program in Python 3.14.7, using seeds 8400, 8401, 8402. Retain model, n, R, seed, runtime, all sample summaries and all interval endpoints. The program sets a random source once per run and does not discard small observed SDs or inconvenient misses. Replaying a seed repeats the same run; distinct documented runs are needed for additional simulation evidence

What does the table’s empirical coverage measure?

Read the known and estimated results separately

For the normal n=4 run, the observed known-sigma coverage is 0.805, 0.905, 0.955, 0.990 across the four levels. These are close to, but need not equal, the nominal levels. For the skew n=49 run, the nominal 95% estimated-sigma z procedure covers 0.933 of the 1000 cases; the corresponding known-sigma figure is 0.956. These finite results illustrate both simulation variation and the need to assess approximations. They do not prove exact coverage for either non-normal construction

In this 90% interval plot, which quantity stays fixed across samples?

What changes in the 99% plot of the same 12 samples?

Derive and interpret the interval procedure, ungraded

Starting from the exact normal distribution of X̄ for iid normal observations and known σ, derive a C-level two-sided mean interval. Identify what is fixed and random before sampling, and explain why the statement differs from a probability distribution assigned to an unknown parameter after data

Model: invert the pivot while keeping μ fixed

Choose z=Φ⁻¹((1+C)/2). Then Pr(-z≤(X̄-μ)/(σ/√n)≤z)=C. Multiplying by the positive SE and rearranging gives Pr(X̄-zSE≤μ≤X̄+zSE)=C. The population mean and known σ are fixed, while X̄ and both endpoints are random. For the realised sample, substitute x̄ to obtain numerical endpoints. The interval either covers the fixed μ or misses it. Its confidence label records the long-run procedure, without by itself specifying a posterior probability for μ. Endpoint rounding is a reporting approximation to the derived unrounded procedure

Implement and assess repeated interval coverage, ungraded

Run the supplied experiment or implement it independently. Verify sample means and SDs, retain all runs, check the fixed-μ containment rule, and compare coverage across levels and between known and estimated σ. Plot predeclared interval indices on common axes and explain why empirical coverage need not equal the nominal level

Model: audit the procedure and report all outcomes

Check each s uses denominator n-1; known SE uses σ/√n and estimated SE uses s/√n. Compute each critical value from (1+C)/2 with full precision. A record covers when its lower endpoint ≤ the fixed model μ ≤ its upper endpoint. For the same sample, higher confidence produces nested intervals, so empirical coverage cannot decrease across the four levels. Report all 1000 intervals per run and all stated levels. The normal known-sigma model has exact nominal coverage before reporting rounding; the skew known-sigma and large-sample estimated-sigma z methods are approximations. A finite difference from the nominal level can reflect simulation variation and approximation error. Replaying the identical stream is not a new independent check

Explain precision and sample-size planning, ungraded

A study uses known σ=5 and wants a 95% mean interval with margin≤1. Derive its minimum sample size and explain how the width changes if confidence increases or n quadruples. Discuss what must be reconsidered if σ comes from a pilot sample or observations are dependent

Model: keep the formula’s assumptions attached

From zσ/√n≤1 obtain n≥(5 z)²≈96.0365 and minimum 97. Check 96 fails the unrounded inequality. At fixed σ, n, higher confidence increases z and width; quadrupling n at fixed confidence halves width. A pilot s is uncertain, so substituting it for σ makes the planning calculation approximate. Dependence changes the variance of the mean; it cannot be repaired just by treating the same observations as an iid sample. Selection bias and the target population need separate justification

Interpret an approximate reported interval, ungraded

An iid sample of 100 measurements has x̄=15.2, s=6 and reported nominal 95% z interval [14.02, 16.38]. Explain the target, construction, approximation conditions and limits of the conclusion. State whether it claims that 95% of individual measurements lie in that interval

Model: report an interval for the mean, not individual spread

The target is the specified population mean μ. Estimated SE is 6/√100=0.6; using z≈1.959964 gives unrounded endpoints 14.024022… and 16.375978…. The nominal 95% estimated-sigma z construction relies on appropriate independent sampling and an adequate normal approximation for X̄. Its coverage is approximate, not an unconditional small-sample theorem. The method’s confidence level concerns repeated interval coverage of μ. It does not place 95% of individual measurements inside the interval, establish representativeness or remove systematic measurement error

Assessment boundary

Practice grades defined calculations, critical-area choices, finite coverage counts and explicit interpretation alternatives. Full proof, independent simulation, contextual inference and validation of sampling assumptions remain ungraded. Hypothesis testing next uses the same standard-error and tail-probability foundations