Predict interval coverage and plan a width reduction

Use the completed interval foundation

Use iid normal observations and known positive population SD so the unrounded interval procedure has its stated coverage. Approximate procedures need their own coverage qualification

Add indicator expectations

For each repetition, record whether its interval contains the fixed population mean. Add those indicators to count the intervals that contain it. Linearity of expectation gives the expected covering and missing counts. Independence between repetitions is not needed for these expectations; a binomial distribution for the covering count would require additional joint assumptions

Work a fractional expected count

With 135 repetitions of an exact 98% interval procedure, calculate the expected covering and missing counts. Every realised count is an integer, but its probability-weighted average need not be. Do not round the expected covering count to 132 unless a different reporting instruction explicitly asks for an approximation

An exact 98% interval procedure is repeated 135 times. Give the exact expected number of intervals containing the population mean

An exact 98% interval procedure is repeated 135 times. Give the exact expected number of intervals missing the population mean

Which claim follows from exact 98% coverage?

Work backwards from the remaining width

At fixed confidence and known population SD, use the fraction of width remaining after the reduction to find the new size threshold. Take the ceiling of the exact threshold. Distinguish the new total size from the increase over the original size

Work a non-square minimum

Starting from a sample size of 40, a reduction of at least 25% leaves three quarters of the original width. The minimum new total is 72 and the increase is 32: 71 fails the exact bound and 72 passes. The achieved reduction can exceed 25% because the new size must be an integer

Starting from a sample size of 40, give the minimum new total size that reduces interval width by at least 25% at unchanged confidence and known population SD

Starting from a sample size of 40, give the minimum increase that reduces interval width by at least 25% at unchanged confidence and known population SD

What does the table establish?

Retain an exact integer boundary

For a starting size of 63 and a required 40% width reduction, the exact threshold is an integer. The minimum is 175, not 176. Using a rounded square root or always adding 1 after division can give an unnecessary observation

Starting from a sample size of 63, give the minimum new total size for a 40% interval-width reduction at unchanged confidence and known population SD

Explain what the expectation does and does not establish, ungraded

A valid interval method has constant coverage C. Derive the expected covering count over R repetitions using indicators. Explain why the expectation may be fractional, why it is not the actual count and why independence is unnecessary for its mean. State what changes if the actual coverage is only approximately C.

Model: separate an expectation from a guarantee

Each indicator has mean C, so linearity gives RC and expected misses R(1−C), regardless of dependence between indicators. K takes integer values, but E(K) averages them with probability weights. It need not be attained in one run. If actual coverage is C* rather than the nominal C, the expectation is RC*; RC is only approximate when C*≈C. A binomial model needs suitable independent equal-probability indicators

Derive and audit a precision plan, ungraded

Derive N≥n₀/q² from the width formula with 0<q<1. Prove that the ceiling is minimal, distinguish total and additional sample sizes, and explain why this is a design comparison rather than a guarantee about realised estimation error. Discuss changing confidence or using an observed sample SD

Model: preserve the fixed inputs and the positive inequality

Cancel the common positive 2zσ, obtaining √(n₀/N)≤q. Squaring and rearranging with positive N and q gives N≥n₀/q². For N=ceil(n₀/q²), N satisfies the bound while N−1 does not. The increase is N−n₀. With changed z or σ the cancelled factor is no longer common; if σ is estimated by different realised s values, larger n alone does not guarantee every observed interval is shorter. Width controls the planned interval construction, not the realised difference between x̄ and μ