INF-2: Solutions — Confidence Intervals for a Population Mean (Large Sample)

Problem 1 — Constructing a Confidence Interval (Variants A–E)

Each variant follows the same four-step workflow: check conditions → identify → compute SE and → build the interval.

• Variant A (coffee shop, , n = 64, , 95% CI): , → 95% CI: .
• Variant B (exam scores, , n = 36, , 99% CI): , → 99% CI: .
• Variant C (delivery times, , n = 49, , 90% CI): , → 90% CI: minutes.
• Variant D (heart rate, , n = 100, , 95% CI): , → 95% CI: bpm.
• Variant E (battery life, , n = 81, , 99% CI): , → 99% CI: hours.

Problem 2 — Correct Interpretation (Variants A–E)

All five variants test the same idea: a confidence interval describes a property of the procedure, not a probability statement about the fixed parameter .

• Variant A (step count CI (8,240, 9,760)): Correct — “we used a procedure that captures the true mean in about 95% of all samples drawn this way.” Wrong — assigning 95% probability to ‘s location (it’s fixed) or claiming 95% of individuals fall in the range.
• Variant B (exam CI (71.3, 76.7)): “90% sure” implies has a 90% chance of being inside. Correct frequentist phrasing: “90% of intervals from this procedure capture .”
• Variant C (wait time CI (14.2, 17.8)): False — the CI is about the mean, not individual customers, whose wait times vary far more widely.
• Variant D (temperature CI (36.1, 36.9)): Correct — “if we repeated this sampling procedure, about 99% of the resulting intervals would contain the true mean.” The others wrongly assign probability to a fixed .
• Variant E (probability is 0 or 1): True. Once the interval is computed, is either inside (probability 1) or not (probability 0). The 95% is the long-run frequency of the procedure, before any specific interval is computed.

Problem 3 — Required Sample Size (Generated Problems)

The formula is — always round up. Critical values: (90%), 1.96 (95%), 2.576 (99%).

Example (, , 95%): .

Problem 4 — Width and Confidence-Level Trade-Off

.

(a) 90%: → CI . 99%: → CI .

(b) Width of 90% CI: . Width of 99% CI: . The 99% CI is ≈ 2.53 units (about 57%) wider for a 9-percentage-point increase in confidence.

(c) “More confidence is always better” ignores the cost in precision. A very wide 99% CI may be too uninformative to be useful. The right level depends on the decision being made and how costly a wrong call is — 90% may suit exploratory work; 99% (or higher) suits safety-critical applications.

Problem 1 — Large-Sample CI Using s (Variants A–E)

When is unknown and , substitute for in the SE formula. This is an approximation — inf-3 covers the exact t-distribution method.

• Variant A (study hours, n = 40, , , 95%): , → 95% CI: hours.
• Variant B (commute, n = 50, , , 90%): , → 90% CI: minutes.
• Variant C (electricity, n = 60, , , 99%): , → 99% CI: ($133.19, $151.81).
• Variant D (hospital stay, n = 80, , , 95%): , → 95% CI: days.
• Variant E (student height, n = 35, , , 90%): , → 90% CI: cm.

Problem 2 — Full CI (Generated Problems)

The workflow for every generated CI problem:

1. Compute .
2. Identify : 1.645 (90%), 1.96 (95%), 2.576 (99%).
3. Compute .
4. Interval: .

Common errors: wrong (mismatched level), using instead of SE in the margin, and rounding SE before multiplying by (compounds rounding — round only the final endpoints).

Problem 3 — Sample Size and the Quadruple Rule (Generated Problems)

Halving the margin of error requires multiplying by 4, because sits under a square root in the SE formula: Replacing with : . Exact in theory; with ceiling rounding, the ratio is ≈ 4 in practice.

Problem 4 — Find the False Statement (Variants A–E)

• Variant A: False is C — “there is a 95% probability that .” is a fixed constant, not a random variable.
• Variant B: False is B — “99% of people consume between 1,840 and 2,160 calories.” CIs locate the mean , not individual observations.
• Variant C: False is D — “the second CI has a higher probability of capturing .” Both are 95% CIs with identical coverage; narrower means more precise, not more likely to capture .
• Variant D: False is D — “doubling n halves E.” It reduces by a factor of ; to halve you must quadruple .
• Variant E: The more problematic comment is the first — “95% confident that ” mixes CI logic with hypothesis-testing language. ” is likely near 60” is a reasonable informal reading.

Problem 5 — Sodium Intake Synthesis

(a) mg, mg. 95% CI: mg/day.

(b) The lower bound (2,321 mg) is entirely above the recommended 2,300 mg. The data provide statistical evidence that mean sodium intake exceeds the recommended level at the 95% confidence level.

(c) . More than 3× the original sample of 50, because the target margin (50 mg) is far tighter than what 50 students achieve (88.7 mg).

Problem 1 — Apply: Light Bulb CI

Given: n = 64, hours, hours. , . 95% CI: hours.

Problem 2 — Analyze: Two Errors

A student used instead of .

Error 1 (computational): the margin should be , not . Correct interval: (82.04, 85.96).

Error 2 (interpretive): “I am 95% confident that is in this interval” sounds like a probability statement about the fixed . Correct: “this interval was constructed by a method that captures in about 95% of all samples.”

Path A — The Analyst (Physical Activity Data)

Given: n = 60, steps, steps. steps.

1. Confidence intervals: 95%: → CI . 99%: → CI steps.

2. Interpretation: both lower bounds are far above 10,000 (WHO recommendation). Even at 99% confidence, the data overwhelmingly support that mean daily steps exceed 10,000.

3. Required n for at 95%: .

4. At 99%: . 99% confidence needs 72% more participants than 95% for the same precision — weigh the cost against the decision stakes.

Path B — The Architect (Healthcare Cost Study)

Given: $2,500, $300, per-participant cost = $120.

1. Required sample sizes: 95%: . 99%: .

2. Recruitment costs: 95%: 267 × $120 = $32,040. 99%: 461 × $120 = $55,320.

3. At 90%: → 188 × $120 = $22,560.

4. Recommendation: 95% is the standard in health economics and policy research. At $32,040 vs. $22,560 (90%) and $55,320 (99%), it balances confidence and cost. The extra $9,480 over 90% buys a meaningful reduction in the miss rate (5% → 1%), worthwhile for government policy; the jump to 99% nearly doubles the cost and is justified only if standards require it.

Challenge 1 — Achievability Analysis (Variants A–E)

• Variant A (, , , 95%): required — not achievable with 100. Best achievable .
• Variant B (, , , 99%): required — not achievable with 150. Best achievable .
• Variant C (, , , 90%): required — achievable ().
• Variant D (, , , 95%): required — exactly achievable.
• Variant E (, n = 200, — find minimum confidence): . , so confidence ≈ 95.2% — just above the standard 95%.

Challenge 2 — Confidence Intervals and Hypothesis Tests

(a) is inside → a two-sided test of at would fail to reject.

(b) is outside → the test would reject at .

(c) The 95% CI contains all for which ; the test rejects when that ratio exceeds 1.96. These are exact complements — the CI and the test are mathematically equivalent formulations of the same evidence.

Challenge 3 — Conservative Sample Size

(a) Range = 40 − 0 = 40. hours.

(b) .

(c) “Conservative” means the procedure slightly overestimates so the target is met even if is a bit smaller than estimated. Overestimating → larger → smaller than required (safe); underestimating → smaller → larger (failed objective). The range/4 heuristic errs toward caution.