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INF-2 Solutions: Confidence Intervals for a Population Mean

Solutions Reference · ← Back to Lesson INF-2

Section 5 — Guided Practice Solutions

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GP1 — Constructing a Confidence Interval (Variants A–E)

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

Variant A (coffee shop, σ = 1.6, n = 64, x̄ = 7.8, 95% CI):

Variant B (exam scores, σ = 12, n = 36, x̄ = 74, 99% CI):

Variant C (delivery times, σ = 8.4, n = 49, x̄ = 31.6, 90% CI):

Variant D (heart rate, σ = 10, n = 100, x̄ = 68, 95% CI):

Variant E (battery life, σ = 25, n = 81, x̄ = 312, 99% CI):

Critical value reference: 90% → z* = 1.645; 95% → z* = 1.96; 99% → z* = 2.576. The single most common error is mixing up which z* corresponds to which level. Remember: higher confidence requires a larger z*, producing a wider interval.

GP2 — Correct Interpretation (Variants A–E)

All five variants test the same core idea: confidence intervals describe a property of the procedure, not a probability statement about the fixed parameter \(\mu\).

Variant A (step count CI: (8,240, 9,760)): Correct statement — "We used a procedure that captures the true mean in about 95% of all samples drawn this way." Wrong: assigning 95% probability to μ's location (μ is fixed), or claiming 95% of individuals fall in the range.

Variant B (exam score CI: (71.3, 76.7)): The phrase "90% sure" is problematic because it implies μ has a 90% chance of being inside. The frequentist correct phrasing: "90% of intervals from this procedure capture μ."

Variant C (wait time CI: (14.2, 17.8)): False — the CI is about the mean, not individual customers. Individual wait times have much higher variability; most customers fall outside this range in either direction.

Variant D (temperature CI: (36.1, 36.9)): Correct statement — "If we used this same sampling procedure repeatedly, about 99% of the resulting confidence intervals would contain the true mean." The other options mistakenly assign probability to the fixed μ.

Variant E (probability is 0 or 1): True. After the interval is computed, μ is either inside it (probability 1) or not (probability 0). The 95% refers to the long-run frequency of the procedure before any specific interval is computed.

GP3 — Required Sample Size (Generated Problems)

The formula is \(n = \lceil (z^* \sigma / E)^2 \rceil\). Always round up. The three critical values are z* = 1.645 (90%), 1.96 (95%), 2.576 (99%). Plug in σ and E for the specific generated problem, compute the ratio, square it, and take the ceiling.

Example walk-through (σ = 12, E = 2, 95% confidence):

GP4 — Width and Confidence Level Trade-Off

SE = 9.1/√45 ≈ 1.357.

(a)

(b) Width of 90% CI: \(2 \times 2.232 = 4.464\). Width of 99% CI: \(2 \times 3.496 = 6.992\). The 99% CI is approximately 2.53 units wider — 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 so uninformative as to be practically useless. The right confidence level depends on what decision will be made from the interval and how costly a wrong decision is. In exploratory work, 90% may be fine; in safety-critical applications, 99% (or even higher) may be warranted.

Section 6 — Independent Practice Solutions

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IP1 — Large-Sample CI Using s (Variants A–E)

When σ is unknown and n ≥ 30, substitute s for σ in the SE formula. This is an approximation — INF-3 covers the exact method using the t-distribution.

Variant A (study hours, n = 40, x̄ = 18.2, s = 4.6, 95% CI):

Variant B (commute, n = 50, x̄ = 38.4, s = 11.2, 90% CI):

Variant C (electricity, n = 60, x̄ = 142.50, s = 28.00, 99% CI):

Variant D (hospital stay, n = 80, x̄ = 4.3, s = 1.9, 95% CI):

Variant E (student height, n = 35, x̄ = 170.4, s = 8.1, 90% CI):

IP2 — Full CI (Generated Problems)

The workflow for every generated CI problem:

  1. Compute SE = σ/√n.
  2. Identify z*: 1.645 (90%), 1.96 (95%), 2.576 (99%).
  3. Compute E = z* × SE.
  4. Interval: (x̄ − E, x̄ + E).

Check the specific generated values against this workflow. Common errors: wrong z* (mismatched confidence level), using σ instead of SE in the margin of error, rounding SE before multiplying by z* (which compounds rounding error — better to round only the final interval endpoints).

IP3 — Sample Size and the Quadruple Rule (Generated Problems)

The key result: halving the margin of error requires multiplying n by 4 (because n appears under a square root in the SE formula). Formally:

\[ E = z^* \sigma / \sqrt{n} \implies n = (z^* \sigma / E)^2 \]

Replace \(E\) with \(E/2\): \(n_{\text{new}} = (z^* \sigma / (E/2))^2 = 4 \cdot (z^* \sigma / E)^2 = 4 n_{\text{old}}\).

This relationship holds exactly in theory; with ceiling rounding, the ratio is approximately 4 in practice.

IP4 — Find the False Statement (Variants A–E)

Variant A: False statement is C — "There is a 95% probability that μ = 45.0." μ is a fixed constant, not a random variable. It doesn't have a probability of equaling any particular value.

Variant B: False statement is B — "99% of people in the study consume between 1,840 and 2,160 calories." CIs are about the location of the population mean μ, not the distribution of individual observations.

Variant C: False statement is D — "The second CI has a higher probability of capturing μ." Both are 95% CIs with identical coverage probability. Narrower means more precise, not more likely to capture μ.

Variant D: False statement is D — "Doubling n halves E." It reduces E by factor \(1/\sqrt{2} \approx 0.707\), not by half. To halve E, you must quadruple n.

Variant E: The more problematic comment is the first — "95% confident that μ ≠ 50" is a statement mixing CI logic and hypothesis testing language imprecisely. The second comment ("μ is likely near 60") is a reasonable informal interpretation.

IP5 — Sodium Intake Synthesis

Part (a):

\[ \text{SE} = 320/\sqrt{50} \approx 45.25 \text{ mg}, \quad E = 1.96 × 45.25 ≈ 88.7 \text{ mg} \]

\begin{gather*} \text{95\% CI:} \quad 2410 \pm 88.7 \\ \Rightarrow (2321.3,\ 2498.7) \text{ mg/day} \end{gather*}

Part (b): The lower bound of the 95% CI is 2,321 mg — entirely above the recommended 2,300 mg. The data provide statistical evidence that mean sodium intake in this population exceeds the recommended level at the 95% confidence level.

Part (c):

\[ n = \left\lceil \left(\frac{1.96 × 320}{50}\right)^2 \right\rceil = \left\lceil (12.544)^2 \right\rceil = \lceil 157.35 \rceil = \mathbf{158} \]

158 students are required — more than 3× the original sample of 50 — because the target margin of error (50 mg) is far tighter than what 50 students can achieve (88.7 mg).

Section 7 — Mastery Check Solutions

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Apply Question — Light Bulb CI

Given: n = 64, x̄ = 1215 hours, σ = 80 hours.

\[ \text{SE} = 80/\sqrt{64} = 10, \quad E = 1.96 × 10 = 19.6 \]

\begin{gather*} \text{95\% CI:} \quad 1215 \pm 19.6 \\ \Rightarrow (1195.4,\ 1234.6) \text{ hours} \end{gather*}

Analyze — Two Errors

Student used σ = 10 instead of SE = σ/√n = 10/√100 = 1.

Error 1 (computational): Margin of error should be \(E = 1.96 × 1 = 1.96\), not \(1.96 × 10 = 19.6\). Correct interval: (82.04, 85.96).

Error 2 (interpretive): "I am 95% confident that μ is in this interval" sounds like a probability statement about the fixed parameter μ. Correct phrasing: "This interval was constructed using a method that captures μ in about 95% of all samples."

The computational error (using σ instead of SE) is the most consequential — it produced an interval nearly 10× too wide, which would badly misrepresent the precision of the estimate.

Section 8 — Boss Fight Solutions

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Path A — The Analyst (Physical Activity Data)

Given: n = 60, x̄ = 28,400 steps, σ = 7,200 steps.

\[ \text{SE} = 7200/\sqrt{60} \approx 929.6 \text{ steps} \]

1. Confidence intervals:

95%: \(E = 1.96 \times 929.6 \approx 1822\). CI: (26,578, 30,222) steps.

99%: \(E = 2.576 \times 929.6 \approx 2394\). CI: (26,006, 30,794) steps.

2. Interpretation: Both CIs have lower bounds far above 10,000 (WHO recommendation). Even at 99% confidence, the data provide overwhelming evidence that mean daily steps in this city exceeds 10,000.

3. Required n for E ≤ 1,500 at 95%:

\[ n = \lceil (1.96 × 7200/1500)^2 \rceil = \lceil (9.408)^2 \rceil = \lceil 88.51 \rceil = \mathbf{89} \]

4. At 99% confidence:

\[ n = \lceil (2.576 × 7200/1500)^2 \rceil = \lceil (12.365)^2 \rceil = \lceil 152.9 \rceil = \mathbf{153} \]

99% confidence requires 72% more participants than 95% for the same precision. The agency must weigh whether the extra cost is warranted by the decision stakes.

Path B — The Architect (Healthcare Cost Study)

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

1. Required sample sizes:

95%: \[ n = \lceil (1.96 \times 2500/300)^2 \rceil = \lceil (16.333)^2 \rceil = \lceil 266.77 \rceil = \mathbf{267} \]

99%: \[ n = \lceil (2.576 \times 2500/300)^2 \rceil = \lceil (21.467)^2 \rceil = \lceil 460.8 \rceil = \mathbf{461} \]

2. Recruitment costs:

3. At 90% confidence:

90%: \[ n = \lceil (1.645 \times 2500/300)^2 \rceil = \lceil (13.708)^2 \rceil = \lceil 187.9 \rceil = \mathbf{188} \]

188 × $120 = $22,560

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

Section 9 — Challenge Problem Solutions

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C1 — Achievability Analysis (Variants A–E)

Variant A (σ = 30, E ≤ 5, n_max = 100, 95%):

Variant B (σ = 10, E ≤ 2, n_max = 150, 99%):

Variant C (σ = 15, E ≤ 3, n_max = 70, 90%):

Variant D (σ = 8, E ≤ 1.5, n_max = 110, 95%):

Variant E (σ = 50, n = 200, E ≤ 7 — find minimum confidence level):

C2 — Confidence Intervals and Hypothesis Tests

(a) \(\mu_0 = 40\) is inside (38.2, 45.8) → a two-sided test of \(H_0: \mu = 40\) at α = 0.05 would fail to reject.

(b) \(\mu_0 = 37\) is outside (38.2, 45.8) → the test would reject \(H_0: \mu = 37\) at α = 0.05.

(c) The 95% CI contains all \(\mu_0\) for which \(|\bar{x} - \mu_0|/\text{SE} \leq 1.96\). The hypothesis test rejects when this ratio exceeds 1.96. These conditions are exact complements — the CI and the test are mathematically equivalent formulations of the same evidence.

C3 — Conservative Sample Size

(a) Range = 40 − 0 = 40. \(\sigma \approx 40/4 = 10\) hours.

(b)

\[ n = \lceil (1.96 × 10/1.5)^2 \rceil = \lceil (13.07)^2 \rceil = \lceil 170.7 \rceil = \mathbf{171} \]

(c) "Conservative" means the formula is designed to overestimate n slightly, ensuring the target E is met even if σ is a bit smaller than estimated. Overestimating σ → larger n → wider interval → E smaller than required (safe). Underestimating σ → smaller n → E larger than required (failed objective). The range/4 heuristic errs on the side of caution.