Confidence Intervals — Worked Solutions — Grade 12 Specialist Maths

Q1 — Construct a 95% CI

x̄ = 84, σ = 12, n = 36. Construct a 95% CI for μ.

Step 1: Compute the standard error.

SE = σ/√n = 12/√36 = 12/6 = 2

Step 2: Identify the critical value.

For a 95% CI: z_0.025 = 1.96.

Step 3: Calculate margin of error.

E = 1.96 × 2 = 3.92

Step 4: Construct the interval.

CI: (84 − 3.92, 84 + 3.92) = (80.08, 87.92)

Conclusion: We are 95% confident that the true population mean lies between 80.08 and 87.92.

Q2 — 90% CI with Unknown σ

x̄ = 150, s = 20, n = 64. Construct a 90% CI for μ.

Step 1: Since n = 64 ≥ 30 and σ is unknown, substitute s for σ.

SE = s/√n = 20/√64 = 20/8 = 2.5

Step 2: For a 90% CI: z_0.05 = 1.645.

Step 3: E = 1.645 × 2.5 = 4.1125.

Step 4: CI: (150 − 4.11, 150 + 4.11) = (145.89, 154.11).

Conclusion: We are 90% confident the population mean lies between 145.89 and 154.11.

Note: This approximation is valid because n is large. For small n with unknown σ, a t-interval would be needed.

Q3 — Extract x̄ and E from a Given CI

A 95% CI is (45.2, 54.8). Find x̄ and E.

The confidence interval is symmetric about the sample mean x̄.

Sample mean (midpoint):

x̄ = (lower + upper)/2 = (45.2 + 54.8)/2 = 100/2 = 50

Margin of error (half-width):

E = (upper − lower)/2 = (54.8 − 45.2)/2 = 9.6/2 = 4.8

So the interval was constructed as 50 ± 4.8.

Q4 — Construct a 99% CI

x̄ = 200, σ = 30, n = 100. Construct a 99% CI for μ.

Step 1: SE = 30/√100 = 30/10 = 3.

Step 2: For 99% CI: z_0.005 = 2.576.

Step 3: E = 2.576 × 3 = 7.728.

Step 4: CI: (200 − 7.728, 200 + 7.728) = (192.27, 207.73).

Conclusion: We are 99% confident that the population mean lies between 192.27 and 207.73.

Note that the 99% CI is wider than the 95% CI would be for the same data — greater confidence requires a wider interval.

Q5 — Required Sample Size

E ≤ 3, 95% confidence, σ = 15. Find minimum n.

Use the sample size formula: n = (zσ/E)².

For 95% confidence: z = 1.96.

n = (1.96 × 15/3)² = (1.96 × 5)² = (9.8)² = 96.04

Rounding up: n = 97.

Verification: SE = 15/√97 ≈ 1.523. E = 1.96 × 1.523 ≈ 2.98 ≤ 3. ✓

With n = 96: SE = 15/√96 ≈ 1.531. E = 1.96 × 1.531 ≈ 3.00, which is borderline. To be safe (and ensure E strictly < 3), n = 97 is required.

Q6 — Confidence Level vs. Probability of Containment

Explain the difference between the confidence level and the probability that μ is in a specific CI.

Confidence level (e.g., 95%): This is a long-run frequency property of the estimation procedure. If we construct 95% CIs from many independent samples, approximately 95% of those intervals will contain the true μ. The confidence level describes the reliability of the method, not any single interval.

Probability for a specific, computed interval: Once a specific interval (e.g., (48.2, 53.8)) has been computed from observed data, μ is a fixed (unknown) constant. A frequentist interpretation gives the probability that μ lies in this specific interval as either 0 or 1 — we simply don’t know which. It is incorrect to say “there is a 95% probability that μ is between 48.2 and 53.8.”

Analogy: Imagine flipping a coin that is known to either show heads or tails (it’s already flipped but covered). The probability of heads is either 0 or 1; the 50% applies to the process before the flip, not after observing the covered result.

Q7 — Survey CI and Interpretation

n = 100, x̄ = 35 years, s = 8 years. Construct a 95% CI and interpret.

Since n = 100 ≥ 30, use s in place of σ.

SE = 8/√100 = 8/10 = 0.8.

For 95% CI: z = 1.96. E = 1.96 × 0.8 = 1.568.

CI: (35 − 1.568, 35 + 1.568) = (33.43, 36.57).

Interpretation: We are 95% confident that the true mean age of the population from which this sample was drawn lies between 33.43 years and 36.57 years. This interval was constructed using a procedure that captures the true mean in 95% of repeated samples of size 100.

Q8 — Effect on CI Width

(a) n quadrupled; (b) confidence level increases from 90% to 99%. How does width change?

CI width = 2E = 2 × z_α/2 × σ/√n.

(a) n → 4n:

New width = 2zσ/√(4n) = 2zσ/(2√n) = original width / 2.

The CI is halved in width. Quadrupling the sample size reduces the margin of error by 50%.

(b) 90% → 99%:

z_90% = 1.645, z_99% = 2.576.

Width scales by the ratio 2.576/1.645 ≈ 1.566.

The CI width increases by approximately 56.6%. Greater confidence demands a wider interval.

These two effects illustrate the trade-off in interval estimation: wider intervals give more confidence, larger samples give tighter intervals.

Q9 — Two CIs from the Same Population

Two independent samples give different 95% CIs. Does this mean one is wrong?

No — both intervals can be correct.

Each random sample produces a different x̄, so the two CIs have different centres and different intervals. This is expected — sampling variability is inherent to statistical inference.

What “95% confidence” means in this context:

Each CI was constructed by the same 95% procedure, so each independently has a 95% chance of containing μ. In the long run, 95% of all such CIs will contain μ. For any two specific CIs:

P(both contain μ) ≈ 0.95 × 0.95 = 0.9025
P(exactly one contains μ) ≈ 2 × 0.95 × 0.05 = 0.095
P(neither contains μ) ≈ 0.05 × 0.05 = 0.0025

Different intervals are a feature, not a bug. If both samples gave identical intervals, that would indicate the samples were not truly random and independent.

Q10 — Sample Size and Achievable Confidence Level

(a) Find n for E = 5 at 99% confidence with σ = 40. (b) With n = 100, what confidence level is achievable?

(a) Required sample size:

n = (zσ/E)² = (2.576 × 40/5)² = (2.576 × 8)² = (20.608)² = 424.69…

Round up: n = 425.

(b) Achievable confidence with n = 100:

SE = 40/√100 = 40/10 = 4.

For E = 5: E = z × SE ⇒ 5 = z × 4 ⇒ z = 1.25.

Confidence level = P(−1.25 ≤ Z ≤ 1.25) = 2Φ(1.25) − 1 = 2(0.8944) − 1 = 0.7888 ≈ 78.9%.

Conclusion: With only n = 100, the researcher achieves only about 78.9% confidence within E = 5 units. To achieve 99% confidence, 425 observations are required. This illustrates the significant cost of high confidence with tight precision requirements.

Confidence Intervals for Population Mean — Worked Solutions