Practice Maths

Confidence Intervals for Population Mean

Key Terms

A confidence interval (CI) for μ gives a range of plausible values based on sample data.
For a known σ, a (1 − α) × 100% CI is: x̄ ± zα/2 × σ/√n.
Critical values: 90% CI → z = 1.645;   95% CI → z = 1.96;   99% CI → z = 2.576.
Margin of error
E = zα/2 × σ/√n.
Required sample size
n = (zα/2 × σ / E)² (round up to next integer).
If σ is unknown and n is large (≥ 30), replace σ with the sample standard deviation s.
Confidence Interval Formula:
  • CI: x̄ − zα/2σ/√n < μ < x̄ + zα/2σ/√n
  • Centre: x̄ (the sample mean)
  • Half-width (margin of error): E = zα/2 × σ/√n
  • 90% CI: z = 1.645  |  95% CI: z = 1.96  |  99% CI: z = 2.576
  • To achieve margin of error E: n ≥ (zα/2σ/E)²
Hot Tip A 95% confidence level does NOT mean “there is a 95% probability that μ lies in this interval.” Once the interval is calculated, μ either is or is not in it — it is fixed (we just don’t know it). The correct interpretation is: if we repeated this procedure many times, 95% of the resulting intervals would contain μ.

Worked Example 1 — Constructing a 95% CI

A random sample of n = 49 observations gives x̄ = 120. The population standard deviation is σ = 14. Construct a 95% CI for μ.

SE = σ/√n = 14/√49 = 14/7 = 2.

For 95% CI, zα/2 = 1.96.

E = 1.96 × 2 = 3.92.

CI: (120 − 3.92, 120 + 3.92) = (116.08, 123.92).

We are 95% confident that the true population mean lies between 116.08 and 123.92.

Worked Example 2 — Finding Required Sample Size

A researcher wants to estimate μ to within E = 4 units at 99% confidence. Historical data suggests σ = 20. Find the minimum sample size.

For 99% CI: zα/2 = 2.576.

n = (zσ/E)² = (2.576 × 20/4)² = (12.88)² = 165.89…

Round up: n = 166.

What is a Confidence Interval?

A confidence interval provides a range of values that we believe contains the true (unknown) population mean μ, based on sample data. Rather than giving a single point estimate x̄ (which will almost certainly not equal μ exactly), a CI acknowledges our uncertainty and provides a plausible range.

The interval is constructed so that, in repeated sampling, a specified percentage of intervals will contain μ. This percentage is called the confidence level. Common choices are 90%, 95%, and 99%.

Derivation of the Confidence Interval

By the CLT (or if the population is normal), X̄ ∼ N(μ, σ²/n). Standardising:

Z = (X̄ − μ) / (σ/√n) ∼ N(0, 1)

For a 95% CI, P(−1.96 ≤ Z ≤ 1.96) = 0.95. Substituting and rearranging:

P(X̄ − 1.96σ/√n ≤ μ ≤ X̄ + 1.96σ/√n) = 0.95

Once we observe x̄ from our sample, the 95% CI is: (x̄ − 1.96σ/√n, x̄ + 1.96σ/√n).

Critical Values for Common Confidence Levels

The critical value zα/2 is the z-value that cuts off α/2 in each tail of the standard normal distribution.

  • 90% CI: α = 0.10, zα/2 = z0.05 = 1.645
  • 95% CI: α = 0.05, zα/2 = z0.025 = 1.96
  • 99% CI: α = 0.01, zα/2 = z0.005 = 2.576

Margin of Error and Width

The margin of error E = zα/2 × σ/√n is the half-width of the CI. The total width of the CI is 2E. To reduce E:

  • Increase n (the sample size). Doubling n reduces E by a factor of √2.
  • Decrease the confidence level (reduces zα/2). But this makes us less confident!
  • Reduce σ (not usually controllable, but better measurement instruments help).

Determining Required Sample Size

Often we specify E in advance (e.g., “estimate to within 5 units”) and solve for n:

E = zα/2 × σ/√n  ⇒  √n = zα/2 × σ/E  ⇒  n = (zα/2 × σ/E)²

Always round n up to the next integer (rounding down would violate the margin of error requirement).

When σ is Unknown

If σ is not known and n is large (≥ 30), we substitute the sample standard deviation s in place of σ. This gives an approximate CI. For small samples with unknown σ, the t-distribution is used instead (a topic for Methods students, but mentioned here for completeness).

Correct Interpretation

A specific 95% CI such as (48.2, 53.8) does not mean “there is a 95% probability that μ is between 48.2 and 53.8.” The parameter μ is fixed — it either is or is not in this interval. The correct statement is: this interval was constructed by a procedure that produces intervals containing μ in 95% of repeated samples. We are confident (at the 95% level) that μ lies in the interval.

Exam Tip: In exam questions, state the CI in the form (lower, upper), then write a conclusion sentence like “We are 95% confident that the true population mean lies between [lower] and [upper].”
Exam Tip: When finding n, the formula requires knowledge of σ. If only s is given in the question, use s as an estimate for σ in the formula. Always round n up.

Mastery Practice

See Answers ➔

  1. Construct a 95% CI. Fluency

    A random sample of n = 36 gives x̄ = 84. The population standard deviation is σ = 12. Construct a 95% confidence interval for the population mean μ.

  2. Construct a 90% CI with large n. Fluency

    A sample of n = 64 gives x̄ = 150 and s = 20 (population σ unknown). Construct a 90% CI for μ.

  3. Extract x̄ and E from a given CI. Fluency

    A 95% CI is reported as (45.2, 54.8). Find the sample mean x̄ and the margin of error E.

  4. Construct a 99% CI. Fluency

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

  5. Required sample size. Understanding

    A researcher wants a margin of error E ≤ 3 at 95% confidence, with σ = 15. Find the minimum sample size n.

  6. Confidence level vs. probability of containment. Understanding

    Explain the difference between the confidence level (e.g., 95%) and the probability that μ is in a specific, already-calculated confidence interval.

  7. Survey CI and interpretation. Understanding

    A survey of n = 100 people gave a mean age of 35 years with a sample standard deviation of s = 8 years. Construct a 95% CI for the true mean age and interpret it.

  8. Effect on CI width. Understanding

    How does the CI width change if (a) n is quadrupled, and (b) the confidence level increases from 90% to 99%?

  9. Two CIs from the same population. Problem Solving

    Two independent samples from the same population give different 95% CIs. Does this mean one is wrong? Explain.

  10. Sample size and achievable confidence level. Problem Solving

    A researcher wants to estimate μ to within E = 5 units with 99% confidence. Historical data shows σ ≈ 40. (a) What sample size is needed? (b) If the budget limits the researcher to n = 100, what confidence level can be achieved?