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Confidence Interval Example

Medium DifficultyFE Mathematics and Statistics

Construct a 95% confidence interval for the population mean when the population standard deviation is unknown, using the sample mean, sample standard deviation, and the $t$ -distribution.

Concept

When the population is approximately normal and the population standard deviation is unknown, a confidence interval for the mean μ is based on the t -distribution with n - 1 degrees of freedom. The margin of error depends on the confidence level (e.g. 95% \to α = 0.05) and the standard error s / n . \overset{x}{ˉ} \pm t_{α /2, n - 1} \frac{s}{n} Here t_{α /2, n - 1} is the critical value such that P (∣ T ∣ \leq t_{α /2}) = 1 - α for T \sim t (n - 1) .

Problem

Five measurements of compressive strength (MPa) from a concrete batch are: 52, 48, 55, 49, 51. Assume the population is approximately normal. Find: Sample mean and sample standard deviation A 95% confidence interval for the true mean compressive strength Brief interpretation of the interval

Given

Data: 52, 48, 55, 49, 51 (MPa) n = 5 Confidence level: 95% Population approximately normal; σ unknown

Sample mean and standard deviation

\overset{x}{ˉ} = \frac{52 + 48 + 55 + 49 + 51}{5} = \frac{255}{5} = 51 s^{2} = \frac{( 52 - 51 ) ^{2} + ( 48 - 51 ) ^{2} + ( 55 - 51 ) ^{2} + ( 49 - 51 ) ^{2} + ( 51 - 51 ) ^{2}}{4} = \frac{1 + 9 + 16 + 4 + 0}{4} = 7.5 s = 7.5 \approx 2.74

Critical value and margin of error

For 95% CI, α = 0.05, so α /2 = 0.025 . Degrees of freedom df = n - 1 = 4 . From a t -table: t_{0.025, 4} \approx 2.776 (95% CI, df = 4) E = t_{α /2, n - 1} \frac{s}{n} E = 2.776 \cdot \frac{2.74}{5} \approx 2.776 \cdot 1.225 \approx 3.40

Confidence interval

\overset{x}{ˉ} \pm t_{α /2, n - 1} \frac{s}{n} 51 \pm 3.40 \Rightarrow [47.6, 54.4]

Interpretation

We are 95% confident that the true mean μ lies between 47.6 and 54.4.

Final Answer

(1) Sample stats \overset{x}{ˉ} = 51 MPa, s \approx 2.74 MPa (2) 95% confidence interval 51 \pm 3.40 \Rightarrow [47.6, 54.4] (3) Interpretation We are 95% confident that the true mean μ lies between 47.6 and 54.4.

Key Formulas

CI for μ : \overset{x}{ˉ} \pm t_{α /2, n - 1} \frac{s}{n} Use t when σ is unknown; use z when σ is known. For 95% CI with df = 4, t_{0.025, 4} \approx 2.776 .

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