Confidence Intervals

A confidence interval provides a range of plausible values for an unknown population parameter. Unlike a hypothesis test, which gives a yes/no decision, a confidence interval quantifies the precision of an estimate.

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The Concept

A 95% confidence interval for a parameter $\theta$θ means: if we repeated the sampling process many times, approximately 95% of the intervals constructed would contain the true value of $\theta$θ.

Important: A 95% confidence interval does NOT mean there is a 95% probability that $\theta$θ lies in the interval. The interval is fixed once calculated; $\theta$θ either is or is not in it. The 95% refers to the long-run success rate of the procedure.

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Confidence Interval for the Mean ($\sigma$σ Known)

If $X_1, \ldots, X_n \sim N(\mu, \sigma^2)$X1​,…,Xn​∼N(μ,σ2) and $\sigma$σ is known:

$\bar{X} \pm z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}$Xˉ±zα/2​⋅n​σ​

Confidence level	$z_{\alpha/2}$zα/2​
90%	1.645
95%	1.960
99%	2.576

Worked Example

A machine fills bottles with mean volume $\mu$μ and known standard deviation $\sigma = 5$σ=5 ml. A sample of 25 bottles gives $\bar{x} = 502$xˉ=502. Find a 95% confidence interval for $\mu$μ.

$502 \pm 1.960 \times \frac{5}{\sqrt{25}} = 502 \pm 1.960 \times 1 = 502 \pm 1.96$502±1.960×25​5​=502±1.960×1=502±1.96

95% CI: $(500.04, 503.96)$(500.04,503.96).

We are 95% confident that the true mean volume lies between 500.04 ml and 503.96 ml.

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Confidence Interval for the Mean ($\sigma$σ Unknown)

When $\sigma$σ is unknown, replace it with $s$s and use the t-distribution:

$\bar{x} \pm t_{n-1, \alpha/2} \cdot \frac{s}{\sqrt{n}}$xˉ±tn−1,α/2​⋅n​s​

Worked Example

A sample of 10 measurements gives $\bar{x} = 48.3$xˉ=48.3 and $s = 2.1$s=2.1. Find a 95% confidence interval for $\mu$μ.

$t_{9, 0.025} = 2.262$t9,0.025​=2.262.

$48.3 \pm 2.262 \times \frac{2.1}{\sqrt{10}} = 48.3 \pm 2.262 \times 0.664 = 48.3 \pm 1.50$48.3±2.262×10​2.1​=48.3±2.262×0.664=48.3±1.50

95% CI: $(46.80, 49.80)$(46.80,49.80).

Note that the interval is wider than if $\sigma$σ were known, because the t-distribution has heavier tails.

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Width of a Confidence Interval

The width of a CI is $2 \times z_{\alpha/2} \times \sigma/\sqrt{n}$2×zα/2​×σ/n​ (for the $\sigma$σ known case).

To make the interval narrower:

• Increase the sample size $n$n (most practical option)
• Decrease the confidence level (but this reduces reliability)
• Reduce $\sigma$σ (not usually within your control)

Sample Size Determination

To achieve a confidence interval of half-width $w$w: