Confidence Intervals for the Mean — z and t Intervals

Foundations of Statistics

Estimating Means with Precision

Confidence intervals provide a range of plausible values for the population mean, offering more information than point estimates alone. Choosing between z and t intervals depends on whether σ is known, a critical distinction in practice.

Clinical Trials — Estimating treatment effect sizes with quantified uncertainty
Business Planning — Forecasting average customer lifetime value with confidence bounds
Policy Analysis — Measuring intervention effects with appropriate uncertainty

Confidence intervals turn point estimates into informed decisions.

Core Concepts

A confidence interval gives a range of plausible values for the population mean $\mu$ , based on sample data. The choice between z-interval and t-interval depends on whether $\sigma$ is known.

DfConfidence Interval

A $(1-\alpha)\times 100\%$ confidence interval for $\mu$ is an interval $[L, U]$ computed from sample data such that $P(L \leq \mu \leq U) = 1-\alpha$ in repeated sampling. Equivalently, if we were to construct such intervals from infinitely many samples, $(1-\alpha)\times 100\%$ of them would contain the true $\mu$ .

Correct Interpretation

A 95% CI does not mean "there is a 95% probability that $\mu$ is in this interval." In the frequentist framework, $\mu$ is a fixed (but unknown) constant — it is either in the interval or it is not. The 95% refers to the long-run coverage rate of the procedure, not to any single interval.

z-Interval ( $\sigma$ Known)

z-Interval (σ known)

\bar{x} \pm z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}

Here,

$\bar{x}$ =Sample mean
$z_{\alpha/2}$ =Critical value from standard normal
$\sigma$ =Known population standard deviation
$n$ =Sample size

ThDerivation of the z-Interval

Since $\bar{X} \sim N(\mu, \sigma^2/n)$ , we have $Z = \frac{\bar{X} - \mu}{\sigma/\sqrt{n}} \sim N(0,1)$ . The central $(1-\alpha)$ region satisfies:

P(-z_{\alpha/2} \leq Z \leq z_{\alpha/2}) = 1 - \alpha

Substituting and rearranging:

P\left(\bar{X} - z_{\alpha/2}\frac{\sigma}{\sqrt{n}} \leq \mu \leq \bar{X} + z_{\alpha/2}\frac{\sigma}{\sqrt{n}}\right) = 1 - \alpha

t-Interval ( $\sigma$ Unknown)

t-Interval (σ unknown)

\bar{x} \pm t_{\alpha/2, n-1} \cdot \frac{s}{\sqrt{n}}

Here,

$\bar{x}$ =Sample mean
$t_{\alpha/2, n-1}$ =Critical value from t-distribution with n-1 df
$s$ =Sample standard deviation
$n$ =Sample size

ThDerivation of the t-Interval

When $\sigma$ is unknown, we replace it with $s$ . The statistic $T = \frac{\bar{X} - \mu}{s/\sqrt{n}} \sim t_{n-1}$ (by Fisher's theorem). The central region:

P(-t_{\alpha/2, n-1} \leq T \leq t_{\alpha/2, n-1}) = 1 - \alpha

Rearranging yields the interval $\bar{x} \pm t_{\alpha/2, n-1} \cdot s/\sqrt{n}$ . The t-interval is wider than the z-interval because $t_{\alpha/2, n-1} > z_{\alpha/2}$ for finite $n$ .

Factors Affecting Width

Interval Width

The width of a confidence interval is $2 \times \text{margin of error} = 2 \cdot c \cdot \frac{s}{\sqrt{n}}$ , where $c$ is the critical value. The width depends on:

Confidence level: higher confidence $\to$ larger $c$ $\to$ wider interval
Sample size: larger $n$ $\to$ smaller $s/\sqrt{n}$ $\to$ narrower interval
Variability: larger $s$ $\to$ wider interval

Sample Size Formula

To achieve a margin of error $E$ at confidence level $1-\alpha$ :

n = \left(\frac{z_{\alpha/2} \cdot \sigma}{E}\right)^2

For example, to estimate $\mu$ within $\pm 1$ unit with 95% confidence when $\sigma = 5$ : $n = (1.96 \times 5 / 1)^2 = 96.04$ , so $n = 97$ .

Worked Example: z-Interval

A factory produces bolts with known $\sigma = 0.5$ mm. A sample of $n = 36$ bolts has $\bar{x} = 12.3$ mm. Construct a 95% CI for $\mu$ .

Step 1. The critical value: $z_{0.025} = 1.96$ .

Step 2. The standard error: $\text{SE} = \sigma/\sqrt{n} = 0.5/6 = 0.0833$ .

Step 3. The margin of error: $E = 1.96 \times 0.0833 = 0.163$ .

Step 4. The 95% CI: $12.3 \pm 0.163 = (12.137, 12.463)$ mm.

Interpretation: We are 95% confident that the true mean bolt length is between 12.14 and 12.46 mm.

Worked Example: t-Interval

A pharmacologist measures drug half-life in $n = 9$ patients: $\bar{x} = 4.2$ hours, $s = 0.8$ hours. Construct a 95% CI for $\mu$ .

Step 1. With $\nu = 8$ degrees of freedom: $t_{0.025, 8} = 2.306$ .

Step 2. Standard error: $\text{SE} = s/\sqrt{n} = 0.8/3 = 0.267$ .

Step 3. Margin of error: $E = 2.306 \times 0.267 = 0.615$ .

Step 4. The 95% CI: $4.2 \pm 0.615 = (3.585, 4.815)$ hours.

Comparison: If we had (incorrectly) used the z-interval: $4.2 \pm 1.96 \times 0.267 = 4.2 \pm 0.523 = (3.677, 4.723)$ . The t-interval is wider, correctly reflecting the additional uncertainty from estimating $\sigma$ with $s$ from only 9 observations.

Confidence Level and Coverage

Confidence Level Comparison

Confidence Level	$z_{\alpha/2}$	$t_{0.025, 10}$	Relative Width
90%	1.645	1.812	1.00
95%	1.960	2.228	1.22
99%	2.576	3.169	1.75

Higher confidence requires wider intervals. The cost of being "more sure" is less precision.

Key Takeaways

Summary: Confidence Intervals for the Mean

z-interval ( $\sigma$ known): $\bar{x} \pm z_{\alpha/2} \cdot \sigma/\sqrt{n}$
t-interval ( $\sigma$ unknown): $\bar{x} \pm t_{\alpha/2, n-1} \cdot s/\sqrt{n}$
Higher confidence or smaller $n$ $\to$ wider interval
Larger $n$ $\to$ narrower interval (smaller SE)
Interpretation: "In repeated sampling, 95% of such intervals contain $\mu$"
For large $n$ , t-critical $\approx$ z-critical, so the intervals are nearly identical

Confidence Intervals for the Mean — z and t Intervals

Confidence Intervals for the Mean — z and t Intervals

Estimating Means with Precision

Core Concepts

DfConfidence Interval

z-Interval ( $\sigma$ Known)

z-Interval (σ known)

ThDerivation of the z-Interval

t-Interval ( $\sigma$ Unknown)

t-Interval (σ unknown)

ThDerivation of the t-Interval

Factors Affecting Width

Worked Example: z-Interval

Worked Example: t-Interval

Confidence Level and Coverage

Key Takeaways

Summary: Confidence Intervals for the Mean

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Confidence Intervals for the Mean — z and t Intervals

Confidence Intervals for the Mean — z and t Intervals

Estimating Means with Precision

Core Concepts

DfConfidence Interval

z-Interval (σ\sigmaσ Known)

z-Interval (σ known)

ThDerivation of the z-Interval

t-Interval (σ\sigmaσ Unknown)

t-Interval (σ unknown)

ThDerivation of the t-Interval

Factors Affecting Width

Worked Example: z-Interval

Worked Example: t-Interval

Confidence Level and Coverage

Key Takeaways

Summary: Confidence Intervals for the Mean

Premium Content

Need Expert Statistics Help?

z-Interval ( $\sigma$ Known)

t-Interval ( $\sigma$ Unknown)