Confidence Interval for μ Using the t Distribution
In many real studies, the population standard deviation σ is not known. In that case, we estimate it using the
sample standard deviation s and use the t distribution to build a confidence interval for the population mean μ.
When to use the t interval
- σ is unknown (typical in practice), so we use \(s\).
- If n ≥ 30, the t method is generally reliable (especially with technology).
- If n < 30, it is important that the population distribution is approximately normal (no strong skew/outliers).
- If the population is clearly not normal and n is small, a nonparametric approach may be more appropriate.
Key ideas
The t distribution is bell-shaped and symmetric around 0, but it is a bit flatter (heavier tails) than the standard normal curve.
The exact shape depends on the degrees of freedom.
\[
\begin{aligned}
df &= n - 1
\end{aligned}
\]
As \(df\) increases (larger samples), the t distribution becomes closer to the standard normal distribution.
Standard error of the mean
When σ is unknown, the standard deviation of the sampling distribution of \(\bar{x}\) is estimated by:
\[
\begin{aligned}
s_{\bar{x}} &= \frac{s}{\sqrt{n}}
\end{aligned}
\]
The (1 − α)100% confidence interval
A two-sided confidence interval for μ uses a critical value t* that places total area α in the two tails,
so each tail has area α/2. The interval is:
\[
\begin{aligned}
\text{CI for }\mu &= \bar{x} \pm t^{*}\cdot s_{\bar{x}} \\
&= \bar{x} \pm t^{*}\cdot \frac{s}{\sqrt{n}}
\end{aligned}
\]
Margin of error
The margin of error is the amount added/subtracted from \(\bar{x}\):
\[
\begin{aligned}
E &= t^{*}\cdot s_{\bar{x}} = t^{*}\cdot\frac{s}{\sqrt{n}}
\end{aligned}
\]
How to construct the interval step-by-step
- Compute the sample size \(n\), the sample mean \(\bar{x}\), and the sample standard deviation \(s\).
- Compute degrees of freedom: \(df = n - 1\).
- Choose the confidence level (so \(\alpha = 1 - \text{confidence}\) and \(\alpha/2\) is in each tail).
- Find \(t^{*}\) for \(df\) and tail area \(\alpha/2\) (via a table or technology).
- Compute \(s_{\bar{x}} = s / \sqrt{n}\) and then \(E = t^{*}\, s_{\bar{x}}\).
- Report: \((\bar{x} - E,\ \bar{x} + E)\).
Interpretation
If we repeatedly took samples in the same way and built an interval each time, then about
(1 − α)100% of those intervals would contain the true population mean μ.
The confidence level describes the long-run success rate of the method (not the probability that a specific computed interval contains μ).
Note on large sample sizes
If you are restricted to a printed t-table that only lists values up to some maximum \(df\),
common options are to use the last row (sometimes labeled "∞") or use the normal approximation.
With technology, you can compute \(t^{*}\) directly for the actual \(df\), so you can continue using the t method even when \(n\) is very large.
