Table Critical Values of t
The table of critical values of t (often called a t-table) provides cutoffs \(t_{\alpha,df}\) for the Student t distribution. These cutoffs are used as \(t^*\) to build confidence intervals and to define rejection regions in hypothesis tests when the population standard deviation \(\sigma\) is unknown.
Step 1: Identify Degrees of Freedom
For one-sample inference about a population mean using a sample standard deviation \(s\), the degrees of freedom are:
The t distribution depends on \(df\): smaller \(df\) produces heavier tails, so critical values are larger in magnitude.
Step 2: Decide Which Tail Probability the Table Uses
Many t-tables label columns by a right-tail area \(\alpha\), giving \(t_{\alpha,df}\) such that \(P(T \ge t_{\alpha,df})=\alpha\). Some tables label columns by a two-tail area \(2\alpha\). The correct column depends on whether the procedure is one-sided or two-sided.
For a two-sided confidence interval with confidence level \(C\), the total tail area is \(\alpha=1-C\), so each tail has area \(\alpha/2\). The needed critical value is \(t_{\alpha/2,df}\).
Step 3: Use the Correct Critical Value
| Goal | Probability statement | Critical value to read from the t-table |
|---|---|---|
| Two-sided confidence interval at level \(C\) | \(P(-t^* \le T \le t^*)=C\) | \(t^* = t_{\alpha/2,df}\) where \(\alpha = 1-C\) |
| Two-sided hypothesis test at significance \(\alpha\) | Reject if \(|t| \ge t^*\) | \(t^* = t_{\alpha/2,df}\) |
| Right-tailed hypothesis test at significance \(\alpha\) | Reject if \(t \ge t^*\) | \(t^* = t_{\alpha,df}\) |
| Left-tailed hypothesis test at significance \(\alpha\) | Reject if \(t \le -t^*\) | \(t^* = t_{\alpha,df}\) and use \(-t^*\) |
How the Critical t Value Enters a Confidence Interval
For a one-sample confidence interval for a population mean (with \(\sigma\) unknown), the margin of error uses \(t^*\):
Worked Example Using a t-Table
A sample of size \(n=12\) is used to estimate a population mean with a 95% confidence interval. Here \(C=0.95\), so \(\alpha=1-0.95=0.05\) and \(\alpha/2=0.025\). Degrees of freedom:
From the table of critical values of t, locate the row \(df=11\) and the column for right-tail area \(0.025\) (equivalently, a two-tail area of \(0.05\)). The critical value is:
This \(t^*\) is then substituted into the margin of error \(t^*\cdot \dfrac{s}{\sqrt{n}}\).
Visualization: Two-Tail Critical Regions for \(t_{\alpha/2,df}\)
Quick Checklist Before Reading a t-Table
- Compute \(df\) (often \(n-1\) for one-sample mean inference).
- Determine whether the problem is two-sided (\(\alpha/2\) per tail) or one-sided (\(\alpha\) in one tail).
- Match the table’s column definition (right-tail \(\alpha\) vs two-tail \(\alpha\)) to the problem.
- Use symmetry: left-tail critical values are the negatives of right-tail values.
A common two-sided mistake is using \(t_{\alpha,df}\) instead of \(t_{\alpha/2,df}\). For confidence level \(C\) or two-sided significance \(\alpha\), the correct column uses \(\alpha/2\).