Finding t* Critical Values With a Table or invT

What a t Critical Value Represents

In “The t Distribution and Degrees of Freedom,” you saw that a one-sample t procedure uses \(df=n-1\), and that the t distribution’s tails depend on the degrees of freedom. This tutorial focuses on finding a cutoff, called the t critical value, for a specified confidence level and \(df\).

A t critical value is a point on the horizontal axis of a t distribution. For a two-sided confidence level, the positive critical value \(t^*\) leaves the stated confidence level in the middle of the curve, with equal areas in the two tails. The corresponding negative cutoff is \(-t^*\). For example, a central 95% area leaves 5% outside the central region: 2.5% in each tail.

The significance level associated with a central confidence level is \(\alpha=1-C\). For a two-sided critical value, each tail has area \(\alpha/2\). Because calculator commands such as invT use the cumulative area to the left of a cutoff, the left-tail area at the positive cutoff is \(1-\alpha/2\), or equivalently \((1+C)/2\).

Keep the central confidence area, the area in one tail, and the cumulative area entered into invT distinct. For a central 90% area, \(C=0.90\), so \(\alpha=1-0.90=0.10\). Each tail has area \(0.10/2=0.05\), and the positive cutoff has \(1-0.05=0.95\) to its left. Thus the central area is 0.90, while the invT input for the positive cutoff is 0.95.

Finding t* in a t Table

A t table organizes critical values by degrees of freedom and tail area. Tables do not all use the same column headings: a table may label columns by one-tail area, two-tail area, or confidence level. Read the heading before selecting a column. For a central confidence level \(C\), the two-tail area is \(1-C\), and the one-tail area is \((1-C)/2\).

For example, a central 95% critical value corresponds to two-tail area \(0.05\), or one-tail area \(0.025\). A central 90% critical value corresponds to two-tail area \(0.10\), or one-tail area \(0.05\). These are two ways of describing the same pair of tails.

To look up \(t^*\), first find the row for \(df\), then find the column that represents the correct tail area or confidence level. The entry is the positive cutoff. For a one-sample t procedure with sample size \(n\), use \(df=n-1\), as established in the earlier tutorial on degrees of freedom.

Some printed tables do not include every possible \(df\). If a table gives only selected rows, a common conservative approach for a confidence interval is to use the next smaller listed \(df\). Because smaller \(df\) gives a larger \(t^*\) at the same confidence level, this choice does not understate the critical value. If the table provides the exact row, use it rather than substituting another row.

Finding t* with invT

A calculator’s invT function returns a t cutoff from a cumulative area to the left and a degrees-of-freedom value. For a two-sided confidence level, use the cumulative area at the positive cutoff—not the central confidence level and not the area in one tail.

On a TI-84, the command is commonly written as invT(area, df). For instance, invT(0.975, 14) finds the t cutoff with area 0.975 to its left and 14 degrees of freedom. If a task asks for the negative cutoff instead, the distribution’s symmetry means it is \(-t^*\). A confidence interval uses the positive critical value as a multiplier.

Worked Examples: Finding t Critical Values

Common Mistakes and AP Exam Tips

• Entering the confidence level directly into invT. For a central 95% critical value, enter 0.975, not 0.95. For a central 90% critical value, enter 0.95, not 0.90. First split the area outside the center between the two tails.
• Confusing central area with total tail area. For a central 90% area, the central area is 0.90 and the total tail area is 0.10. Each tail contains 0.05. State which area you mean instead of calling all of them “the tail.”
• Using the wrong table column. Check whether the heading gives one-tail area, two-tail area, or confidence level. For a central 95% critical value, the matching one-tail area is 0.025 and the matching two-tail area is 0.05.
• Using \(n\) for the degrees of freedom. A one-sample t procedure uses \(df=n-1\). A sample of 15 has 14 degrees of freedom, not 15.
• Forgetting that invT uses area to the left. The positive cutoff has most of the distribution to its left. For a central 99% area, the correct input is 0.995, not the 0.005 area in the upper tail.
• Mixing up a central confidence level and a one-sided percentile. invT(0.90, \(df\)) gives the 90th percentile, with 90% to the left. It is not the positive critical value for a central 90% area; that requires invT(0.95, \(df\)).
• Reporting only a calculator result. Show \(df\), the tail-area reasoning, and the invT inputs or the table row and column. For example: “With \(n=15\), \(df=14\). A central 95% level leaves 0.025 in each tail, so the left area at \(t^*\) is 0.975; therefore \(t^*\approx2.145\).”

A quick reasonableness check can catch many lookup errors. At a fixed confidence level, smaller \(df\) gives a larger \(t^*\), because the t distribution has heavier tails. At fixed \(df\), a higher confidence level gives a larger \(t^*\), because the cutoff must move farther from zero to capture more central area. A result that goes in the opposite direction is a reason to recheck the row, column, and tail area.