How to Find Right-Tail Values and Confidence Intervals Using the t-Table

You can use a t-table to find right-tail probabilities and p-values for hypothesis tests and to find t*-values (critical values) for a confidence interval involving t. Unlike a Z-distribution, a t-distribution is not classified by its mean and standard deviation, but by the sample size of the data set being used (n). Unfortunately, there is no single "standard t-distribution" that you can use to transform the numbers and find probabilities on a table.

Because it wouldn't be humanly possible to create a table of probabilities and corresponding t-values for every possible t-distribution, statisticians created the t-table, which shows certain values of t-distributions for a selection of degrees of freedom and a selection of probabilities.

Each row of the t-table represents a different t-distribution, classified by its degrees of freedom (df). The columns represent various common greater-than probabilities, such as 0.40, 0.25, 0.10, and 0.05. The numbers across a row indicate the values on the t-distribution (the t-values) corresponding to the greater-than probabilities shown at the top of the columns. Rows are arranged by degrees of freedom.

Another term for greater-than probability is right-tail probability, which indicates that such probabilities represent areas on the right-most end (tail) of the t-distribution.

For example, the second row of the t-table is for the t₂ distribution (2 degrees of freedom, pronounced tee sub-two). You see that the second number, 0.816, is the value on the t₂ distribution whose area to its right (its right-tail probability) is 0.25 (see the heading for column 2). In other words, the probability that t₂ is greater than 0.816 equals 0.25. In probability notation, that means p(t₂ > 0.816) = 0.25.

The next number in row two of the t-table is 1.886, which lies in the 0.10 column. This means the probability of being greater than 1.886 on the t₂ distribution is 0.10. Because 1.886 falls to the right of 0.816, its right-tail probability is lower.

Follow these steps to use the t-table to find right-tail probabilities and p-values for hypothesis tests involving t:

1. Find the t-value for which you want the right-tail probability (call it t), and find the sample size (for example, n).

2. Find the row corresponding to the degrees of freedom (df) for your problem (for example, n – 1). Go across that row to find the two t-values between which your t falls.

   For example, if your t is 1.60 and your n is 7, you look in the row for df = 7 – 1 = 6. Across that row, you find your t lies between t-values 1.44 and 1.94.

3. Go to the top of the columns containing the two t-values from Step 2.

   The right-tail (greater-than) probability for your t-value is somewhere between the two values at the top of these columns. For example, your t = 1.60 is between t-values 1.44 and 1.94 (df = 6); so the right tail probability for your t is between 0.10 (column heading for t = 1.44) and 0.05 (column heading for t = 1.94).

The row near the bottom with Z in the df column gives right-tail (greater-than) probabilities from the Z-distribution.

Here's how to use the t-table to find t*-values (critical values) for a confidence interval involving t:

1. Determine the confidence level you need (as a percentage).

2. Determine the sample size (for example, n).

3. Look at the bottom row of the table where the percentages are shown.

   Find your % confidence level there.

4. Intersect this column with the row representing your degrees of freedom (df).

   This is the t-value you need for your confidence interval.

   For example, a 95% confidence interval with df = 6 has t * = 2.45. (Find 95% on the last line and go up to row 6.)