How to Use the T-table to Solve Statistics Problems

The t-table (for the t-distribution) is different from the z-table (for the z-distribution). Make sure you understand the values in the first and last rows. Finding probabilities for various t-distributions, using the t-table, is a valuable statistics skill.

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

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

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.)

Practice solving problems using the t-table sample questions below

1. For a study involving one population and a sample size of 18 (assuming you have a t-distribution), what row of the t-table will you use to find the right-tail (“greater than”) probability affiliated with the study results?

   Answer: df = 17

   The study involving one population and a sample size of 18 has n – 1 = 18 – 1 = 17 degrees of freedom.

2. For a study involving a paired design with a total of 44 observations, with the results assuming a t-distribution, what row of the table will you use to find the probability affiliated with the study results?

   Answer: df = 21

   A matched-pairs design with 44 total observations has 22 pairs. The degrees of freedom is one less than the number of pairs: n – 1 = 22 – 1 = 21.

3. A t-value of 2.35, from a t-distribution with 14 degrees of freedom, has an upper-tail (“greater than”) probability between which two values on the t-table?

   Answer: 0.025 and 0.01

   Using the t-table, locate the row with 14 degrees of freedom and look for 2.35. However, this exact value doesn’t lie in this row, so look for the values on either side of it: 2.14479 and 2.62449. The upper-tail probabilities appear in the column headings; the column heading for 2.14479 is 0.025, and the column heading for 2.62449 is 0.01.

   Hence, the upper-tail probability for a t-value of 2.35 must lie between 0.025 and 0.01.