Calculating t-scores

 TSCORE.83p TSCORE.86p tscore.89p

Let T(n) be the t-distribution with n degrees of freedom. For a given probability p, we often need to know the value t such that P(T(n) <= t) = p, or P(t <= T(n)) = p, or P(-t <= T(n) <= t) = p. Such a value t is called a t-score, or critical t value.

Using the TSCORE Program

We can calculate these or any other critical t values using the TSCORE program. To execute the program, first enter 1, 2, or 3 to specify that you want the critical value from a left probability, or the critical value from a right probability, or the critical values from a middle probability.

If you enter 1, then the program will find the value t such that P(T(n) <= t) = p. If you enter 2, then the program will find the value t such that P(t <= T(n)) = p. If you enter 3, then the program will give the values -t and t such that P(-t <= T(n) <= t) = p. In all cases, the t-scores are rounded to three decimal places.

After specifying either 1, 2, or 3, simply enter the desired probability p and the degrees of freedom n.

Example. Find the critical value t that satisfies:

(a) P(T(10) <= t) = 0.05
(b) P(t <= T(15)) = 0.025
(c) P(-t <= T(20) <= t) = 0.98

Solution. (a) After calling up the TSCORE program, enter 1 to specify a left probability, then enter .05 for this left probability and enter 10 for DEG. OF FREEDOM. We receive a critical value of -1.812. Thus, P(T(10) <= -1.812) = 0.05.

(b) First enter 2 to specify a right probability, then enter .025 for this right probability and enter 15 for the degrees of freedom. We receive a critical value of 2.131. Hence, P(2.131 <= T(15)) = 0.035

(c) After entering 3 to specify a middle probability, we enter .98 and 20 to receive the critical values -2.528 and 2.528. Hence, P(-2.528 <= T(20) <= 2.528) = 0.98.