The t-Distributions


The t-distributions can be used to analyze the mean of a normally distributed population. In particular, we can construct confidence intervals and perform hypothesis tests on such means with this distribution.

A t-distribution is based on a parameter known as the degrees of freedom n, where n is an integer greater than or equal to 1. Such a random variable is denoted by X ~ t(n). The probability density function (pdf) is given by

f(x) = C / [ 1 + (x^2) / n) ^ ((n+1) / 2) ] ,

for all x, where C is a constant that depends on n.

As with a normal distribution, the pdf of a t-distribution creates a "Bell-Shaped" curve. This curve is symmetric about the point x = 0. In fact, as the degrees of freedom n increase, the t-Distributions converge in probability to the standard normal distribution Z (having mean 0 and variance 1).

Using the TDIST Program

The TDIST program can be used to compute probabilities such as P(t(n) <=k), P(t(n) >= k), or P(j <= t(n) <= k). To execute the program, first enter 1, 2, or 3 to designate the type of probability you wish to compute. Next, enter the degrees of freedom followed by the lower and upper bounds j and k (or just the single bound k for a tail probability). The program next asks if you want to see a shaded graph of the pdf. If so, enter 1. If not, then enter 0. After the graph, we receive a display of the desired probability.

Example. Let X ~ t(24). Find the probability that X lies between -0.45 and 0.82.

Solution. After calling up the TDIST program, enter 3 for MIDDLE PROB, then enter 24 for DEG. OF FREEDOM, followed -.45 for LOWER BOUND (use the negative sign, not the minus sign), and .82 for UPPER BOUND. We see that P(-.45 <= t(24) <= .82) = 0.4614827.

Other Features:

1. If you entered 1 to see a graph, then the partially shaded graph of the Bell-Curve initially appears, but then is replaced by the display of the computed probability. To see the graph again, press GRAPH. Then to see the probability value again, press CLEAR. On the TI-86, press CLEAR twice, or just press EXIT to remove the graph. To exit the graph on the TI-89, press HOME. Then press F5 to see the program output again.

2. If you initially enter either 1 or 2 to compute a tail probability, then both the left-tail and right-tail probabilities will be displayed in either case. However only the desired region will be shaded in the graph.


1. Compute P(t(19) <= 1.05) and P(t(19) > 1.05).

2. Approximate P(t(75) >= 1) with a standard normal distribution.


1. After calling up the TDIST program, enter 1 for LEFT PROB, then enter 19 for DEG. OF FREEDOM, followed by 1.05 for BOUND. We see that P(t(19) <= 1.05) = 0.846554 and P(X > 1.05) = 0.153445.

2. For large n, we can approximate t-distribution probabilities with a Standard Normal Distribution. In the NORMDIST program, enter 2 for RIGHT PROB, then enter 0 for MEAN, enter 1 for STANDARD DEV., and enter 1 for BOUND. We find that P(t(75) >= 1) ~ 0.158655.

Note: With the TDIST program, we can compute the value of P(T(75) >= 1) more precisely as 0.160263.

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