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

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).

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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