# Student's t Distribution

We have just one more topic to tackle in this lesson, namely, **Student's ****t**** distribution**. Let's just jump right in and define it!

\(T=\dfrac{Z}{\sqrt{U/r}}\) follows a \(f(t)=\dfrac{\Gamma((r+1)/2)}{\sqrt{\pi r} \Gamma(r/2)} \cdot \dfrac{1}{(1+t^2/r)^{(r+1)/2}}\) for −∞ < |

By the way, the *t* distribution was first discovered by a man named W.S. Gosset. He discovered the distribution when working for an Irish brewery. Because he published under the pseudonym Student, the *t* distribution is often called Student's *t* distribution.

History aside, the above definition is probably not particularly enlightening. Let's try to get a feel for the *t* distribution by way of simulation. Let's randomly generate 1000 standard normal values (*Z*) and 1000 chi-square(3) values (*U*). Then, the above definition tells us that, if we take those randomly generated values, calculate:

\(T=\dfrac{Z}{\sqrt{U/3}}\)

and create a histogram of the 1000 resulting *T* values, we should get a histogram that looks like a *t* distribution with 3 degrees of freedom. Well, here's a subset of the resulting values from one such simulation:

Note, for example, in the first row:

\(T(3)=\dfrac{-2.60481}{\sqrt{10.2497/3}}=-1.4092\)

Here's what the resulting histogram of the 1000 randomly generated *T*(3) values looks like, with a standard *N*(0,1) curve superimposed:

Hmmm. The *t*-distribution seems to be quite similar to the standard normal distribution. Using the formula given above for the p.d.f. of *T*, we can plot the density curve of various *t* random variables, say when *r* = 1, *r* = 4, and *r* = 7, to see that that is indeed the case:

In fact, it looks as if, as the degrees of freedom *r* increases, the *t* density curve gets closer and closer to the standard normal curve. Let's summarize what we've learned in our little investigation about the **characteristics of the ****t**** distribution**:

(1) The support appears to be −∞ <* t* < ∞. (It is!)

(2) The probability distribution appears to be symmetric about *t* = 0. (It is!)

(3) The probability distribution appears to be bell-shaped. (It is!)

(4) The density curve looks like a standard normal curve, but the tails of the* t*-distribution are "heavier" than the tails of the normal distribution. That is, we are more likely to get extreme *t*-values than extreme* z*-values.

(5) As the degrees of freedom *r* increases, the *t*-distribution appears to approach the standard normal *z*-distribution. (It does!)

As you'll soon see, we'll need to look up *t*-values, as well as probabilities concerning *T* random variables, quite often in Stat 415. Therefore, we better make sure we know how to read a *t* table.

### The *t* Table

If you take a look at **Table VI** in the back of your text book, you'll find what looks like a typical *t* table. Here's what the top of Table VI looks like (well, minus the shading that I've added):

The *t*-table is similar to the chi-square table in that the inside of the *t*-table (shaded in **purple**) contains the *t*-values for various cumulative probabilities (shaded in **red**), such as 0.60, 0.75, 0.90, 0.95, 0.975, 0.99, and 0.995, and for various *t* distributions with *r *degrees of freedom (shaded in **blue**). The row shaded in **green** indicates the upper *α* probability that corresponds to the 1−*α* cumulative probability. For example, if you're interested in either a cumulative probability of 0.60, or an upper probability of 0.40, you'll want to look for the *t*-value in the first column.

Let's use the *t*-table to read a few probabilities and *t*-values off of the table:

Let's take a look at a few more examples.

### Example

Let* T* follow a* t*-distribution with *r* = 8 *df. *What is the probability that the absolute value of *T* is less than 2.306?

**Solution.** The probability calculation is quite similar to a calculation we'd have to make for a normal random variable. First, rewriting the probability in terms of *T* instead of the absolute value of *T*, we get:

*P*(|*T* | < 2.306) = *P*(*−*2.306 < *T* < 2.306)

Then, we have to rewrite the probability in terms of cumulative probabilities that we can actually find, that is:

*P*(|*T* | < 2.306) = *P*(*T* < 2.306) *−* *P*(*T* < *−*2.306)

Pictorially, the probability we are looking for looks something like this:

But the *t*-table doesn't contain negative *t*-values, so we'll have to take advantage of the symmetry of the *T* distribution. That is:

*P*(|*T* | < 2.306) = *P*(*T* < 2.306) *−* *P*(*T* > 2.306)

Can you find the necessary *t*-values on the *t*-table?

The *t*-table tells us that *P*(*T* < 2.306) = 0.975 and *P*(*T* > 2.306) = 0.025. Therefore:

* P(|T | < 2.306) = 0.975 −* 0.025 = 0.95

What is *t*_{0.05}(8)?

**Solution.** The value *t*_{0.05}(8) is the value *t*_{0.05 }such that the probability that a *T* random variable with 8 degrees of freedom is greater than the value *t*_{0.05} is 0.05. That is:

Can you find the value *t*_{0.05 }on the *t*-table?

We have determined that the probability that a *T* random variable with 8 degrees of freedom is greater than the value 1.860 is 0.05.

### Why will we encounter a *T* random variable?

Given a random sample *X*_{1}, *X*_{2}, ..., *X*_{n} from a normal distribution, we know that:

\(Z=\dfrac{\bar{X}-\mu}{\sigma/\sqrt{n}}\sim N(0,1)\)

Earlier in this lesson, we learned that:

\(U=\dfrac{(n-1)S^2}{\sigma^2}\)

follows a chi-square distribution with *n*−1 degrees of freedom. We also learned that *Z* and *U* are independent. Therefore, using the definition of a *T* random variable, we get:

It is the resulting quantity, that is:

\(T=\dfrac{\bar{X}-\mu}{s/\sqrt{n}}\)

that will help us, in Stat 415, to use a mean from a random sample, that is \(\bar{X}\), to learn, with confidence, something about the population mean *μ*.