Is "test statistic" a value or a random variable?

Question

I am a student taking my first Statistics course now. I am confused by the term "test statistic".

In the following (I saw this in some textbooks), $t$ seems to be a specific value calculated from a specific sample. $$ t=\frac{\overline{x} - \mu_0}{s / \sqrt{n}} $$

However, in the following (I saw this in some other textbooks), $T$ seems to be a random variable. $$ T=\frac{\overline{X} - \mu_0}{S / \sqrt{n}} $$

So, does the term "test statistic" mean a specific value or a random variable, or both?

Answer 1

The short answer is "yes".

The tradition in notation is to use an upper case letter (T in the above) to represent a random variable, and a lower case letter (t) to represent a specific value computed or observed of that random variable.

T is a random variable because it represents the results of calculating from a sample chosen randomly. Once you take the sample (and the randomness is over) then you can calculate t, the specific value, and make conclusions based on how t compares to the distribution of T.

So the test statistic is a random variable when we think about all the values it could take on based on all the different samples we could collect. But once we collect a single sample, we calculate a specific value of the test statistic.

Answer 2

A test statistic is a statistic used in making a decision about the null hypothesis.

A statistic is a realized value (e.g. t): A statistic is a numerical value that states something about a sample. As statistics are used to estimate the value of a population parameter, they are themselves values. Because (long enough) samples are different all the time, the statistics (the numerical statements about the samples) will differ. A probability distribution of a statistic obtained through a large number of samples drawn from a specific population is called its sampling distribution --- a distribution of that statistic, considered as a random variable.

A statistic is a random variable (e.g. T): A statistic is any function of the data (unchanged from sample to sample). The data are described by random variables (of some suitable dimension). As any function of a random variable is itself a random variable, a statistic is a random variable.

It almost always clear from context what meaning is intended, especially when the upper/low-case convention is observed.

Answer 3

A test statistic is an observation specific to your observed data that follows a probability distribution under a given assumption. This assumption is usually called the $H_0$.

For instance, in your sample the test statistic (called t-statistic) depends on the observed data ($\bar{x}$ and $s$ are both derived from the data).

Under the assumption that your mean is $\mu_0$, the statistic you computed will follow a certain distribution. The probability of this value of the statistic occurring is then determined under the assumption. If that value is deemed to be low, the assumption ($H_0$) is rejected.

If we reject the $H_0$ assumption, this does not mean that the assumption we made was guaranteed to be untrue. If it was true and we rejected it because of the low probability of the test statistic under $H_0$, we call it a type I error.

On the other hand, if we accept the assumption this does not mean that our assumption for sure was true. If the assumption was untrue and we accepted it because it had high enough probability under our wrong assumption, this is called type II error.

The statistic is a specific value and it is only if we accept certain assumptions as given that we can assume it follows a specific probability distribution.

This principle holds for all test statistics, not just for the t-statistic you mention here.