How are t and p Related? You have learned that a t-test produces a t-value, which is then converted into a probability (or p-value). You may wonder why we don't just report the value of t. The main reason is that p-values are much easier to understand. A p-value answers this question: If my null hypothesis is true, what is the probability of getting a t-value at least as big as mine?. It tells you the probability of seeing the difference you found (or larger) due to sampling error if there is really no difference. It doesn't need all the extra interpretation that a t-value needs, such as knowing how many degrees of freedom or tails there are. It is worth noting that it is NOT the probability of the null hypothesis being true. It is specifically the probability of seeing a difference at least as large as yours between two samples from a single population (i.e.where there is no difference).

Here are the key relationships between p-values and t-values:

As t-values get bigger, p-values get smaller;

T-values can be negative, but p-values are always positive. The p-value for a negative t-value is the same as that for the positive version of that t-value;

A two tailed test will produce a p-value that is twice as large as a one tailed test would produce;

A two tailed test requires a higher value of t to produce the same p-value as a one tailed test would;

If you want to know where the t-tables come from, you can read the help topic below.

Exploration

The game below shows how t-values and p-values are related. The curve shows values of t across the bottom (this curve is for very large data sets - i.e. with many degrees of freedom). Notice how the value of t can be between -4 and +4. Obviously, t-values can be much larger than 4, but only if the samples are from different populations. Its is very rare indeed for two large samples from the same population to produce a t-value greater than 4.

Indeed, the t-tables, and the curve below tell you how rare (i.e. how low the probability is) any given t-value would be if the samples were taken from the same population.

The curve is highest where the difference is zero, indicating that the average difference between two random samples from the same population is zero. As the differences get bigger, the probability of two samples showing such a difference gets smaller. The red shaded area on the curve shows you where the largest differences lie. You can choose to see the largest 1%, 5% or 10% of possible differences and you can choose whether you want to see that percentage in one direction (one-tailed) or two. The shaded area corresponds to the p-values at the top of t-tables.

Which value changes as you move along the horizontal axis of the graph?

Which value changes as you move up the vertical axis of the graph?

As the t-value moves towards zero, what happens to the p-value?

How does the p-value for t=-2 compare to the p-value for t=2?

To reject the null hypothesis, should your t-value be in the red zone or the white?

When p is in the red zone, is p very large or very small?

By reducing the significance level from 10% to 5%, what effect do you have on the size of t-value that would lead you to reject the null hypothesis?

Application

When you carried out a one-tailed t-test on your data, you found a t-value of t=2.13 and a p-value of p<0.021. Look at the chart in the game above and answer these questions about your data.

Working at the 5% significance level, where does your t-value fall on the horizontal axis of the chart?

Image that somebody repeated your experiment with different random samples from the same population and they got a t-value of 1.598

Should the fact that their t-value is different from yours make you think that a mistake has been made?

Their t-value produces a p-value of 0.06. Is their conclusion different from yours at the 5% level?