## P-Values and T-Tables

 Getting Started General Instructions | Introduction to Your Study | Experimental Design | Stating a Hypothesis Descriptive Statistics Histograms | Central Tendency | Standard Deviation | Confidence Intervals Comparing Two Samples Samples and Populations | Choosing a T-Test | Paired T-Test | P-Values and T-Tables Important Concepts The Normal Distribution | Z Scores | Probability Distributions Levels You are currently on P-Values and T-Tables at level 1. Level 1 | Level 2 | Level 3 Next Topic Paired T-Test | The Normal Distribution

### Explanation

Understanding p-Values and t-Values
When you learned about the t-test (independent, paired, or both) you saw that results are reported as a probability of the differences in the data occuring from the same population by chance. This probability is called the p-value. The p-value is not produced directly by the t-test, it is calculated in one further step, using the outcome of the t-test (which is called the t-value!).

These two values are related and they are explained in this section.

T-Values
Most software packages report the t-value as part of the summary of the results of a t-test. The value of t is the result of putting the sample data through the formula for the t-test.

The t-value is related to the size of the difference between the means of the two samples you are comparing. The larger t is, the larger the difference. The t-value is not the most useful result to report, which is why we also report p-values.

P-Values
A p-value answers this question: If my null hypothesis were true, what is the probability of getting a t-value at least as big as mine?. Obviously, the lower this value is, the less likely it is that you would find a difference like yours by chance.

The p-value helps you decide whether or not to accept the null hypothesis. You make this decision by deciding how low your p-value should be before you will reject the null hypothesis. This cut-off point is called the significance level and is usually set at 0.05 or 0.01.

Sometimes you will see the p value reported as an exact figure, for example, p=0.023. In other places, you will see a less than sign (<), for example, p<0.05. This is because computer programs can calculate the exact value of p, whereas looking up p-values in tables (which is covered at level two of this topic) only allows you to say that the p-value is below one of a few set values. Either method is acceptable, but we will use the less than sign because we use tables at level two. Note also, that software will sometimes report very low p-values, 0.00001, for example, or even p=0. Never report that p=0, this is a side effect of the limited accuracy of some software. If p is less than 0.001, then report p<0.001 rather than the exact p value.

Degrees of Freedom
The degrees of freedom (df) of a set of data relates to the number of values there are in that data.

• For paired samples (within subjects design), for a paired t-test, df is the number of People minus 1;
• For independent samples (between subjects design), for an independent t-test, df is the number of People in sample 1 plus the number of People in sample 2 minus 2.
The more data you have, the smaller your sampling error is likely to be. The degrees of freedom value takes this into account when calculating the p-value.

Reporting p-Values and t-Values
You report the results of a t-test in the following way:

t(df)=t, p<p

Where df is the degrees of freedom of your data, t is the t-value you found and p is the p-value you found.

Here is an example: t(20)=2.8, p<0.01, which means that a t-test with 20 degrees of freedom produced a t-value of 2.8, which means that the probability of the difference being due to chance is less than 0.01. As we mentioned above, you can also report the exact p value, for example, t(20)=2.8, p=0.007

### Exploration

In this section, we will present you with some imaginary results in a two different formats. Your job is to interpret what they mean.

1. Output From Statistical Software
Imagine your statistics software reported the following:
 `Paired T-Test Resultsdf=16t=1.746p<0.05tails=1`
The probability that the differences in the data were due to sampling error is less than what value?
How many subjects were in the study?
Which of these is the correct way to report these results?

2. Results in a Journal Article
Imagine you read the following in a journal article:
 ResultsThe sample means were compared using an independent t-test with the following results:t(24)=1.9, p<0.05
How many subjects were tested in this study?
Was the difference significant at the 5% level?
Can the researchers reject their null hypothesis?

### Application

Imagine a software package has performed a paired t-test on your data. It would report the following values:

t = -2.13
p = 0.021
df = 29
Tails = 1
What is the probability that the difference between your samples is due to sampling error?
How many degrees of freedom does the data have?
What value of t did the t-test produce?
Which of these would be the correct way to report these results?
Taking p<0.05 to be the significance level, can you reject the null hypothesis, which is: There is no difference in Pre/Post Test Scores?

 Paired T-Test | The Normal Distribution