## P-Values and T-Tables

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### Explanation

Using T-Tables
We have learned the following things about a t-test:
• The t-test produces a single value, t, which grows larger as the difference between the means of two samples grows larger;
• t does not cover a fixed range such as 0 to 1 like probabilities do;
• You can convert a t-value into a probability, called a p-value;
• The p-value is always between 0 and 1 and it tells you the probability of the difference in your data being due to sampling error;
• The p-value should be lower than a chosen significance level (0.05 for example) before you can reject your null hypothesis.
Converting from t-values to p-values is usually done by software but you can do it by hand by looking values up in a table. This page explains how.

T-tables are filled with t-values. Each t-value is in a column that is specific to a given significance level. The t-values shown in the table are known as critical values. If your t-value is greater than or equal to the critical value in the table, then you can conclude that your p-value is less than the significance level you have chosen. The procedure is simple. Once you have chosen your significance level you look to see whether your t-value is larger than the critical t-value shown in the column relating to your chosen significance level. If it is, then you can say that p is less than your chosen significance level.

The t-table has more than one row of t-values. Each row corresponds to a given number of degrees of freedom. Degrees of freedom are explained at level on of this topic. Use the row of the table that corresponds to the degrees of freedom in your data.

The final thing that you need to know about using t-tables is that you must read them differently depending on whether your test is one-tailed or two-tailed. The rule is simple enough:

The p-value for a two tailed test is twice what it would be for a one tailed test.

Some tables (such as the ones on this page) show a p-value heading for both one and two-tailed tests to make it easier. If your test is one tailed, find the p-value you need in the top row. If your test is two tailed, find your p-value in the second row. Notice how the p-values for the two tailed test are simply double those for the one tailed test.

Unfortunately, some tables show only one-tailed values and some show only two-tailed values. If you are using a table from a book or from the internet, make sure you know what it shows. You can easily convert from one to two or two to one by remembering the p for two-tailed tests is twice what it is for one-tailed tests.

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.

### Exploration

Converting from a t-value to a p-value requires some tricky maths, so statisticians use pre-calculated tables to make it easy. These tables are called t-tables.

A t-table is shown below. Notice that it has a number of columns, each showing a different significance level (p). T-tables do not tell you the exact value of p. They list a few key values of p and tell you what value of t is required to produce a p-value less than the listed value. It has many rows and each row is marked with a number showing the degrees of freedom (df). Once you have chosen your significance level, p, calculated a value for t, and worked out how many degrees of freedom you have, you can find the entry in the t-table that you need as follows:

• Look down the column that corresponds to your chosen value for p
• Find the row that corresponds to your degrees of freedom, and where they meet, you will find the value you need.
This value is called the critical value. The final thing to do is compare this value with your value of t
• If your t-value is greater than or equal to this value, then t is significant and you have found a difference
• If your t-value is less than this value is then t is not significant.
Here are some examples for you to work out using the table below.
p<0.05
df=4
t=3.143
Tails=1
Critical value:
Significance:
p<0.01
df=12
t=3.143
Tails=1
Critical value:
Significance:
p<0.05
df=8
t=3.143
Tails=2
Critical value:
Significance:
p<0.01
df=7
t=3.143
Tails=2
Critical value:
Significance:
One-tailed p0.10.050.0250.010.005
Two-tailed p
df
0.20.10.050.020.01
13.0786.31412.70631.82163.657
21.8862.924.3036.9659.925
31.6382.3533.1824.5415.841
41.5332.1322.7763.7474.604
51.4762.0152.5713.3654.032
61.441.9432.4473.1433.707
71.4151.8952.3652.9983.499
81.3971.862.3062.8963.355
91.3831.8332.2622.8213.25
101.3721.8122.2282.7643.169
111.3631.7962.2012.7183.106
121.3561.7822.1792.6813.055
131.351.7712.162.653.012
141.3451.7612.1452.6242.977
151.3411.7532.1312.6022.947

### Application

Now lets look at your own data. The first step is to decide on the number of degrees of freedom you have. This depends on whether your samples are paired or independent.
• Your experiment compares when is Pre-Test with when is Post-Test. is measured from the same People under both conditions, Pre-Test and Post-Test.
• You measured 30 People in each condition.
Should you be looking in the one-tailed or two-tailed column?
How many degrees of freedom does your data have?

Your t-value is -2.13. We will ignore the minus sign and just use 2.13, as the values in the t-tables are all positive. Using a p-value of 0.05, look in the section of the t-table shown below and find the critical value for your data.

One-tailed p0.10.050.0250.010.005
Two-tailed p
df
0.20.10.050.020.01
241.3181.7112.0642.4922.797
251.3161.7082.062.4852.787
261.3151.7062.0562.4792.779
271.3141.7032.0522.4732.771
281.3131.7012.0482.4672.763
291.3111.6992.0452.4622.756
301.311.6972.0422.4572.75
311.3091.6962.042.4532.744
321.3091.6942.0372.4492.738
331.3081.6922.0352.4452.733
341.3071.6912.0322.4412.728
p<0.05
df=29
t=2.13
Tails=1
Critical value:
Significance:
Which of these would be the correct way to report this result?
Now look up your t-values at the 0.01 level.
p<0.01
df=29
t=2.13
Tails=1
Critical value:
Significance:
Which of these would be the correct way to report this result?

 Paired T-Test | The Normal Distribution