In the realm of statistics, a t table is a critical tool used extensively in hypothesis testing. The t distribution appears when the mean of a normally distributed population is estimated in scenarios with a small sample size and an unknown population standard deviation. In such cases, the t table provides critical values for t tests, which are used to determine whether the difference between sample means and a population mean is statistically significant. Find an in-depth insight in this article.
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t Table — In a Nutshell
- The t table in statistics lists the critical values of the t distribution.
- Its columns contain all the t distribution probabilities denoted by the letter “p.”
- The rows of the t table contain the degrees of freedom denoted by the symbol “df.”
- There are two types of t tables in statistics: the one-tail t table and the two-tail t table.
Definition: t table
A t table is a reference statistical table that contains critical values of the t distribution, also known as the t score or t value. The t value explains the significance threshold for specific tests in statistics and the upper or lower confines of confidence intervals for explicit estimates.
The t table is used in statistics when the sample size is small, or when you don’t know the population’s standard deviation. You can also use the t table during a t-test. A t-test is a statistical test used to liken the means of two sets or groups of data.
It is also used in hypothesis testing. You can also use the t table to test the difference between two means, if two variable quantities are significantly correlated, and to calculate the confidence intervals of statistical means or lapse/regression coefficients.
Using the t table
You can use a t table to determine a critical value of t to execute statistical tests or find a confidence interval.
Example
Assume you are testing a new pharmaceutical product for treating acne. You group the participants randomly into a treatment group and a control group. The treatment group receives the acne medication, while the control group receives a placebo cream (unmedicated). You must compare the mean of the number of pimples the partakers in the treatment and placebo sample group have in the treatment to discover the effectiveness of the acne treatment. You use an autonomous samples t-test.
Null hypothesis (H0):
The treatment and placebo sample group have the same mean value of pimples.
Alternative hypothesis (H1):
The treatment and placebo sample groups do not have the same mean quantity of pimples.
After calculating the t value of the sample, you can compare it to the critical value of t to determine if you can discard the null hypothesis or not.
Step 1: One-tailed and two-tailed tests
Firstly, you must determine if you want to use a two or one-tailed test. Below are guidelines for determining when to use one-tailed and two-tailed tests.
Use a one-tailed test when you have a directional alternative hypothesis. A directional premise emphasizes that the population parameter (like mean or reversion constant) is more or a smaller amount than a specific value, like zero.
A directional alternative hypothesis features words like greater than, less than, increases, or decreases. So, a hypothesis that does not feature these words is usually non-directional.
Use a two-tailed test when the alternate premise is non-directional. This type of hypothesis states that the mean or regression coefficient (population parameter) is unequal to a certain value, like zero.
You can also use two-tailed t-tests when calculating a confidence interval. Many studies use two-tailed tests.
Example
Alternative hypothesis (H1): The treatment and control groups differentiate in the mean number of pimples.
This alternative hypothesis is non-directional because it does not state whether the treatment group’s mean is more or less than the control group’s.
In conclusion, since the alternate (premise) hypothesis is non-directional, you should use a two-tailed test.
Step 2: Calculating the degrees of freedom
Once you have decided to use a two-tailed test, the next step is calculating the degrees of freedom.
You can calculate the (df) degree of freedom from the general trial size (n). Also, the type of equation you need will depend on the test you decide to perform.
| Test Type or Procedure | df Equation |
| One-sample t-test Confidence interval of a mean |
 |
| Independent samples t-test |
= sample size of the 1st group = sample size of the 2nd group |
| Dependent samples t-test | |
| Linear regression Pearson correlation Spearman rank correlation Confidence interval of a regression |
|
Example
The df equation for autonomous t-tests is:
So, if you steered a tentative treatment trial with 15 partakers in the control group (placebo) and 19 participants in the treatment group, then the degree of freedom is:
Step 3: Choosing a significance level
Traditionally, the level of significance, denoted as, is 0.05. However, in certain situations, you can decrease the α to reduce the chances of Type I errors or increase the α to decrease the risk of Type II errors.
So, you select the confidence interval depending on your selected confidence level. So, α = 1 – confidence level. The confidence interval is usually 0.95 since the most prevalent confidence interval is usually 0.05.
It is worth noting that the α section is typically highlighted in the t table as it is the most prevalently applied as a significance level.
Example
For our sample above, you can choose an α of 0.05 to test your hypothesis, as it is the most common significance level used by researchers.
Step 4: Finding the critical value
Now that you have all the data you require to apply the t table, you can find the critical value:
- Apply the first t table when using a two-tailed test or finding a confidence interval.
- Apply the second table when using a one-tailed test.
- The dfs are listed on the left side of the table.
- Find the row with the df you found in the second step.
- Round the number down to the nearest smallest if it is not listed.
- Find the level of significance at the top of the t table.
- Find the significance level you chose in step three.
- Determine the critical value of t for your statistical test where the row and the column intersect.
Example
Using the t table, you will discover that in a two-tailed test featuring a
and
,
the critical value is .
Compare the critical value of t you find from the table to the t value you previously calculated for your sample to determine whether to reject your null hypothesis.
FAQs
The t table helps you determine the critical value of your sample and compare it with your t value to determine whether to reject your null hypothesis.
The two types of t-tests are one-tailed and two-tailed tests.
Use a one-tailed test when you have a directional alternative hypothesis. In contrast, use a two-tailed test when the alternate premise is non-directional.
Firstly, you must choose between one-tailed or two-tailed tests, find the degrees of freedom, and choose a significance level. You can use this information to determine the critical value of t in the t table.
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