What is the t-Distribution?
The t-distribution (or Student's t-distribution) is a probability distribution primarily used when:
- The sample size is small
- The population standard deviation is unknown
- The data are normally or nearly normally distributed
Structure of the t-Distribution Table
The t-distribution table contains critical values of the t-distribution for various significance levels (α) and degrees of freedom (df).
Degrees of Freedom (df)
The number of degrees of freedom in a t-table is typically n-1, where n is the sample size.
Significance Level (α)
The significance level represents the probability of rejecting the null hypothesis when it is true (Type I error).
Examples of Using the Table
Example 1: Finding a Critical Value
Suppose we have a sample size of 10 (df = 9) and a significance level of 0.05 for a two-tailed test.
In the table, find the row for df=9 and the column for α=0.05 (two-tailed).
The table value is 2.262.
Example 2: Comparing a Calculated t-Value
Assume we calculated a t-value of 2.5 for df=15 and a significance level of 0.05 for a two-tailed test.
The critical value from the table is 2.131.
Since 2.5 > 2.131, we reject the null hypothesis.
Example 3: Calculating a Confidence Interval
To calculate a 95% confidence interval for the mean, with df=24:
The critical value from the table is 2.064.
Confidence Interval = sample mean ± (2.064 × standard error)
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Bonus: Interactive t-Distribution Table
| df | 0.10 | 0.05 | 0.025 | 0.01 | 0.005 |
|---|
Values in the table represent critical values for a two-tailed test