t-Distribution Table

8.2. t-Distribution Table#

In the Section 6.3 section, we explored how to use a t-distribution table.

Brief Instructions for Using the t-Distribution Table

To use the t-distribution table, where the top row indicates different significance levels (\(\alpha\)) corresponding to the right-hand side area of the t-distribution and the left column lists degrees of freedom (df), follow these steps:

  1. Identify Degrees of Freedom (df): Determine the degrees of freedom for your t-test. For example, if \(\text{df} = 20\), locate the row labeled “20”.

  2. Select the Significance Level (\(\alpha\)): Choose the significance level for your hypothesis test, typically predetermined based on the desired confidence level. For instance, if \(\alpha = 0.1\), find the column labeled “0.1”.

  3. Locate the Intersection: The intersection of the row for df = 20 and the column for \(\alpha = 0.1\) provides the critical t-value (\(t_{0.1, 20}\)) from the t-distribution table.

  4. Read the Critical t-value: In the table, the critical t-value for df = 20 and \(\alpha = 0.1\) is 1.325.

../_images/t_score_right_hand_side.png

Fig. 8.3 t-Distribution curve illustrating the right-hand side area#

0.1 0.05 0.025 0.01 0.005 0.001
df
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The t-distribution table provides critical values of the t-distribution for various significance levels (\(\alpha\)) and degrees of freedom (df). This table is essential for conducting hypothesis tests and constructing confidence intervals when the sample size is small or the population variance is unknown.

Table Structure

  • Rows: Represent the degrees of freedom (df), which are calculated as \(n - 1\) for a one-sample t-test, where \(n\) is the sample size.

  • Columns: Indicate the significance levels (\(\alpha\)) for one-tailed and two-tailed tests. The values in the table represent the critical t-values corresponding to these significance levels.

Significance Levels

  • \(\alpha = 0.1\): 10% significance level

  • \(\alpha = 0.05\): 5% significance level

  • \(\alpha = 0.025\): 2.5% significance level (two-tailed)

  • \(\alpha = 0.01\): 1% significance level

  • \(\alpha = 0.005\): 0.5% significance level

  • \(\alpha = 0.001\): 0.1% significance level

Example

Scenario: Suppose you are conducting a two-tailed t-test with a sample size of 15. You want to determine the critical t-value for a significance level of \(\alpha = 0.05\).

  1. Calculate Degrees of Freedom:

    \[\begin{equation*} df = n - 1 = 15 - 1 = 14 \end{equation*}\]

  2. Locate the Critical t-value:

    • From the table, find the row for \(df = 14\) and the column for \(\alpha = 0.05\).

    • The intersection gives the critical t-value: 2.145.

  3. Interpretation:

    • If your calculated t-statistic exceeds 2.145 or is less than -2.145, you would reject the null hypothesis at the 5% significance level.