# 5.4.2 - The t-distribution

5.4.2 - The t-distributionIn 1908, William Sealy Gosset from Guinness Breweries discovered the *t*-distribution. His pen-name was Student and thus it is called the "Student's *t*-distribution."

The *t*-distribution is different for different sample size, *n*. Thus, tables, as detailed as the standard normal table, are not provided in the usual statistics books. The graph below shows the t-distribution for degrees of freedom of 10 (blue) and 30 (red dashed).

### Properties of the t-distribution

- t is symmetric about 0
- t-distribution is more variable than the Standard Normal distribution
- t-distributions are different for different degrees of freedom (d.f.).
- The larger $n$ gets (or as $n$ goes to infinity), the closer the $t$-distribution is to the $z$.
- The meaning of $t_\alpha$ is the $t$-value having the area "$\alpha$" to the right of it.

## Example 5-5: Finding t-values

Use this t-table or the one in your text to find following the example.

Find \(t_{0.05}\) where the degree of freedom is 20.

In a t-distribution table below the top row represents the upper tail area, while the first column are the degrees of freedom.

The \(t_{0.05}\) where the degree of freedom is 20 is 1.725 .

df | 0.40 | 0.25 | 0.10 | 0.05 | 0.025 | 0.01 | 0.005 | 0.001 | .0005 |
---|---|---|---|---|---|---|---|---|---|

... | ... | ... | ... | ... | ... | ... | ... | ... | ... |

18 | 0.257 | 0.688 | 1.330 | 1.734 | 2.101 | 2.552 | 2.878 | 3.610 | 3.922 |

19 | 0.257 | 0.688 | 1.328 | 1.729 | 2.093 | 2.539 | 2.861 | 3.579 | 3.883 |

20 | 0.257 | 0.687 | 1.325 | 1.725 | 2.086 | 2.528 | 2.845 | 3.552 | 3.850 |

21 | 0.257 | 0.686 | 1.323 | 1.721 | 2.080 | 2.518 | 2.831 | 3.527 | 3.819 |

The graph shows that the \(\alpha\) values at the top of this table are the upper tail areas of the distribution.

**Note!**When the corresponding degree of freedom is not given in the table, you can use the value for the closest degree of freedom that is

**smaller**than the given one. We use this approach since it is better to err in a conservative manner (get a

*t*-value that is slightly larger than the precise

*t*-value).

Find \(t_{0.05}\) where the degree of freedom is 34.

What do we do when the degrees of freedom are not on the table? The t-table degrees of freedom run continuously from 1 to 30, then go by intervals after 30 (e.g. after 30 we have 35). In such cases, we can use software such as Minitab to find a more exact value for the multiplier as opposed to using a degrees of freedom that is "close".

To find the t-value in Minitab...

- From the Minitab Menu select
`Calc`>`Probability Distributions`>`t...` - Choose inverse cumulative probability
- Enter the degrees of freedom
- Set the input constant as 0.95 (1 - 0.05).
- Choose
`OK`

The output from Minitab gives us \(t_{0.05}\) with df= 34 as 1.69092.

P (X \(\le\) x) | x |
---|---|

0.95 | 1.69092 |

The t-value for an \(\alpha\) of .05 and df of 30 is 1.697.

df | 0.40 | 0.25 | 0.10 | 0.05 | 0.025 | 0.01 | 0.005 | 0.001 | .0005 |
---|---|---|---|---|---|---|---|---|---|

... | ... | ... | ... | ... | ... | ... | ... | ... | ... |

27 | 0.256 | 0.684 | 1.314 | 1.703 | 2.052 | 2.473 | 2.771 | 3.421 | 3.690 |

28 | 0.256 | 0.683 | 1.313 | 1.701 | 2.048 | 2.467 | 2.763 | 3.408 | 3.674 |

29 | 0.256 | 0.683 | 1.311 | 1.699 | 2.045 | 2.462 | 2.756 | 3.396 | 3.659 |

30 | 0.256 | 0.683 | 1.310 | 1.697 | 2.042 | 2.457 | 2.750 | 3.385 | 3.646 |

**Note!** When the sample size is larger than 30, the t-values are not that different from the z-values. Thus, a crude estimate for \(t_{0.05}\) with 34 degrees of freedom is \(z_{0.05} = 1.645\). Although it is a crude estimate, when software is available, it is best to find the $t$ values rather than use the $z$.