**Part 1: Introduction**

**Definition,** The T.INV function in Microsoft Excel returns the left-tailed inverse of the Student’s t-distribution.

**Purpose** The function is used to find the t-value given a certain probability and degrees of freedom. This is particularly useful in hypothesis testing and confidence interval construction.

**Syntax & Arguments**

```
T.INV(probability, deg_freedom)
```

The T.INV function syntax has the following arguments:

**probability**: Required. The chance associated with the Student’s t-distribution.**deg_freedom**: Required. The number of degrees of freedom with which to characterize the distribution.

**Return value** The T.INV function returns the left-tailed inverse of the Student’s t-distribution.

**Remarks**

- If either argument is non-numeric, T.INV returns the #VALUE! Error value.
- If probability <= 0 or if probability > 1, T.INV returns the #NUM! Error value.
- If deg_freedom is not an integer, it is truncated.
- If deg_freedom < 1, T.INV returns the #NUM! Error value.

**Part 2: Examples**

**Example 1**

*Purpose of Example*: To calculate the left-tailed inverse of the Student’s t-distribution for a given probability and degrees of freedom.

A | B | C | D | |
---|---|---|---|---|

1 | Probability | Degrees of Freedom | Formula | Result |

2 | 0.75 | 2 | `=T.INV(A2, B2)` | 0.8165 |

*Explanation*: In this example, we use the T.INV function to calculate the left-tailed inverse of the Student’s t-distribution for a probability of 0.75 and 2 degrees of freedom.

**Example 2**

*Purpose of Example*: To calculate the left-tailed inverse of the Student’s t-distribution for a different probability and degrees of freedom.

A | B | C | D | |
---|---|---|---|---|

1 | Probability | Degrees of Freedom | Formula | Result |

2 | 0.80 | 3 | `=T.INV(A2, B2)` | 0.9785 |

*Explanation*: In this example, we use the T.INV function to calculate the left-tailed inverse of the Student’s t-distribution for a probability of 0.80 and 3 degrees of freedom.

**Example 3**

*Purpose of Example*: To calculate the left-tailed inverse of the Student’s t-distribution for a different probability and degrees of freedom.

A | B | C | D | |
---|---|---|---|---|

1 | Probability | Degrees of Freedom | Formula | Result |

2 | 0.85 | 4 | `=T.INV(A2, B2)` | 1.0639 |

*Explanation*: In this example, we use the T.INV function to calculate the left-tailed inverse of the Student’s t-distribution for a probability of 0.85 and 4 degrees of freedom.

**Example 4**

*Purpose of Example*: To calculate the left-tailed inverse of the Student’s t-distribution for a different probability and degrees of freedom.

A | B | C | D | |
---|---|---|---|---|

1 | Probability | Degrees of Freedom | Formula | Result |

2 | 0.90 | 5 | `=T.INV(A2, B2)` | 1.4759 |

*Explanation*: In this example, we use the T.INV function to calculate the left-tailed inverse of the Student’s t-distribution for a probability of 0.90 and 5 degrees of freedom.

**Example 5**

*Purpose of Example*: To calculate the left-tailed inverse of the Student’s t-distribution for a different probability and degrees of freedom.

A | B | C | D | |
---|---|---|---|---|

1 | Probability | Degrees of Freedom | Formula | Result |

2 | 0.95 | 6 | `=T.INV(A2, B2)` | 1.9432 |

*Explanation*: In this example, we use the T.INV function to calculate the left-tailed inverse of the Student’s t-distribution for a probability of 0.95 and 6 degrees of freedom.

**Example 6**

*Purpose of Example*: To calculate the left-tailed inverse of the Student’s t-distribution for a different probability and degrees of freedom.

A | B | C | D | |
---|---|---|---|---|

1 | Probability | Degrees of Freedom | Formula | Result |

2 | 0.99 | 7 | `=T.INV(A2, B2)` | 2.9979 |

*Explanation*: In this example, we use the T.INV function to calculate the left-tailed inverse of the Student’s t-distribution for a probability of 0.99 and 7 degrees of freedom.

**Example 7**

*Purpose of Example*: To calculate the left-tailed inverse of the Student’s t-distribution for a different probability and degrees of freedom.

A | B | C | D | |
---|---|---|---|---|

1 | Probability | Degrees of Freedom | Formula | Result |

2 | 0.995 | 8 | `=T.INV(A2, B2)` | 3.4995 |

*Explanation*: In this example, we use the T.INV function to calculate the left-tailed inverse of the Student’s t-distribution for a probability of 0.995 and 8 degrees of freedom.

**Part 3: Tips and Tricks**

- Always ensure that the degree of freedom is at least 1. If it’s less than 1, the T.INV function will return an error.
- The T.INV function can be used instead of a table of critical values for the t-distribution.
- If probability <= 0 or if probability > 1, T.INV returns the #NUM! Error value.
- If any argument is non-numeric, T.INV returns the #VALUE! Error value. Always ensure that your arguments are numeric.
- The T.INV function is helpful in hypothesis testing small sample data sets and constructing confidence intervals.