# One-sample t-test

**One sample t-test**

Comparison of sample mean with population mean when standard deviation of the population is estimated from the sample.

**Requirements: **

Dependent variable is at least interval scaled and stems from a normal distributed population.

**Hypothesis:**

Nondirectional:

H0:

H1:

Directional:

H0: or

H1: or

The term is t-distributed with n-1 degrees of freedom.

whereas

and

The obtained t-value is then compared with the critical t-value to make the decision of significance.

However, for sufficient large samples (n > 100)* one can compute the z-Value even if σ is estimated from the sample:

* for large samples (n > 100) the t-distribution approximates the standard normal distribution sufficiently well.

**Example of a one sample t-test**

A random sample of 22 fifth grade pupils have a grade point average of 5.0 in maths with a standard deviation of 0.452, whereas marks range from 1 (worst) to 6 (excellent). The grade point average (GPA) of all fifth grade pupils of the last five years is 4.7. Is the GPA of the 22 pupils different from the populations’ GPA?

Pupil |
Grade points |

1 |
5 |

2 |
5.5 |

3 |
4.5 |

4 |
5 |

5 |
5 |

6 |
6 |

7 |
5 |

8 |
5 |

9 |
4.5 |

10 |
5 |

11 |
5 |

12 |
4.5 |

13 |
4.5 |

14 |
5.5 |

15 |
4 |

16 |
5 |

17 |
5 |

18 |
5.5 |

19 |
4.5 |

20 |
5.5 |

21 |
5 |

22 |
5.5 |

Mean |
5.0 |

Variance |
0.2045 |

We estimate the standard deviation of the population:

and we get the standard deviation of the mean:

now we can compute the t-value:

for the non directed Hypothesis we have a critical t-value of t_{(df=21, alpha=0.05)}=2.080

for the directed Hypothesis we have a critical t-value of t_{(df=21, alpha=0.05)}=1,721

The obtained t-value is greater that the critical t-value. The grade point average of the 22 pupils is different from the populations’ GPA.

BrightStat output of one sample t-test example

This is a fictitious example.