# AP Statistics Curriculum 2007 Fisher F

### From Socr

## Contents |

## General Advance-Placement (AP) Statistics Curriculum - Fisher's F Distribution

### Fisher's F Distribution

Commonly used as the null distribution of a test statistic, such as in analysis of variance (ANOVA). Relationship to the t-distribution and [beta Distribution].

**PDF**:

**CDF**:

**Mean**:

for *d*_{2} > 2

**Median**:

None

**Variance**:

for *d*_{2} > 4

**Support**:

### Applications

### Example

We want to examine the effect of three different brands of gasoline on gas mileage using an alpha value of 0.05. We will have 6 observations for each of the 3 gasoline brands. Gas mileage figures are as follows:

Brand A | Brand B | Brand C |
---|---|---|

29 | 30 | 28 |

30 | 31 | 29 |

29 | 32 | 28 |

28 | 29 | 26 |

30 | 31 | 30 |

28 | 33 | 29 |

Our null hypothesis, *H*_{0}, is that the three brands of gasoline will yield the same amount of gas mileage, on average.

First, we find the F-ratio:

**Step 1:** Calculate the mean for each brand:

Brand A:

Brand B:

Brand C:

**Step 2:** Calculate the overall mean:

**Step 3:** Calculate the Between-Group Sum of Squares:

Where n is the number of observations per group.

The between-group degrees of freedom is one less than the number of groups: 3-1=2.

Therefore, the between-group mean square value, *M**S*_{B}, is

**Step 4:** Calculate the Within-Group Sum of Squares:

We start by subtracting each observation by its group mean:

Brand A | Brand B | Brand C |
---|---|---|

29-29=0 | 30-31=-1 | 28-28=0 |

30-29=1 | 31-31=0 | 29-28=1 |

29-29=0 | 32-31=1 | 28-28=0 |

28-29=-1 | 29-31=-2 | 26-28=-2 |

30-29=1 | 31-31=0 | 30-28=2 |

28-29=-1 | 33-31=2 | 29-28=1 |

The Within-Group Sum of Squares, *S**S*_{w}, is the sum of the squares of the values in the previous table:

0 + 1 + 0 + 1 + 0 + 1 + 0 + 1 + 0 + 1 + 4 + 4 + 1 + 0 + 4 + 1 + 4 + 1 = 24

The Within-Group degrees of freedom is the number of groups times 1 less the number of observations per group:

3(6 − 1) = 15

The Within-Group Mean Square Value, *M**S*_{W} is:

**Step 5:** Finally, the F-Ratio is:

The F critical value is the value that the test statistic must exceed in order to reject the *H*_{0}. In this case, *F*_{c}*r**i**t*(2,15) = 3.68 at α = 0.05. Since F=9.39>3.68, we reject *H*_{0} at the 5% significance level, concluding that there is a difference in gas mileage between the gasoline brands.

We can find the critical F-value using the SOCR F Distribution Calculator:

### SOCR Links

http://www.distributome.org/ -> SOCR -> Distributions -> Fisher’s F

http://www.distributome.org/ -> SOCR -> Distributions -> Fisher’s F Distribution

http://www.distributome.org/ -> SOCR -> Functors -> Fisher’s F Distribution

http://www.distributome.org/ -> SOCR -> Analyses -> ANOVA – One Way

http://www.distributome.org/ -> SOCR -> Analyses -> ANOVA – Two Way

SOCR F-Distribution Calculator (http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html)