# AP Statistics Curriculum 2007 Fisher F

### From Socr

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===Fisher's F Distribution=== | ===Fisher's F Distribution=== | ||

- | Commonly used as the null distribution of a test statistic, such as in analysis of variance [http:// | + | Commonly used as the null distribution of a test statistic, such as in analysis of variance [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_ANOVA_1Way ANOVA]. Relationship to the [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_StudentsT t-distribution] and [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Beta beta Distribution]. |

'''PDF''': <br> | '''PDF''': <br> | ||

Line 17: | Line 17: | ||

'''Median''': <br> | '''Median''': <br> | ||

None | None | ||

+ | |||

+ | '''Mode''': <br> | ||

+ | <math>\frac{d_1-2}{d_1}\;\frac{d_2}{d_2+2}\!</math> for <math>d_1 > 2</math> | ||

'''Variance''': <br> | '''Variance''': <br> | ||

Line 24: | Line 27: | ||

<math>x \in [0, +\infty)\!</math> | <math>x \in [0, +\infty)\!</math> | ||

- | ''' | + | '''Moment Generating Function''' <br> |

+ | Does Not Exist | ||

+ | |||

+ | ===Applications=== | ||

+ | [http://en.wikipedia.org/wiki/ANOVA ANOVA] | ||

+ | |||

+ | ===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: | ||

+ | |||

+ | {| class="wikitable" | ||

+ | |- | ||

+ | ! 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, <math>H_0</math>, 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: <br> | ||

+ | |||

+ | Brand A: <math>\overline{Y}_1=\tfrac{29+30+29+28+30+28}{6} = 29</math> | ||

+ | |||

+ | Brand B: <math>\overline{Y}_2\tfrac{30+31+32+29+31+33}{6} = 31</math> | ||

+ | |||

+ | Brand C: <math>\overline{Y}_3\tfrac{28+29+28+26+30+29}{6} = 28</math> | ||

+ | |||

+ | |||

+ | '''Step 2:''' Calculate the overall mean: <br> | ||

+ | |||

+ | ===<math>\overline{Y}=29+31+28=29.67</math>=== | ||

+ | |||

+ | '''Step 3:''' Calculate the Between-Group Sum of Squares: <br> | ||

+ | |||

+ | <math> | ||

+ | \begin{align} | ||

+ | SS_b &= n(\overline{Y}_1-\overline{Y})^2+n(\overline{Y}_2-\overline{Y})^2+n(\overline{Y}_3-\overline{Y})^2\\ | ||

+ | &= 6(29-29.67)^2+6(31-29.67)^2+6(28-29.67)^2=30.04 | ||

+ | \end{align} | ||

+ | </math> | ||

+ | |||

+ | 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, <math>MS_B</math>, is <math>\tfrac{30.04}{2}=15.02</math> | ||

+ | |||

+ | '''Step 4:''' Calculate the Within-Group Sum of Squares: <br> | ||

+ | |||

+ | We start by subtracting each observation by its group mean: | ||

+ | |||

+ | {| class="wikitable" | ||

+ | |- | ||

+ | ! 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, <math>SS_w</math>, is the sum of the squares of the values in the previous table: | ||

+ | |||

+ | <math>0+1+0+1+0+1+0+1+0+1+4+4+1+0+4+1+4+1=24</math> | ||

+ | |||

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

+ | |||

+ | <math>3(6-1)=15</math> | ||

+ | |||

+ | The Within-Group Mean Square Value, <math>MS_W</math> is: <math>\tfrac{24}{15}=1.6</math> | ||

+ | |||

+ | '''Step 5:''' Finally, the F-Ratio is: | ||

+ | |||

+ | <math>\tfrac{MS_B}{MS_W}=\tfrac{15.02}{1.6}=9.39</math> | ||

+ | |||

+ | The F critical value is the value that the test statistic must exceed in order to reject the <math>H_0</math>. In this case, <math>F_crit(2,15)=3.68</math> at <math>\alpha=0.05</math>. Since F=9.39>3.68, we reject <math>H_0</math> 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: | ||

+ | |||

+ | [[File:F.png]] | ||

+ | |||

+ | ===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) | |

+ | <hr> | ||

+ | * SOCR Home page: http://www.socr.ucla.edu | ||

- | + | {{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Fisher_F}} |

## Current revision as of 22:42, 18 July 2011

## 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

**Mode**:

for *d*_{1} > 2

**Variance**:

for *d*_{2} > 4

**Support**:

**Moment Generating Function**

Does Not Exist

### 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)

- SOCR Home page: http://www.socr.ucla.edu

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