# AP Statistics Curriculum 2007 Fisher F

## 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:
$$\frac{\sqrt{\frac{(d_1\,x)^{d_1}\,\,d_2^{d_2}} {(d_1\,x+d_2)^{d_1+d_2}}}} {x\,\mathrm{B}\!\left(\frac{d_1}{2},\frac{d_2}{2}\right)}\!$$

CDF:
$$I_{\frac{d_1 x}{d_1 x + d_2}}(d_1/2, d_2/2)\!$$

Mean:
$$\frac{d_2}{d_2-2}\!$$ for $$d_2 > 2$$

Median:
None

Mode:
$$\frac{d_1-2}{d_1}\;\frac{d_2}{d_2+2}\!$$ for $$d_1 > 2$$

Variance:
$$\frac{2\,d_2^2\,(d_1+d_2-2)}{d_1 (d_2-2)^2 (d_2-4)}\!$$ for $$d_2 > 4$$

Support:
$$x \in [0, +\infty)\!$$

Moment Generating Function
Does Not Exist

### 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$\overline{Y}_1=\tfrac{29+30+29+28+30+28}{6} = 29$

Brand B$\overline{Y}_2\tfrac{30+31+32+29+31+33}{6} = 31$

Brand C$\overline{Y}_3\tfrac{28+29+28+26+30+29}{6} = 28$

Step 2: Calculate the overall mean:

### $$\overline{Y}=29+31+28=29.67$$

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

\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}

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, $$MS_B$$, is $$\tfrac{30.04}{2}=15.02$$

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, $$SS_w$$, 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, $$MS_W$$ is$\tfrac{24}{15}=1.6$

Step 5: Finally, the F-Ratio is$\tfrac{MS_B}{MS_W}=\tfrac{15.02}{1.6}=9.39$

The F critical value is the value that the test statistic must exceed in order to reject the $$H_0$$. In this case, $$F_crit(2,15)=3.68$$ at $$\alpha=0.05$$. 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:

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)