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

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:
\(x \in [0, +\infty)\!\)

Applications

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:

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:

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)