AP Statistics Curriculum 2007 Fisher F

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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

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:

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)




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