AP Statistics Curriculum 2007 Fisher F

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<math>x \in [0, +\infty)\!</math>
<math>x \in [0, +\infty)\!</math>
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'''1st Moment''': <br>
+
===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:
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'''2nd Moment''': <br>
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{| 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.
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'''Kth Moment''': <br>
+
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>
 +
 
 +
 
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'''Step 2:''' Calculate the overall mean: <br>
 +
 
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===<math>\overline{Y}=29+31+28=29.67</math>===
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'''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
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\end{align}
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</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]]
 +
 
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===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)

Revision as of 06:12, 3 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:
\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 d2 > 2

Median:
None

Variance:
\frac{2\,d_2^2\,(d_1+d_2-2)}{d_1 (d_2-2)^2 (d_2-4)}\! for d2 > 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, H0, 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, MSB, 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, SSw, 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, MSW 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 H0. In this case, Fcrit(2,15) = 3.68 at α = 0.05. Since F=9.39>3.68, we reject H0 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)

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