AP Statistics Curriculum 2007 Fisher F

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 Revision as of 22:23, 2 July 2011 (view source)JayZzz (Talk | contribs) (Created page with '== General Advance-Placement (AP) Statistics Curriculum - Fisher's F Distribution== ===Fisher's F Distribution=== Commonly used as the null di…')← Older edit Revision as of 06:12, 3 July 2011 (view source)JayZzz (Talk | contribs) Newer edit → Line 24: Line 24: $x \in [0, +\infty)\!$ $x \in [0, +\infty)\!$ - '''1st Moment''':
+ ===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: - '''2nd Moment''':
+ {| 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, $H_0$, is that the three brands of gasoline will yield the same amount of gas mileage, on average. - '''Kth Moment''':
+ 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: + + {| 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, $SS_w$, 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: + + [[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)

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

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:

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)