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AP Statistics Curriculum 2007 Fisher F - Revision history
2024-03-28T20:22:41Z
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http://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_Fisher_F&diff=11199&oldid=prev
JayZzz: /* Fisher's F Distribution */
2011-07-18T22:48:57Z
<p><span class="autocomment">Fisher's F Distribution</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===Fisher's F Distribution===</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===Fisher's F Distribution===</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>Commonly used as the null distribution of a test statistic, such as in analysis of variance [http://<del class="diffchange diffchange-inline">en</del>.<del class="diffchange diffchange-inline">wikipedia</del>.<del class="diffchange diffchange-inline">org</del>/<del class="diffchange diffchange-inline">wiki</del>/<del class="diffchange diffchange-inline">Anova (</del>ANOVA<del class="diffchange diffchange-inline">)</del>]. Relationship to the [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_StudentsT t-distribution] and [beta Distribution].</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>Commonly used as the null distribution of a test statistic, such as in analysis of variance [http://<ins class="diffchange diffchange-inline">wiki</ins>.<ins class="diffchange diffchange-inline">stat</ins>.<ins class="diffchange diffchange-inline">ucla.edu</ins>/<ins class="diffchange diffchange-inline">socr</ins>/<ins class="diffchange diffchange-inline">index.php/AP_Statistics_Curriculum_2007_ANOVA_1Way </ins>ANOVA]. Relationship to the [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_StudentsT t-distribution] and [<ins class="diffchange diffchange-inline">http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Beta </ins>beta Distribution].</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>'''PDF''': <br></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>'''PDF''': <br></div></td></tr>
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JayZzz
http://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_Fisher_F&diff=11111&oldid=prev
JayZzz at 04:36, 7 July 2011
2011-07-07T04:36:50Z
<p></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>'''Median''': <br></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>'''Median''': <br></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>None</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>None</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">'''Mode''': <br></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"><math>\frac{d_1-2}{d_1}\;\frac{d_2}{d_2+2}\!</math> for <math>d_1 > 2</math></ins></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>'''Variance''': <br></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>'''Variance''': <br></div></td></tr>
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JayZzz
http://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_Fisher_F&diff=11106&oldid=prev
JayZzz at 04:24, 7 July 2011
2011-07-07T04:24:18Z
<p></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>'''Support''': <br></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>'''Support''': <br></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>x \in [0, +\infty)\!</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>x \in [0, +\infty)\!</math></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">'''Moment Generating Function''' <br></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Does Not Exist</ins></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===Applications===</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===Applications===</div></td></tr>
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JayZzz
http://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_Fisher_F&diff=11093&oldid=prev
JayZzz at 06:21, 3 July 2011
2011-07-03T06:21:03Z
<p></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>* SOCR Home page: http://www.socr.ucla.edu</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>* SOCR Home page: http://www.socr.ucla.edu</div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php/<del class="diffchange diffchange-inline">AP_Statistics_Curriculum_2007_Cauchy</del>}}</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php/<ins class="diffchange diffchange-inline">AP_Statistics_Curriculum_2007_Fisher_F</ins>}}</div></td></tr>
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JayZzz
http://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_Fisher_F&diff=11092&oldid=prev
JayZzz at 06:20, 3 July 2011
2011-07-03T06:20:47Z
<p></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>SOCR F-Distribution Calculator (http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html)</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>SOCR F-Distribution Calculator (http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html)</div></td></tr>
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<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">* SOCR Home page: http://www.socr.ucla.edu</ins></div></td></tr>
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JayZzz
http://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_Fisher_F&diff=11091&oldid=prev
JayZzz at 06:20, 3 July 2011
2011-07-03T06:20:11Z
<p></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>x \in [0, +\infty)\!</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>x \in [0, +\infty)\!</math></div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">'''1st Moment'''</del>: <del class="diffchange diffchange-inline"><br></del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">===Applications===</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">[http</ins>:<ins class="diffchange diffchange-inline">//en.wikipedia.org/wiki/ANOVA ANOVA]</ins></div></td></tr>
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<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">===Example===</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">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:</ins></div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">'''2nd Moment''': <br></del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">{| class="wikitable"</ins></div></td></tr>
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<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">! Brand A</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">! Brand B</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">! Brand C</ins></div></td></tr>
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<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">| 29</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">| 30</ins></div></td></tr>
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<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">|}</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Our null hypothesis, <math>H_0</math>, is that the three brands of gasoline will yield the same amount of gas mileage, on average.</ins></div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>'''<del class="diffchange diffchange-inline">Kth Moment</del>''': <br></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">First, we find the F-ratio:</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>'''<ins class="diffchange diffchange-inline">Step 1:</ins>''' <ins class="diffchange diffchange-inline">Calculate the mean for each brand</ins>: <br></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">Brand A: <math>\overline{Y}_1=\tfrac{29+30+29+28+30+28}{6} = 29</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">Brand B: <math>\overline{Y}_2\tfrac{30+31+32+29+31+33}{6} = 31</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">Brand C: <math>\overline{Y}_3\tfrac{28+29+28+26+30+29}{6} = 28</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">'''Step 2:''' Calculate the overall mean: <br></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">===<math>\overline{Y}=29+31+28=29.67</math>===</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">'''Step 3:''' Calculate the Between-Group Sum of Squares: <br></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline"><math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">\begin{align}</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">SS_b &= n(\overline{Y}_1-\overline{Y})^2+n(\overline{Y}_2-\overline{Y})^2+n(\overline{Y}_3-\overline{Y})^2\\</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">&= 6(29-29.67)^2+6(31-29.67)^2+6(28-29.67)^2=30.04</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">\end{align}</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline"></math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">Where n is the number of observations per group.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">The between-group degrees of freedom is one less than the number of groups: 3-1=2.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">Therefore, the between-group mean square value, <math>MS_B</math>, is <math>\tfrac{30.04}{2}=15.02</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">'''Step 4:''' Calculate the Within-Group Sum of Squares: <br></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">We start by subtracting each observation by its group mean:</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">{| class="wikitable"</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">|-</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">! Brand A</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">! Brand B</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">! Brand C</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">|-</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">| 29-29=0</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">| 30-31=-1</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">| 28-28=0</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">|-</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">| 30-29=1</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">| 31-31=0</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">| 29-28=1</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">|-</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">| 29-29=0</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">| 32-31=1</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">| 28-28=0</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">|-</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">| 28-29=-1</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">| 29-31=-2</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">| 26-28=-2</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">|-</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">| 30-29=1</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">| 31-31=0</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">| 30-28=2</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">|-</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">| 28-29=-1</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">| 33-31=2</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">| 29-28=1</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">|}</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">The Within-Group Sum of Squares, <math>SS_w</math>, is the sum of the squares of the values in the previous table:</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline"><math>0+1+0+1+0+1+0+1+0+1+4+4+1+0+4+1+4+1=24</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">The Within-Group degrees of freedom is the number of groups times 1 less the number of observations per group:</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline"><math>3(6-1)=15</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">The Within-Group Mean Square Value, <math>MS_W</math> is: <math>\tfrac{24}{15}=1.6</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">'''Step 5:''' Finally, the F-Ratio is:</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline"><math>\tfrac{MS_B}{MS_W}=\tfrac{15.02}{1.6}=9.39</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">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.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">We can find the critical F-value using the SOCR F Distribution Calculator:</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">[[File:F.png]]</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">===SOCR Links===</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">http://www.distributome.org/ -> SOCR -> Distributions -> Fisher’s F</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">http://www.distributome.org/ -> SOCR -> Distributions -> Fisher’s F Distribution</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">http://www.distributome.org/ -> SOCR -> Functors -> Fisher’s F Distribution</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">http://www.distributome.org/ -> SOCR -> Analyses -> ANOVA – One Way</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">http://www.distributome.org/ -> SOCR -> Analyses -> ANOVA – Two Way</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">SOCR F-Distribution Calculator (http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html)</ins></div></td></tr>
<!-- diff generator: internal 2024-03-28 20:22:41 -->
</table>
JayZzz
http://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_Fisher_F&diff=11088&oldid=prev
JayZzz: Created page with '== General Advance-Placement (AP) Statistics Curriculum - Fisher's F Distribution== ===Fisher's F Distribution=== Commonly used as the null di…'
2011-07-02T22:31:32Z
<p>Created page with '==<a href="/socr/index.php/AP_Statistics_Curriculum_2007" class="mw-redirect" title="AP Statistics Curriculum 2007"> General Advance-Placement (AP) Statistics Curriculum</a> - Fisher's F Distribution== ===Fisher's F Distribution=== Commonly used as the null di…'</p>
<p><b>New page</b></p><div>==[[AP_Statistics_Curriculum_2007 | General Advance-Placement (AP) Statistics Curriculum]] - Fisher's F Distribution==<br />
<br />
===Fisher's F Distribution===<br />
Commonly used as the null distribution of a test statistic, such as in analysis of variance [http://en.wikipedia.org/wiki/Anova (ANOVA)]. Relationship to the [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_StudentsT t-distribution] and [beta Distribution].<br />
<br />
'''PDF''': <br><br />
<math>\frac{\sqrt{\frac{(d_1\,x)^{d_1}\,\,d_2^{d_2}}<br />
{(d_1\,x+d_2)^{d_1+d_2}}}}<br />
{x\,\mathrm{B}\!\left(\frac{d_1}{2},\frac{d_2}{2}\right)}\!</math><br />
<br />
'''CDF''': <br><br />
<math>I_{\frac{d_1 x}{d_1 x + d_2}}(d_1/2, d_2/2)\!</math><br />
<br />
'''Mean''': <br><br />
<math>\frac{d_2}{d_2-2}\!</math> for <math>d_2 > 2</math><br />
<br />
'''Median''': <br><br />
None<br />
<br />
'''Variance''': <br><br />
<math>\frac{2\,d_2^2\,(d_1+d_2-2)}{d_1 (d_2-2)^2 (d_2-4)}\!</math> for <math>d_2 > 4</math><br />
<br />
'''Support''': <br><br />
<math>x \in [0, +\infty)\!</math><br />
<br />
'''1st Moment''': <br><br />
<br />
<br />
'''2nd Moment''': <br><br />
<br />
<br />
'''Kth Moment''': <br></div>
JayZzz