http://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_Infer_BiVar&feed=atom&action=historyAP Statistics Curriculum 2007 Infer BiVar - Revision history2024-03-29T02:39:12ZRevision history for this page on the wikiMediaWiki 1.15.1http://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_Infer_BiVar&diff=12372&oldid=prevIvoDinov: /* Comparing Two Variances (\sigma_1^2 = \sigma_2^2?) */2014-02-14T14:50:48Z<p><span class="autocomment">Comparing Two Variances (\sigma_1^2 = \sigma_2^2?)</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>: <math>\chi_o^2 = {(n-1)s^2 \over \sigma^2} \sim \Chi_{(df=n-1)}^2</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>: <math>\chi_o^2 = {(n-1)s^2 \over \sigma^2} \sim \Chi_{(df=n-1)}^2</math></div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>===Comparing Two Variances (<del class="diffchange diffchange-inline"><math></del>\sigma_1^2 = \sigma_2^2<del class="diffchange diffchange-inline"></math></del>?)===</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>===Comparing Two Variances (<ins class="diffchange diffchange-inline">\(</ins>\sigma_1^2 = \sigma_2^2<ins class="diffchange diffchange-inline">\)</ins>?)===</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Suppose we study two populations which are approximately Normally distributed, and we take a random sample from each population, {<math>X_1, X_2, X_3, \cdots, X_n</math>} and {<math>Y_1, Y_2, Y_3, \cdots, Y_k</math>}. Recall that <math>{(n-1) s_1^2 \over \sigma_1^2}</math> and <math>{(n-1) s_2^2 \over \sigma_2^2}</math> have <math>\Chi^2_{(df=n - 1)}</math> and <math>\Chi^2_{(df=k - 1)}</math> distributions. We are interested in assessing <math>H_o: \sigma_1^2 = \sigma_2^2</math> vs. <math>H_1: \sigma_1^2 \not= \sigma_2^2</math>, where <math>s_1</math> and <math>\sigma_1</math>, and <math>s_2</math> and <math>\sigma_2</math> and the sample and the population standard deviations for the two populations/samples, respectively.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Suppose we study two populations which are approximately Normally distributed, and we take a random sample from each population, {<math>X_1, X_2, X_3, \cdots, X_n</math>} and {<math>Y_1, Y_2, Y_3, \cdots, Y_k</math>}. Recall that <math>{(n-1) s_1^2 \over \sigma_1^2}</math> and <math>{(n-1) s_2^2 \over \sigma_2^2}</math> have <math>\Chi^2_{(df=n - 1)}</math> and <math>\Chi^2_{(df=k - 1)}</math> distributions. We are interested in assessing <math>H_o: \sigma_1^2 = \sigma_2^2</math> vs. <math>H_1: \sigma_1^2 \not= \sigma_2^2</math>, where <math>s_1</math> and <math>\sigma_1</math>, and <math>s_2</math> and <math>\sigma_2</math> and the sample and the population standard deviations for the two populations/samples, respectively.</div></td></tr>
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</table>IvoDinovhttp://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_Infer_BiVar&diff=10401&oldid=prevJenny at 20:49, 28 June 20102010-06-28T20:49:13Z<p></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===Comparing Two Standard Deviations (<math>\sigma_1 = \sigma_2</math>?)===</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===Comparing Two Standard Deviations (<math>\sigma_1 = \sigma_2</math>?)===</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>To make inference on whether the standard deviations of two populations are equal we calculate the sample variances and apply the inference on the ratio of the sample variance using the F-test, as described above.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>To make inference on whether the standard deviations of two populations are equal<ins class="diffchange diffchange-inline">, </ins>we calculate the sample variances and apply the inference on the ratio of the sample variance using the F-test, as described above.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===Hands-on activities===</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===Hands-on activities===</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>* Formulate appropriate hypotheses and assess the significance of the evidence to reject the null hypothesis that the variances of the two populations, <del class="diffchange diffchange-inline">that </del>the following data come from, are distinct. Assume the observations below represent random samples (of sizes 6 and 10) from two Normally distributed populations of liquid content (in fluid ounces) of beverage cans. Use (<math>\alpha=0.1</math>).</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>* Formulate appropriate hypotheses and assess the significance of the evidence to reject the null hypothesis that the variances of the two populations, <ins class="diffchange diffchange-inline">where </ins>the following data come from, are distinct. Assume the observations below represent random samples (of sizes 6 and 10) from two Normally distributed populations of liquid content (in fluid ounces) of beverage cans. Use (<math>\alpha=0.1</math>).</div></td></tr>
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</table>Jennyhttp://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_Infer_BiVar&diff=10299&oldid=prevJenny at 17:30, 28 June 20102010-06-28T17:30:00Z<p></p>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>In the image below the left and right critical regions are white with <math>F(\alpha,df_1=n_1-1,df_2=n_2-1)</math> and <math>F(1-\alpha,df_1=n_1-1,df_2=n_2-1)</math> representing the lower and upper, respectively, critical values. In this example of <math>F(df_1=12, df_2=15)</math>, the left and right critical values at <math>\alpha/2=0.025</math> are <math>F(\alpha/2=0.025,df_1=9,df_2=14)=0.314744</math> and <math>F(1-\alpha/2=0.975,df_1=9,df_2=14)=2.96327</math>, respectively.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>In the image below<ins class="diffchange diffchange-inline">, </ins>the left and right critical regions are white with <math>F(\alpha,df_1=n_1-1,df_2=n_2-1)</math> and <math>F(1-\alpha,df_1=n_1-1,df_2=n_2-1)</math> representing the lower and upper, respectively, critical values. In this example of <math>F(df_1=12, df_2=15)</math>, the left and right critical values at <math>\alpha/2=0.025</math> are <math>F(\alpha/2=0.025,df_1=9,df_2=14)=0.314744</math> and <math>F(1-\alpha/2=0.975,df_1=9,df_2=14)=2.96327</math>, respectively.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><center>[[Image:SOCR_EBook_Dinov_Infer_BiVar_021608_Fig1.jpg|500px]]</center></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><center>[[Image:SOCR_EBook_Dinov_Infer_BiVar_021608_Fig1.jpg|500px]]</center></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===Comparing Two Standard Deviations (<math>\sigma_1 = \sigma_2</math>?)===</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===Comparing Two Standard Deviations (<math>\sigma_1 = \sigma_2</math>?)===</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">Two </del>make inference on whether the standard deviations of two populations are equal we calculate the sample variances and apply the inference on the ratio of the sample variance using the F-test, as described above.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">To </ins>make inference on whether the standard deviations of two populations are equal we calculate the sample variances and apply the inference on the ratio of the sample variance using the F-test, as described above.</div></td></tr>
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</table>Jennyhttp://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_Infer_BiVar&diff=8453&oldid=prevIvoDinov at 02:30, 29 November 20082008-11-29T02:30:32Z<p></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>* Use the [[SOCR_012708_ID_Data_HotDogs | hot-dogs dataset]] to formulate and test hypotheses about the difference of the population standard deviations of sodium between the poultry and the meet based hot-dogs. Repeat this with variances of calories between the beef and meet based hot-dogs.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>* Use the [[SOCR_012708_ID_Data_HotDogs | hot-dogs dataset]] to formulate and test hypotheses about the difference of the population standard deviations of sodium between the poultry and the meet based hot-dogs. Repeat this with variances of calories between the beef and meet based hot-dogs.</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;">===See also===</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;">[[AP_Statistics_Curriculum_2007_NonParam_VarIndep | Fligner-Killeen non-parametric test for variance homogeneity]].</ins></div></td></tr>
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</table>IvoDinovhttp://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_Infer_BiVar&diff=8347&oldid=prevIvoDinov: /* Hands-on activities */2008-11-19T19:05:13Z<p><span class="autocomment">Hands-on activities</span></p>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>| '''Sample from Population 1''' || 14.816 || 14.863 || 14.814 || 14.998 || 14.965 || 14.824 </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>| '''Sample from Population 1''' || 14.816 || 14.863 || 14.814 || 14.998 || 14.965 || 14.824 <ins class="diffchange diffchange-inline">|| || || || </ins></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>| '''Sample from Population 2'''|| 14.884 || 14.838 || 14.916 || 15.021 || 14.874 || 14.856 || 14.860 || 14.772 || 14.980 || 14.919</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>| '''Sample from Population 2'''|| 14.884 || 14.838 || 14.916 || 15.021 || 14.874 || 14.856 || 14.860 || 14.772 || 14.980 || 14.919</div></td></tr>
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</table>IvoDinovhttp://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_Infer_BiVar&diff=8215&oldid=prevJenny: /* Comparing Two Variances (\sigma_1^2 = \sigma_2^2?) */2008-11-04T17:38:33Z<p><span class="autocomment">Comparing Two Variances (\sigma_1^2 = \sigma_2^2?)</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>* Test Statistic: <math>F_o = {\sigma_1^2 \over \sigma_2^2}</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>* Test Statistic: <math>F_o = {\sigma_1^2 \over \sigma_2^2}</math></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>The <del class="diffchange diffchange-inline">farter </del>away this ratio is from 1, the stronger the evidence for unequal population variances.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>The <ins class="diffchange diffchange-inline">farther </ins>away this ratio is from 1, the stronger the evidence for unequal population variances.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>* Inference: Suppose we test at significance level <math>\alpha=0.05</math>. Then the hypothesis that the two standard deviations are equal is rejected if the test statistics is outside this interval</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>* Inference: Suppose we test at significance level <math>\alpha=0.05</math>. Then the hypothesis that the two standard deviations are equal is rejected if the test statistics is outside this interval</div></td></tr>
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</table>Jennyhttp://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_Infer_BiVar&diff=6812&oldid=prevPriscillaChui: /* Comparing Two Variances (<math>\sigma_1^2 = \sigma_2^2</math>?) */2008-03-01T23:03:39Z<p><span class="autocomment">Comparing Two Variances (<math>\sigma_1^2 = \sigma_2^2</math>?)</span></p>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>Notice that the [http://mathworld.wolfram.com/Chi-SquaredDistribution.html Chi-<del class="diffchange diffchange-inline">square distribution</del>] is not symmetric (it is positively skewed). You can visualize the Chi-Square distribution and compute all critical values either using the [http://socr.ucla.edu/htmls/SOCR_Distributions.html SOCR Chi-Square Distribution] or using the [http://socr.ucla.edu/Applets.dir/Normal_T_Chi2_F_Tables.htm SOCR Chi-<del class="diffchange diffchange-inline">square distribution calculator</del>].</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>Notice that the [http://mathworld.wolfram.com/Chi-SquaredDistribution.html Chi-<ins class="diffchange diffchange-inline">Square Distribution</ins>] is not symmetric (it is positively skewed). You can visualize the Chi-Square distribution and compute all critical values either using the [http://socr.ucla.edu/htmls/SOCR_Distributions.html SOCR Chi-Square Distribution] or using the [http://socr.ucla.edu/Applets.dir/Normal_T_Chi2_F_Tables.htm SOCR Chi-<ins class="diffchange diffchange-inline">Square Distribution Calculator</ins>].</div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>The [http://mathworld.wolfram.com/F-Distribution.html Fisher's F <del class="diffchange diffchange-inline">distribution</del>], and the corresponding F-test, is used to test if the variances of two populations are equal. Depending on the alternative hypothesis, we can use either a two-tailed test or a one-tailed test. The two-tailed version tests against an alternative that the standard deviations are not equal (<math>H_1: \sigma_1^2 \not= \sigma_2^2</math>). The one-tailed version only tests in one direction (<math>H_1: \sigma_1^2 < \sigma_2^2</math> or <math>H_1: \sigma_1^2 > \sigma_2^2</math>). The choice is determined by the [[AP_Statistics_Curriculum_2007_IntroDesign | study design]] before any data is analyzed. For example, if a modification to an existent medical treatment is proposed, we may only be interested in knowing if the new treatment is more consistent and less variable than the established medical intervention.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>The [http://mathworld.wolfram.com/F-Distribution.html Fisher's F <ins class="diffchange diffchange-inline">Distribution</ins>], and the corresponding F-test, is used to test if the variances of two populations are equal. Depending on the alternative hypothesis, we can use either a two-tailed test or a one-tailed test. The two-tailed version tests against an alternative that the standard deviations are not equal (<math>H_1: \sigma_1^2 \not= \sigma_2^2</math>). The one-tailed version only tests in one direction (<math>H_1: \sigma_1^2 < \sigma_2^2</math> or <math>H_1: \sigma_1^2 > \sigma_2^2</math>). The choice is determined by the [[AP_Statistics_Curriculum_2007_IntroDesign | study design]] before any data is analyzed. For example, if a modification to an existent medical treatment is proposed, we may only be interested in knowing if the new treatment is more consistent and less variable than the established medical intervention.</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">'''</del>Test Statistic<del class="diffchange diffchange-inline">'''</del>: <math>F_o = {\sigma_1^2 \over \sigma_2^2}</math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>* Test Statistic: <math>F_o = {\sigma_1^2 \over \sigma_2^2}</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The farter away this ratio is from 1, the stronger the evidence for unequal population variances.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The farter away this ratio is from 1, the stronger the evidence for unequal population variances.</div></td></tr>
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</table>PriscillaChuihttp://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_Infer_BiVar&diff=6636&oldid=prevPriscillaChui: /* General Advance-Placement (AP) Statistics Curriculum - Comparing Two Variances */2008-02-19T20:53:17Z<p><span class="autocomment"><a href="/socr/index.php/AP_Statistics_Curriculum_2007" class="mw-redirect" title="AP Statistics Curriculum 2007"> General Advance-Placement (AP) Statistics Curriculum</a> - Comparing Two Variances</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==[[AP_Statistics_Curriculum_2007 | General Advance-Placement (AP) Statistics Curriculum]] - Comparing Two Variances==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==[[AP_Statistics_Curriculum_2007 | General Advance-Placement (AP) Statistics Curriculum]] - Comparing Two Variances==</div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>In the [[AP_Statistics_Curriculum_2007_Hypothesis_Var | section on inference about the variance and the standard deviation]] we already learned how to do inference on either of these two population parameters. Now we discuss the comparison of the variances (or standard deviations) using data randomly sampled from two different populations.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>In the [[AP_Statistics_Curriculum_2007_Hypothesis_Var | section on inference about the variance and the standard deviation]]<ins class="diffchange diffchange-inline">, </ins>we already learned how to do inference on either of these two population parameters. Now we discuss the comparison of the variances (or standard deviations) using data randomly sampled from two different populations.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>=== [[AP_Statistics_Curriculum_2007_Estim_Var | Background]]===</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>=== [[AP_Statistics_Curriculum_2007_Estim_Var | Background]]===</div></td></tr>
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</table>PriscillaChuihttp://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_Infer_BiVar&diff=6515&oldid=prevIvoDinov at 22:39, 16 February 20082008-02-16T22:39:08Z<p></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==[[AP_Statistics_Curriculum_2007 | General Advance-Placement (AP) Statistics Curriculum]] - Comparing Two Variances==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==[[AP_Statistics_Curriculum_2007 | General Advance-Placement (AP) Statistics Curriculum]] - Comparing Two Variances==</div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>In the [[AP_Statistics_Curriculum_2007_Hypothesis_Var | section on inference about the variance and the standard deviation]] we already learned how to do inference on either of these two population <del class="diffchange diffchange-inline">paparemters</del>. Now we discuss the comparison of the variances (or standard deviations) using data randomly sampled from two different populations.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>In the [[AP_Statistics_Curriculum_2007_Hypothesis_Var | section on inference about the variance and the standard deviation]] we already learned how to do inference on either of these two population <ins class="diffchange diffchange-inline">parameters</ins>. Now we discuss the comparison of the variances (or standard deviations) using data randomly sampled from two different populations.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>=== [[AP_Statistics_Curriculum_2007_Estim_Var | Background]]===</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>=== [[AP_Statistics_Curriculum_2007_Estim_Var | Background]]===</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===Comparing Two Variances (<math>\sigma_1^2 = \sigma_2^2</math>?)===</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===Comparing Two Variances (<math>\sigma_1^2 = \sigma_2^2</math>?)===</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>Suppose we study two populations which are approximately <del class="diffchange diffchange-inline"> </del>Normally distributed, and we take a random sample from each population, {<math>X_1, X_2, X_3, \cdots, X_n</math>} and {<math>Y_1, Y_2, Y_3, \cdots, Y_k</math>}. Recall that <math>{(n-1) s_1^2 \over \sigma_1^2}</math> and <math>{(n-1) s_2^2 \over \sigma_2^2}</math> have <math>\Chi^2_{(df=n - 1)}</math> and <math>\Chi^2_{(df=k - 1)}</math> distributions. We are interested in assessing <math>H_o: \sigma_1^2 = \sigma_2^2</math> vs. <math>H_1: \sigma_1^2 \not= \sigma_2^2</math>, where <math>s_1</math> and <math>\sigma_1</math>, and <math>s_2</math> and <math>\sigma_2</math> and the sample and the population standard deviations for the two populations/samples, respectively.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>Suppose we study two populations which are approximately Normally distributed, and we take a random sample from each population, {<math>X_1, X_2, X_3, \cdots, X_n</math>} and {<math>Y_1, Y_2, Y_3, \cdots, Y_k</math>}. Recall that <math>{(n-1) s_1^2 \over \sigma_1^2}</math> and <math>{(n-1) s_2^2 \over \sigma_2^2}</math> have <math>\Chi^2_{(df=n - 1)}</math> and <math>\Chi^2_{(df=k - 1)}</math> distributions. We are interested in assessing <math>H_o: \sigma_1^2 = \sigma_2^2</math> vs. <math>H_1: \sigma_1^2 \not= \sigma_2^2</math>, where <math>s_1</math> and <math>\sigma_1</math>, and <math>s_2</math> and <math>\sigma_2</math> and the sample and the population standard deviations for the two populations/samples, respectively.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>* '''Test Statistic''': <math>F_o = {\sigma_1^2 \over \sigma_2^2}</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>* '''Test Statistic''': <math>F_o = {\sigma_1^2 \over \sigma_2^2}</math></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>The <del class="diffchange diffchange-inline">higher the deviation of </del>this ratio <del class="diffchange diffchange-inline">away </del>from 1, the stronger the evidence for unequal population variances.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>The <ins class="diffchange diffchange-inline">farter away </ins>this ratio <ins class="diffchange diffchange-inline">is </ins>from 1, the stronger the evidence for unequal population variances.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>* Inference: Suppose we test at significance level <math>\alpha=0.05</math>. Then the hypothesis that the two standard deviations are equal is rejected if the test statistics is outside this interval</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>* Inference: Suppose we test at significance level <math>\alpha=0.05</math>. Then the hypothesis that the two standard deviations are equal is rejected if the test statistics is outside this interval</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===Hands-on activities===</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===Hands-on activities===</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>* Formulate appropriate hypotheses and assess the significance of the evidence to reject the null hypothesis that the variances of the two populations, that the following data come from, are distinct. Assume the observations below represent random samples (of sizes 6 and 10) from two Normally <del class="diffchange diffchange-inline">distirbuted </del>populations of liquid content (in fluid ounces) of beverage cans. Use (<math>\alpha=0.1</math>).</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>* Formulate appropriate hypotheses and assess the significance of the evidence to reject the null hypothesis that the variances of the two populations, that the following data come from, are distinct. Assume the observations below represent random samples (of sizes 6 and 10) from two Normally <ins class="diffchange diffchange-inline">distributed </ins>populations of liquid content (in fluid ounces) of beverage cans. Use (<math>\alpha=0.1</math>).</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>* Test Statistics: <math>F_o = {\sigma_1^2 \over \sigma_2^2}=1.300406878</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>* Test Statistics: <math>F_o = {\sigma_1^2 \over \sigma_2^2}=1.300406878</math></div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>* Significance Inference: P-value=<math>P(F_{(df_1=5, df_2=9)} > F_o) = 0.328147</math>. This p-value does not indicate strong evidence in the data to reject the null hypothesis. That is, the data does not have power to <del class="diffchange diffchange-inline">discreminate </del>between the population variances of the two populations based on these (small) samples.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>* Significance Inference: P-value=<math>P(F_{(df_1=5, df_2=9)} > F_o) = 0.328147</math>. This p-value does not indicate strong evidence in the data to reject the null hypothesis. That is, the data does not have power to <ins class="diffchange diffchange-inline">discriminate </ins>between the population variances of the two populations based on these (small) samples.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===More examples===</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===More examples===</div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>* Use the [[SOCR_012708_ID_Data_HotDogs | hot-dogs dataset]] to formulate and test hypotheses about the difference of the population standard deviations of sodium between the poultry and the meet based hot-dogs. <del class="diffchange diffchange-inline">Repear </del>this with variances of calories between the beef and meet based hot-dogs.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>* Use the [[SOCR_012708_ID_Data_HotDogs | hot-dogs dataset]] to formulate and test hypotheses about the difference of the population standard deviations of sodium between the poultry and the meet based hot-dogs. <ins class="diffchange diffchange-inline">Repeat </ins>this with variances of calories between the beef and meet based hot-dogs.</div></td></tr>
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</table>IvoDinovhttp://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_Infer_BiVar&diff=6512&oldid=prevIvoDinov: /* Hands-on activities */2008-02-16T22:35:27Z<p><span class="autocomment">Hands-on activities</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>* Identify the degrees of freedom (<math>df_1=6-1=5</math> and <math>df_2=10-1=9</math>).</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>* Identify the degrees of freedom (<math>df_1=6-1=5</math> and <math>df_2=10-1=9</math>).</div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>* Test Statistics: <math>F_o<del class="diffchange diffchange-inline">^2 </del>= {\sigma_1^2 \over \sigma_2^2}=1.300406878</math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>* Test Statistics: <math>F_o = {\sigma_1^2 \over \sigma_2^2}=1.300406878</math></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 class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>* Significance Inference: P-value=<math>P(F_{(df_1=5, df_2=9)} > F_o<del class="diffchange diffchange-inline">^2</del>) = 0.328147</math>. This p-value does not indicate strong evidence in the data to reject the null hypothesis. That is, the data does not have power to discreminate between the population variances of the two populations based on these (small) samples.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>* Significance Inference: P-value=<math>P(F_{(df_1=5, df_2=9)} > F_o) = 0.328147</math>. This p-value does not indicate strong evidence in the data to reject the null hypothesis. That is, the data does not have power to discreminate between the population variances of the two populations based on these (small) samples.</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 class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><center>[[Image:SOCR_EBook_Dinov_Infer_BiVar_021608_Fig3.jpg|500px]]</center></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><center>[[Image:SOCR_EBook_Dinov_Infer_BiVar_021608_Fig3.jpg|500px]]</center></div></td></tr>
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</table>IvoDinov