Difference between revisions of "AP Statistics Curriculum 2007 Infer BiVar"
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===Comparing Two Variances (<math>\sigma_1^2 = \sigma_2^2</math>?)=== | ===Comparing Two Variances (<math>\sigma_1^2 = \sigma_2^2</math>?)=== | ||
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. | 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. | ||
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Notice that the [http://mathworld.wolfram.com/Chi-SquaredDistribution.html Chi-square distribution] 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-square distribution calculator]. | Notice that the [http://mathworld.wolfram.com/Chi-SquaredDistribution.html Chi-square distribution] 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-square distribution calculator]. | ||
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The [http://mathworld.wolfram.com/F-Distribution.html Fisher's F distribution], 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. | The [http://mathworld.wolfram.com/F-Distribution.html Fisher's F distribution], 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. | ||
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: <math>H_1: \sigma_1^2 < \sigma_2^2</math>: If <math>F_o < F(1-\alpha,df_1=n_1-1,df_2=n_2-1)</math> | : <math>H_1: \sigma_1^2 < \sigma_2^2</math>: If <math>F_o < F(1-\alpha,df_1=n_1-1,df_2=n_2-1)</math> | ||
| − | : <math>H_1: \sigma_1^2 \not= \sigma_2^2</math>: If <math>F_o < F(1-\alpha/2,df_1=n_1-1,df_2=n_2-1)</math> or <math>F_o > F(\alpha/2,df_1=n_1-1,df_2=n_2-1)</math>, | + | : <math>H_1: \sigma_1^2 \not= \sigma_2^2</math>: If either <math>F_o < F(1-\alpha/2,df_1=n_1-1,df_2=n_2-1)</math> or <math>F_o > F(\alpha/2,df_1=n_1-1,df_2=n_2-1)</math>, |
| + | where <math>F(\alpha,df_1=n_1-1,df_2=n_2-1)</math> is the critical value of the [http://socr.ucla.edu/htmls/SOCR_Distributions.html F distribution] with ''degrees of freedom for the numerator and denominator'', <math>df_1=n_1-1,df_2=n_2-1</math>, respectively. | ||
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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. | 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. | ||
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===Comparing Two Standard Deviations (<math>\sigma_1 = \sigma_2</math>?)=== | ===Comparing Two Standard Deviations (<math>\sigma_1 = \sigma_2</math>?)=== | ||
| − | + | Two 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. | |
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===Hands-on activities=== | ===Hands-on activities=== | ||
| − | * Formulate appropriate hypotheses and assess the significance of the evidence to reject the null hypothesis | + | * 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 distirbuted populations of liquid content (in fluid ounces) of beverage cans. Use (<math>\alpha=0.1</math>). |
<center> | <center> | ||
{| class="wikitable" style="text-align:center; width:75%" border="1" | {| class="wikitable" style="text-align:center; width:75%" border="1" | ||
|- | |- | ||
| − | | 14.816 || 14.863 || 14.814 || 14.998 || 14.965 || 14.824 || 14.884 || 14.838 || 14.916 || 15.021 || 14.874 || 14.856 || 14.860 || 14.772 || 14.980 || 14.919 | + | | '''Sample from Population 1''' || 14.816 || 14.863 || 14.814 || 14.998 || 14.965 || 14.824 |
| + | |- | ||
| + | | '''Sample from Population 2'''|| 14.884 || 14.838 || 14.916 || 15.021 || 14.874 || 14.856 || 14.860 || 14.772 || 14.980 || 14.919 | ||
|} | |} | ||
</center> | </center> | ||
| − | * Hypotheses: <math>H_o: \ | + | * Hypotheses: <math>H_o: \sigma_1 = \sigma_2</math> vs. <math>H_1: \sigma_1 \not= \sigma_2</math> . |
| − | * Get the sample statistics from [http://socr.ucla.edu/htmls/SOCR_Charts.html SOCR Charts] (e.g., Index Plot); Sample- | + | * Get the sample statistics from [http://socr.ucla.edu/htmls/SOCR_Charts.html SOCR Charts] (e.g., Index Plot); Sample-Mean1=14.8875 and Sample-SD1=0.072700298; Sample-Mean2=14.8875 and Sample-SD2=0.072700298 |
| − | <center>[[Image: | + | <center>[[Image:SOCR_EBook_Dinov_Infer_BiVar_021608_Fig2.jpg|500px]]</center> |
| − | * Identify the degrees of freedom (<math> | + | * Identify the degrees of freedom (<math>df_1=6-1=5</math> and <math>df_2=10-1=9</math>). |
| − | * Test Statistics: <math> | + | * Test Statistics: <math>F_o^2 = {\sigma_1^2 \over \sigma_1^2}=</math> |
* Significance Inference: <math>\chi_o^2 = {15\times 0.005285333 \over 0.06^2}=22.022221</math> | * Significance Inference: <math>\chi_o^2 = {15\times 0.005285333 \over 0.06^2}=22.022221</math> | ||
: P-value=<math>P(\Chi_{(df=n-1)}^2 > \chi_o^2) = 0.107223</math>. This p-value does not indicate strong evidence in the data to reject a possible population standard deviation of 0.06. | : P-value=<math>P(\Chi_{(df=n-1)}^2 > \chi_o^2) = 0.107223</math>. This p-value does not indicate strong evidence in the data to reject a possible population standard deviation of 0.06. | ||
| − | <center>[[Image: | + | <center>[[Image:SOCR_EBook_Dinov_Infer_BiVar_021608_Fig3.jpg|500px]]</center> |
===More examples=== | ===More examples=== | ||
| − | + | * 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. Repear this with variances of calories between the beef and meet based hot-dogs. | |
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| − | * Use the [[SOCR_012708_ID_Data_HotDogs | hot-dogs dataset]] to formulate and test hypotheses about the population standard | ||
<hr> | <hr> | ||
===References=== | ===References=== | ||
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<hr> | <hr> | ||
Revision as of 17:12, 16 February 2008
Contents
General Advance-Placement (AP) Statistics Curriculum - Comparing Two Variances
In the section on inference about the variance and the standard deviation we already learned how to do inference on either of these two population paparemters. Now we discuss the comparison of the variances (or standard deviations) using data randomly sampled from two different populations.
Background
Recall that the sample-variance (s2) is an unbiased point estimate for the population variance \(\sigma^2\), and similarly, the sample-standard-deviation (s) is a point estimate for the population-standard-deviation \(\sigma\).
The sample-variance is roughly Chi-square distributed: \[\chi_o^2 = {(n-1)s^2 \over \sigma^2} \sim \Chi_{(df=n-1)}^2\]
Comparing Two Variances (\(\sigma_1^2 = \sigma_2^2\)?)
Suppose we study two populations which are approximately Normally distributed, and we take a random sample from each population, {\(X_1, X_2, X_3, \cdots, X_n\)} and {\(Y_1, Y_2, Y_3, \cdots, Y_k\)}. Recall that \({(n-1) s_1^2 \over \sigma_1^2}\) and \({(n-1) s_2^2 \over \sigma_2^2}\) have \(\Chi^2_{(df=n - 1)}\) and \(\Chi^2_{(df=k - 1)}\) distributions. We are interested in assessing \(H_o: \sigma_1^2 = \sigma_2^2\) vs. \(H_1: \sigma_1^2 \not= \sigma_2^2\), where \(s_1\) and \(\sigma_1\), and \(s_2\) and \(\sigma_2\) and the sample and the population standard deviations for the two populations/samples, respectively.
Notice that the Chi-square distribution is not symmetric (it is positively skewed). You can visualize the Chi-Square distribution and compute all critical values either using the SOCR Chi-Square Distribution or using the SOCR Chi-square distribution calculator.
The Fisher's F distribution, 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 (\(H_1: \sigma_1^2 \not= \sigma_2^2\)). The one-tailed version only tests in one direction (\(H_1: \sigma_1^2 < \sigma_2^2\) or \(H_1: \sigma_1^2 > \sigma_2^2\)). The choice is determined by the 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.
- Test Statistic\[F_o = {\sigma_1^2 \over \sigma_2^2}\]
The higher the deviation of this ratio away from 1, the stronger the evidence for unequal population variances.
- Inference: Suppose we test at significance level \(\alpha=0.05\). Then the hypothesis that the two standard deviations are equal is rejected if the test statistics is outside this interval
\[H_1: \sigma_1^2 > \sigma_2^2\]: If \(F_o > F(\alpha,df_1=n_1-1,df_2=n_2-1)\)
\[H_1: \sigma_1^2 < \sigma_2^2\]: If \(F_o < F(1-\alpha,df_1=n_1-1,df_2=n_2-1)\)
\[H_1: \sigma_1^2 \not= \sigma_2^2\]: If either \(F_o < F(1-\alpha/2,df_1=n_1-1,df_2=n_2-1)\) or \(F_o > F(\alpha/2,df_1=n_1-1,df_2=n_2-1)\), where \(F(\alpha,df_1=n_1-1,df_2=n_2-1)\) is the critical value of the F distribution with degrees of freedom for the numerator and denominator, \(df_1=n_1-1,df_2=n_2-1\), respectively.
In the image below the left and right critical regions are white with \(F(\alpha,df_1=n_1-1,df_2=n_2-1)\) and \(F(1-\alpha,df_1=n_1-1,df_2=n_2-1)\) representing the lower and upper, respectively, critical values. In this example of \(F(df_1=12, df_2=15)\), the left and right critical values at \(\alpha/2=0.025\) are \(F(\alpha/2=0.025,df_1=9,df_2=14)=0.314744\) and \(F(1-\alpha/2=0.975,df_1=9,df_2=14)=2.96327\), respectively.
Comparing Two Standard Deviations (\(\sigma_1 = \sigma_2\)?)
Two 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.
Hands-on activities
- 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 distirbuted populations of liquid content (in fluid ounces) of beverage cans. Use (\(\alpha=0.1\)).
| Sample from Population 1 | 14.816 | 14.863 | 14.814 | 14.998 | 14.965 | 14.824 | ||||
| Sample from Population 2 | 14.884 | 14.838 | 14.916 | 15.021 | 14.874 | 14.856 | 14.860 | 14.772 | 14.980 | 14.919 |
- Hypotheses\[H_o: \sigma_1 = \sigma_2\] vs. \(H_1: \sigma_1 \not= \sigma_2\) .
- Get the sample statistics from SOCR Charts (e.g., Index Plot); Sample-Mean1=14.8875 and Sample-SD1=0.072700298; Sample-Mean2=14.8875 and Sample-SD2=0.072700298
- Identify the degrees of freedom (\(df_1=6-1=5\) and \(df_2=10-1=9\)).
- Test Statistics\[F_o^2 = {\sigma_1^2 \over \sigma_1^2}=\]
- Significance Inference\[\chi_o^2 = {15\times 0.005285333 \over 0.06^2}=22.022221\]
- P-value=\(P(\Chi_{(df=n-1)}^2 > \chi_o^2) = 0.107223\). This p-value does not indicate strong evidence in the data to reject a possible population standard deviation of 0.06.
More examples
- Use the 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. Repear this with variances of calories between the beef and meet based hot-dogs.
References
- SOCR Home page: http://www.socr.ucla.edu
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