AP Statistics Curriculum 2007 Infer BiVar - Revision history

Tdlee: Text replacement - "{{translate|pageName=http://wiki.stat.ucla.edu/socr/" to ""{{translate|pageName=http://wiki.socr.umich.edu/"

2020-03-03T19:03:46Z

Text replacement - "{{translate|pageName=http://wiki.stat.ucla.edu/socr/" to ""{{translate|pageName=http://wiki.socr.umich.edu/"

Jenny at 20:49, 28 June 2010

2010-06-28T20:49:13Z

Jenny at 17:30, 28 June 2010

2010-06-28T17:30:00Z

IvoDinov at 02:30, 29 November 2008

2008-11-29T02:30:32Z

IvoDinov: /* Hands-on activities */

2008-11-19T19:05:13Z

‎Hands-on activities

Jenny: /* Comparing Two Variances (\sigma_1^2 = \sigma_2^2?) */

2008-11-04T17:38:33Z

‎Comparing Two Variances (\sigma_1^2 = \sigma_2^2?)

PriscillaChui: /* Comparing Two Variances ( $\sigma_1^2 = \sigma_2^2$ ?) */

2008-03-01T23:03:39Z

‎Comparing Two Variances (<math>\sigma_1^2 = \sigma_2^2</math>?)

PriscillaChui: /* General Advance-Placement (AP) Statistics Curriculum - Comparing Two Variances */

2008-02-19T20:53:17Z

‎ General Advance-Placement (AP) Statistics Curriculum - Comparing Two Variances

IvoDinov at 22:39, 16 February 2008

2008-02-16T22:39:08Z

IvoDinov: /* Hands-on activities */

2008-02-16T22:35:27Z

‎Hands-on activities

← Older revision		Revision as of 19:03, 3 March 2020
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	* SOCR Home page: http://www.socr.ucla.edu		* SOCR Home page: http://www.socr.ucla.edu

−	{{translate\|pageName=http://wiki.~~stat~~.~~ucla~~.edu~~/socr~~/index.php?title=AP_Statistics_Curriculum_2007_Infer_BiVar}}	+	"{{translate\|pageName=http://wiki.socr.umich.edu/index.php?title=AP_Statistics_Curriculum_2007_Infer_BiVar}}

@@ Line 35: / Line 35: @@
 ===Comparing Two Standard Deviations (<math>\sigma_1 = \sigma_2</math>?)===
-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.
+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.
 ===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 distributed populations of liquid content (in fluid ounces) of beverage cans. Use (<math>\alpha=0.1</math>).
+* Formulate appropriate hypotheses and assess the significance of the evidence to reject the null hypothesis that the variances of the two populations, where 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>).
 <center>
 {| class="wikitable" style="text-align:center; width:75%" border="1"

@@ Line 30: / Line 30: @@
-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.
 <center>[[Image:SOCR_EBook_Dinov_Infer_BiVar_021608_Fig1.jpg|500px]]</center>
 ===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.
+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.
 ===Hands-on activities===

← Older revision		Revision as of 02:30, 29 November 2008
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	* 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.		* 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.
		+
		+	===See also===
		+	[[AP_Statistics_Curriculum_2007_NonParam_VarIndep \| Fligner-Killeen non-parametric test for variance homogeneity]].

	<hr>		<hr>

@@ Line 42: / Line 42: @@
 {| class="wikitable" style="text-align:center; width:75%" border="1"
 |-
-| '''Sample from Population 1''' || 14.816 || 14.863 || 14.814 || 14.998 || 14.965 || 14.824
+| '''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

@@ Line 19: / Line 19: @@
 * Test Statistic: <math>F_o = {\sigma_1^2 \over \sigma_2^2}</math>
-The farter away this ratio is from 1, the stronger the evidence for unequal population variances.
+The farther away this ratio is from 1, the stronger the evidence for unequal population variances.
 * 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

@@ Line 13: / Line 13: @@
-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].
-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.
-* '''Test Statistic''': <math>F_o = {\sigma_1^2 \over \sigma_2^2}</math>
+* Test Statistic: <math>F_o = {\sigma_1^2 \over \sigma_2^2}</math>
 The farter away this ratio is from 1, the stronger the evidence for unequal population variances.

@@ Line 1: / Line 1: @@
 ==[[AP_Statistics_Curriculum_2007 | General Advance-Placement (AP) Statistics Curriculum]] - Comparing Two Variances==
-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.
+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.
 === [[AP_Statistics_Curriculum_2007_Estim_Var | Background]]===

@@ Line 1: / Line 1: @@
 ==[[AP_Statistics_Curriculum_2007 | General Advance-Placement (AP) Statistics Curriculum]] - Comparing Two Variances==
-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 paparemters. Now we discuss the comparison of the variances (or standard deviations) using data randomly sampled from two different populations.
+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.
 === [[AP_Statistics_Curriculum_2007_Estim_Var | Background]]===
@@ Line 10: / Line 10: @@
 ===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.
@@ Line 19: / Line 19: @@
 * '''Test Statistic''': <math>F_o = {\sigma_1^2 \over \sigma_2^2}</math>
-The higher the deviation of this ratio away from 1, the stronger the evidence for unequal population variances.
+The farter away this ratio is from 1, the stronger the evidence for unequal population variances.
 * 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
@@ Line 38: / Line 38: @@
 ===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 (<math>\alpha=0.1</math>).
+* 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 distributed populations of liquid content (in fluid ounces) of beverage cans. Use (<math>\alpha=0.1</math>).
 <center>
 {| class="wikitable" style="text-align:center; width:75%" border="1"
@@ Line 68: / Line 68: @@
 * Test Statistics: <math>F_o = {\sigma_1^2 \over \sigma_2^2}=1.300406878</math>
-* 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.
+* 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 discriminate between the population variances of the two populations based on these (small) samples.
 <center>[[Image:SOCR_EBook_Dinov_Infer_BiVar_021608_Fig3.jpg|500px]]</center>
@@ Line 74: / Line 74: @@
 ===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.
+* 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.
 <hr>

@@ Line 66: / Line 66: @@
 * Identify the degrees of freedom (<math>df_1=6-1=5</math> and <math>df_2=10-1=9</math>).
-* Test Statistics: <math>F_o^2 = {\sigma_1^2 \over \sigma_2^2}=1.300406878</math>
+* Test Statistics: <math>F_o = {\sigma_1^2 \over \sigma_2^2}=1.300406878</math>
-* Significance Inference: P-value=<math>P(F_{(df_1=5, df_2=9)} > F_o^2) = 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.
+* 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.
 <center>[[Image:SOCR_EBook_Dinov_Infer_BiVar_021608_Fig3.jpg|500px]]</center>

AP Statistics Curriculum 2007 Infer BiVar - Revision history

Tdlee: Text replacement - "{{translate|pageName=http://wiki.stat.ucla.edu/socr/" to ""{{translate|pageName=http://wiki.socr.umich.edu/"

Jenny at 20:49, 28 June 2010

Jenny at 17:30, 28 June 2010

IvoDinov at 02:30, 29 November 2008

IvoDinov: /* Hands-on activities */

Jenny: /* Comparing Two Variances (\sigma_1^2 = \sigma_2^2?) */

PriscillaChui: /* Comparing Two Variances (\sigma_1^2 = \sigma_2^2?) */

PriscillaChui: /* General Advance-Placement (AP) Statistics Curriculum - Comparing Two Variances */

IvoDinov at 22:39, 16 February 2008

IvoDinov: /* Hands-on activities */

PriscillaChui: /* Comparing Two Variances ( $\sigma_1^2 = \sigma_2^2$ ?) */