AP Statistics Curriculum 2007 Infer 2Proportions - Revision history

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

2020-03-03T16:47:14Z

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

Jenny at 17:30, 28 June 2010

2010-06-28T17:30:33Z

IvoDinov: /* Genders of Siblings Example */

2010-05-11T23:11:42Z

‎Genders of Siblings Example

IvoDinov: added a link to the Problems set

2009-10-26T05:19:53Z

added a link to the Problems set

IvoDinov: /* Hypothesis Testing the Difference of Two Proportions */

2008-11-19T19:07:07Z

‎Hypothesis Testing the Difference of Two Proportions

IvoDinov: /* Hypothesis Testing the Difference of Two Proportions */

2008-11-19T19:06:43Z

‎Hypothesis Testing the Difference of Two Proportions

PriscillaChui: /* General Advance-Placement (AP) Statistics Curriculum - Inferences about Two Proportions */

2008-03-01T23:04:29Z

‎ General Advance-Placement (AP) Statistics Curriculum - Inferences about Two Proportions

PriscillaChui: /* Genders of Siblings Example */

2008-02-19T20:55:34Z

‎Genders of Siblings Example

IvoDinov: /* Testing for equality of Two Proportions */

2008-02-17T00:33:26Z

‎Testing for equality of Two Proportions

IvoDinov: /* Testing for equality of Two Proportions */

2008-02-17T00:31:31Z

‎Testing for equality of Two Proportions

← Older revision		Revision as of 16:47, 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_2Proportions}}	+	"{{translate\|pageName=http://wiki.socr.umich.edu/index.php?title=AP_Statistics_Curriculum_2007_Infer_2Proportions}}

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 === Testing for Equality of Two Proportions===
-Suppose we have two populations and we are interested in estimating whether the proportions of subjects that have certain characteristic of interest (e.g., fixed gender) in each population are equal. To make this inference we obtain two samples {<math>X_1, X_2, X_3, \cdots, X_n</math>} and {<math>Y_1, Y_2, Y_3, \cdots, Y_k</math>}, where each <math>X_i</math> and <math>Y_i</math> represents whether the ''i<sup>th</sup>'' observation in the sample had the characteristic of interest. That is
+Suppose we have two populations and we are interested in estimating whether the proportions of subjects that have certain characteristic of interest (e.g., fixed gender) in each population are equal. To make this inference we obtain two samples {<math>X_1, X_2, X_3, \cdots, X_n</math>} and {<math>Y_1, Y_2, Y_3, \cdots, Y_k</math>}, where each <math>X_i</math> and <math>Y_i</math> represents whether the ''i<sup>th</sup>'' observation in the sample has the characteristic of interest. That is
 : <math>X_i = \begin{cases}0,& \texttt{Characteristic-absent},\\
 ,& \texttt{Characteristic-present}.\end{cases}</math> and <math>Y_i = \begin{cases}0,& \texttt{Characteristic-absent},\\

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 * Test Statistics: <math>Z_o = {Estimate-HypothesizedValue\over SE(Estimate)} = {\hat{p}_1 - \hat{p}_2 - 0 \over SE(\hat{p}_1 - \hat{p}_2)} = {\hat{p}_1 - \hat{p}_2 - 0 \over \sqrt{{\hat{p}_1(1-\hat{p}_1)\over n_1} + {\hat{p}_2(1-\hat{p}_2)\over n_2}}} \sim N(0,1)</math> and <math>Z_o=5.4996</math>.
-* <math>P_value = P(Z>Z_o)< 1.9\times 10^{-8}</math>. This small p-values provides extremely strong evidence to reject the null hypothesis that there are no differences between the proportions of mothers that had a girl as a second child but had either boy or girl as their first child. Hence there is strong statistical evidence implying that genders of siblings are not independent.
+* <math>P_{value} = P(Z>Z_o)< 1.9\times 10^{-8}</math>. This small p-values provides extremely strong evidence to reject the null hypothesis that there are no differences between the proportions of mothers that had a girl as a second child but had either boy or girl as their first child. Hence there is strong statistical evidence implying that genders of siblings are not independent.
 * Practical significance: The practical significance of the effect (of the gender of the first child on the gender of the second child, in this case) can only be assessed using [[AP_Statistics_Curriculum_2007#Estimating_a_Population_Proportion |confidence intervals]]. A 95% <math>CI (p_1- p_2) =[0.033; 0.070]</math> is computed by <math>p_1-p_2 \pm 1.96 SE(p_1 - p_2)</math>. Clearly, this is a practically negligible effect and no reasonable person would make important prospective family decisions based on the gender of their (first) child.

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 <hr>
-===References===
+===[[EBook_Problems_Infer_2Proportions|Problems]]===
 <hr>

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 * Null Hypothesis: <math>H_o: p_x=p_y</math>, where <math>p_x</math> and <math>p_x</math> are the sample population proportions of interest.
 * Alternative Research Hypotheses:
-** One sided (uni-directional): <math>H_1: p_x > p_y</math>, or <math>H_o: p_x < p_y</math>
+** One sided (uni-directional): <math>H_1: p_x > p_y</math>, or <math>H_1: p_x < p_y</math>
 ** Double sided: <math>H_1: p_x \not= p_y</math>

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 ** Double sided: <math>H_1: p_x \not= p_y</math>
-* Test Statistics: <math>Z_o={\tilde{p_x} - \tilde{p_y} \over SE_{\tilde{p_x}-\tilde{p_y}}}</math>
+* Test Statistics: <math>Z_o={\tilde{p_x} - \tilde{p_y} \over SE_{(\tilde{p_x}-\tilde{p_y})}} \sim N(0,1)</math>
 ===Genders of Siblings Example===

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 ==[[AP_Statistics_Curriculum_2007 | General Advance-Placement (AP) Statistics Curriculum]] - Inferences about Two Proportions ==
-=== Testing for equality of Two Proportions===
+=== Testing for Equality of Two Proportions===
 Suppose we have two populations and we are interested in estimating whether the proportions of subjects that have certain characteristic of interest (e.g., fixed gender) in each population are equal. To make this inference we obtain two samples {<math>X_1, X_2, X_3, \cdots, X_n</math>} and {<math>Y_1, Y_2, Y_3, \cdots, Y_k</math>}, where each <math>X_i</math> and <math>Y_i</math> represents whether the ''i<sup>th</sup>'' observation in the sample had the characteristic of interest. That is
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 : '''Corrected Proportions''': <math>SE_{\tilde{p_x}-\tilde{p_y}} = \sqrt{SE_{\tilde{p_x}}^2 + SE_{\tilde{p_y}}^2}= \sqrt{ {\tilde{p_x}(1-\tilde{p_x})\over n+z_{\alpha \over 2}^2} + {\tilde{p_y}(1-\tilde{p_y})\over k+z_{\alpha \over 2}^2}}.</math>
-=== Hypothesis Testing the difference of Two Proportions===
+=== Hypothesis Testing the Difference of Two Proportions===
 * Null Hypothesis: <math>H_o: p_x=p_y</math>, where <math>p_x</math> and <math>p_x</math> are the sample population proportions of interest.
 * Alternative Research Hypotheses:
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 * <math>P_value = P(Z>Z_o)< 1.9\times 10^{-8}</math>. This small p-values provides extremely strong evidence to reject the null hypothesis that there are no differences between the proportions of mothers that had a girl as a second child but had either boy or girl as their first child. Hence there is strong statistical evidence implying that genders of siblings are not independent.
-* '''Practical significance''': The practical significance of the effect (of the gender of the first child on the gender of the second child, in this case) can only be assessed using [[AP_Statistics_Curriculum_2007#Estimating_a_Population_Proportion |confidence intervals]]. A 95% <math>CI (p_1- p_2) =[0.033; 0.070]</math> is computed by <math>p_1-p_2 \pm 1.96 SE(p_1 - p_2)</math>. Clearly, this is a practically negligible effect and no reasonable person would make important prospective family decisions based on the gender of their (first) child.
+* Practical significance: The practical significance of the effect (of the gender of the first child on the gender of the second child, in this case) can only be assessed using [[AP_Statistics_Curriculum_2007#Estimating_a_Population_Proportion |confidence intervals]]. A 95% <math>CI (p_1- p_2) =[0.033; 0.070]</math> is computed by <math>p_1-p_2 \pm 1.96 SE(p_1 - p_2)</math>. Clearly, this is a practically negligible effect and no reasonable person would make important prospective family decisions based on the gender of their (first) child.
 * This [[SOCR_EduMaterials_AnalysisActivities_Chi_Contingency | SOCR Analysis Activity]] illustrates how to use the [http://socr.ucla.edu/htmls/SOCR_Analyses.html SOCR Analyses] to compute the p-values and answer the hypothesis testing challenge.

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 ===Genders of Siblings Example===
-Is the gender of a second child influenced by the gender of the first child, in families with >1 kid? Research hypothesis needs to be formulated first before collecting/looking/interpreting the data that will be used to address it. Mothers whose 1<sup>st</sup> child is a girl are more likely to have a girl, as a second child, compared to mothers with boys as 1<sup>st</sup> children. Data: 20 yrs of birth records of 1 Hospital in Auckland, New Zealand.
+Is the gender of a second child influenced by the gender of the first child, in families with >1 child? Research hypothesis needs to be formulated first before collecting/looking/interpreting the data that will be used to address it. Mothers whose 1<sup>st</sup> child is a girl are more likely to have a girl, as a second child, compared to mothers with boys as 1<sup>st</sup> child. Data: 20 yrs of birth records of 1 Hospital in Auckland, New Zealand.
 <center>
 {| class="wikitable" style="text-align:center; width:75%" border="1"

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 ,& \texttt{Characteristic-present}.\end{cases}</math>
-Since the raw sample proportions of observations having the characteristic of interest are
+Since the ''raw sample proportions'' of observations having the characteristic of interest are
 : <math>\hat{p_x}={1 \over n}\sum_{i=1}^n{x_i}</math> and <math>\hat{p_y}={1 \over k}\sum_{i=1}^k{y_i}</math>
-The corrected sample proportions (for small samples) are
+The ''corrected sample proportions'' (for small samples) are
-: <math>\tilde{p_x}={\sum_{i=1}^n{x_i}+0.5z_{\alpha \over 2}^2 \over n+z_{\alpha \over 2}^2},</math>
+: <math>\tilde{p_x}={\sum_{i=1}^n{x_i}+0.5z_{\alpha \over 2}^2 \over n+z_{\alpha \over 2}^2},</math> and <math>\tilde{p_y}={\sum_{i=1}^k{y_i}+0.5z_{\alpha \over 2}^2 \over k+z_{\alpha \over 2}^2},</math>
 where [[AP_Statistics_Curriculum_2007_Normal_Critical | <math>z_{\alpha \over 2}</math> is the normal critical value we saw earlier]].
 By the independence of the samples, the standard error of the difference of the two proportion estimates is:
-: <math>SE_{\hat{p_x}-\hat{p_y}} = \sqrt{SE_{\hat{p_x}}^2 + SE_{\hat{p_y}}^2}= \sqrt{ {\hat{p_x}(1-\hat{p_x})\over n} + {\hat{p_y}(1-\hat{p_y})\over k}}.</math>
+: '''Raw proportions''': <math>SE_{\hat{p_x}-\hat{p_y}} = \sqrt{SE_{\hat{p_x}}^2 + SE_{\hat{p_y}}^2}= \sqrt{ {\hat{p_x}(1-\hat{p_x})\over n} + {\hat{p_y}(1-\hat{p_y})\over k}}.</math>
-: <math>SE_{\tilde{p_x}-\tilde{p_y}} = \sqrt{SE_{\tilde{p_x}}^2 + SE_{\tilde{p_y}}^2}= \sqrt{ {\tilde{p_x}(1-\tilde{p_x})\over n+z_{\alpha \over 2}^2} + {\tilde{p_y}(1-\tilde{p_y})\over k+z_{\alpha \over 2}^2}}.</math>
+: '''Corrected Proportions''': <math>SE_{\tilde{p_x}-\tilde{p_y}} = \sqrt{SE_{\tilde{p_x}}^2 + SE_{\tilde{p_y}}^2}= \sqrt{ {\tilde{p_x}(1-\tilde{p_x})\over n+z_{\alpha \over 2}^2} + {\tilde{p_y}(1-\tilde{p_y})\over k+z_{\alpha \over 2}^2}}.</math>
 === Hypothesis Testing the difference of Two Proportions===