AP Statistics Curriculum 2007 Hypothesis Proportion - Revision history

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

2020-03-03T19:55:46Z

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

IvoDinov at 00:22, 24 April 2013

2013-04-24T00:22:23Z

IvoDinov: /* Genders of Siblings Example */

2013-04-23T23:46:12Z

‎Genders of Siblings Example

IvoDinov: /* Background */ typo

2010-03-01T18:16:40Z

‎Background: typo

IvoDinov: added the link for the Mission Improbable: A Concise and Precise Definition of P-Value discussion

2009-11-18T21:59:23Z

added the link for the Mission Improbable: A Concise and Precise Definition of P-Value discussion

IvoDinov: added a link to the Problems set

2009-10-26T05:14:02Z

added a link to the Problems set

IvoDinov: /* Hypothesis Testing about a Single Sample Proportion */

2008-11-19T18:54:59Z

‎Hypothesis Testing about a Single Sample Proportion

IvoDinov: /* Hypothesis Testing about a Single Sample Proportion */

2008-11-19T18:54:22Z

‎Hypothesis Testing about a Single Sample Proportion

IvoDinov: /* Hypothesis Testing about a Single Sample Proportion */

2008-11-19T18:53:01Z

‎Hypothesis Testing about a Single Sample Proportion

IvoDinov at 18:52, 19 November 2008

2008-11-19T18:52:13Z

← Older revision		Revision as of 19:55, 3 March 2020
Line 84:		Line 84:
	* 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_Hypothesis_Proportion}}	+	"{{translate\|pageName=http://wiki.socr.umich.edu/index.php?title=AP_Statistics_Curriculum_2007_Hypothesis_Proportion}}

@@ Line 76: / Line 76: @@
 ===[[EBook_Problems_Hypothesis_Proportion|Problems]]===
 * [http://sciencenow.sciencemag.org/cgi/content/full/2009/1030/1 Read this Science discussion "Mission Improbable: A Concise and Precise Definition of P-Value"].
 <hr>

@@ Line 66: / Line 66: @@
 * 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.

@@ Line 3: / Line 3: @@
 === Background===
 [[AP_Statistics_Curriculum_2007_Estim_Proportion |Recall that for large samples]], the sampling distribution of the sample proportion <math>\hat{p}</math> is approximately Normal, by [[AP_Statistics_Curriculum_2007_Limits_CLT |CLT]], as the sample proportion may be presented as a [[AP_Statistics_Curriculum_2007_Limits_Norm2Bin |sample average or Bernoulli random variables]]. When the sample size is small, the normal approximation may be inadequate. To accommodate this, we will modify the '''sample-proportion''' <math>\hat{p}</math> slightly and obtain the '''corrected-sample-proportion''' <math>\tilde{p}</math>:
-: <math>\hat{p}={y\over n} \longrightarrow \tilde{y}={y+0.5z_{\alpha \over 2}^2 \over n+z_{\alpha \over 2}^2},</math>
+: <math>\hat{p}={y\over n} \longrightarrow \tilde{p}={y+0.5z_{\alpha \over 2}^2 \over n+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]].

← Older revision		Revision as of 21:59, 18 November 2009
Line 76:		Line 76:

	===[[EBook_Problems_Hypothesis_Proportion\|Problems]]===		===[[EBook_Problems_Hypothesis_Proportion\|Problems]]===
		+	* [http://sciencenow.sciencemag.org/cgi/content/full/2009/1030/1 Read this Science discussion "Mission Improbable: A Concise and Precise Definition of P-Value"].

	<hr>		<hr>

@@ Line 75: / Line 75: @@
 <hr>
-===References===
+===[[EBook_Problems_Hypothesis_Proportion|Problems]]===
 <hr>

@@ Line 15: / Line 15: @@
 ** Double sided: <math>H_1: p \not= p_o</math>
-* Test Statistics: <math>Z_o={\tilde{p} -p_o \over SE_{\tilde{p}}}</math>
+* Test Statistics: <math>Z_o={\tilde{p} -p_o \over SE_{\tilde{p}}} \sim N(0,1).</math>
 ===Example===

@@ Line 10: / Line 10: @@
 === Hypothesis Testing about a Single Sample Proportion===
-* Null Hypothesis: <math>H_o: p=p_o</math> (e.g., <math>p_o=0</math>), where ''p'' is the population proportion of interest.
+* Null Hypothesis: <math>H_o: p=p_o</math> (e.g., <math>p_o={1\over 2}</math>), where ''p'' is the population proportion of interest.
 * Alternative Research Hypotheses:
 ** One sided (uni-directional): <math>H_1: p >p_o</math>, or <math>H_1: p<p_o</math>