AP Statistics Curriculum 2007 Estim Var - Revision history

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

2020-03-03T20:29:22Z

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

IvoDinov: /* Point Estimates of Population Variance and Standard Deviation */

2012-02-27T16:55:22Z

‎Point Estimates of Population Variance and Standard Deviation

Jenny at 20:34, 28 June 2010

2010-06-28T20:34:05Z

IvoDinov: /* Interval Estimates of Population Variance and Standard Deviation */ typo

2010-06-08T23:20:13Z

‎Interval Estimates of Population Variance and Standard Deviation: typo

IvoDinov: /* More Examples */ minor

2010-06-02T17:12:31Z

‎More Examples: minor

IvoDinov: /* More Examples */ added a link to the general CI activity

2010-06-02T17:12:16Z

‎More Examples: added a link to the general CI activity

IvoDinov: added a link to the Problems set

2009-10-26T05:10:11Z

added a link to the Problems set

PriscillaChui: /* General Advance-Placement (AP) Statistics Curriculum - Estimating Population Variance */

2008-03-01T22:27:57Z

‎ General Advance-Placement (AP) Statistics Curriculum - Estimating Population Variance

IvoDinov: /* Hands-on activities */

2008-02-06T23:40:17Z

‎Hands-on activities

IvoDinov: /* Hands-on activities */

2008-02-06T23:34:29Z

‎Hands-on activities

← Older revision		Revision as of 20:29, 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_Estim_Var}}	+	"{{translate\|pageName=http://wiki.socr.umich.edu/index.php?title=AP_Statistics_Curriculum_2007_Estim_Var}}

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 The most unbiased point estimate for the population variance <math>\sigma^2</math> is the [[AP_Statistics_Curriculum_2007_EDA_Var | Sample-Variance (s<sup>2</sup>)]] and the point estimate for the population standard deviation <math>\sigma</math> is the [[AP_Statistics_Curriculum_2007_EDA_Var | Sample Standard Deviation (s)]].
-We use a [http://en.wikipedia.org/wiki/Chi_square_distribution Chi-Square Distribution] to construct confidence intervals for the variance and standard distribution. If the process or phenomenon we study generates a Normal random variable, then computing the following random variable (for a sample of size <math>n>1</math>) has a Chi-Square Distribution
+We use a [http://en.wikipedia.org/wiki/Chi_square_distribution Chi-Square Distribution] to construct confidence intervals for the variance and standard distribution. If the process or phenomenon we study generates a Normal random variable, then computing the following random variable (for a sample of size <math>n>1</math>) has a [[AP_Statistics_Curriculum_2007_Chi-Square|Chi-Square Distribution]]
 : <math>\chi_o^2 = {(n-1)s^2 \over \sigma^2}</math>

@@ Line 11: / Line 11: @@
 ===Chi-Square Distribution Properties===
 * All chi-squares values <math>\chi_o^2 \geq 0</math>.
-* The chi-square distribution is a family of curves, each determined by the degrees of freedom (n-1). See the interactive [http://socr.ucla.edu/htmls/SOCR_Distributions.html Chi-Square distribution].
+* The chi-square distribution is a family of curves, each is determined by the degrees of freedom (n-1). See the interactive [http://socr.ucla.edu/htmls/SOCR_Distributions.html Chi-Square distribution].
 * To form a confidence interval for the variance (<math>\sigma^2</math>), use the <math>\chi^2(df=n-1)</math> distribution with degrees of freedom equal to one less than the sample size.
 * The area under each curve of the Chi-Square Distribution equals one.
@@ Line 55: / Line 55: @@
 ===More Examples===
-* You randomly select and measure the contents of 15 bottles of cough syrup.  The results (in fluid ounces) are shown.  Use a 95% level of confidence to construct a confidence interval for the standard deviation (<math>\sigma</math>) assuming the contents of these cough syrup bottles is Normally distributed. Does this CI(<math>\sigma</math>) suggest that the variation in the bottles is at an acceptable level if the '''population standard deviation''' of the bottle’s contents should be less than 0.025 fluid ounce?
+* You randomly select and measure the contents of 15 bottles of cough syrup.  The results (in fluid ounces) are shown.  Use a 95% level of confidence to construct a confidence interval for the standard deviation (<math>\sigma</math>) assuming the contents of these cough syrup bottles are Normally distributed. Does this CI(<math>\sigma</math>) suggest that the variation in the bottles is at an acceptable level if the '''population standard deviation''' of the bottle’s contents should be less than 0.025 fluid ounce?
 <center>
 {| class="wikitable" style="text-align:center; width:75%" border="1"

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 Notice that the Chi-Square Distribution is '''not''' symmetric (positively skewed) and therefore, there are two critical values for each level of confidence.  The value <math>\chi_L^2</math> represents the left-tail critical value and <math>\chi_R^2</math> represents the right-tail critical value.  For various degrees of freedom and areas, you can compute all critical values either using the [http://socr.ucla.edu/htmls/SOCR_Distributions.html SOCR Distributions] or using the [http://socr.ucla.edu/Applets.dir/Normal_T_Chi2_F_Tables.htm SOCR Chi-square Distribution Calculator].
-* Example: Find the critical values, <math>\chi_L^2</math> and <math>\chi_R^2</math>, for a 95% confidence interval when the sample size is 25. Use the following Protocol:
+* Example: Find the critical values, <math>\chi_L^2</math> and <math>\chi_R^2</math>, for a 90% confidence interval when the sample size is 25. Use the following Protocol:
-** Identify the degrees of freedom (<math>df=n-1=24</math>) and the level of confidence (<math>{\alpha\over 2}=0.025</math>).
+** Identify the degrees of freedom (<math>df=n-1=24</math>) and the level of confidence (<math>{\alpha\over 2}=0.05</math>).
 ** Find the left and right critical values, <math>\chi_L^2=13.848</math> and <math>\chi_R^2=36.415</math>, as in the image below.
 <center>[[Image:SOCR_EBook_Dinov_Estim_Var_020408_Fig2.jpg|500px]]</center>

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 * For the [[SOCR_012708_ID_Data_HotDogs | hot-dogs dataset]] construct 97% CI for the population standard deviation of the calorie and sodium contents, separately.
-* [[SOCR_EduMaterials_Activities_General_CI_Experiment | Also see the General Confidence Interval Activity]]
+* [[SOCR_EduMaterials_Activities_General_CI_Experiment | Also see the General Confidence Interval Activity]].
 <hr>

← Older revision		Revision as of 17:12, 2 June 2010
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	* For the [[SOCR_012708_ID_Data_HotDogs \| hot-dogs dataset]] construct 97% CI for the population standard deviation of the calorie and sodium contents, separately.		* For the [[SOCR_012708_ID_Data_HotDogs \| hot-dogs dataset]] construct 97% CI for the population standard deviation of the calorie and sodium contents, separately.
		+
		+	* [[SOCR_EduMaterials_Activities_General_CI_Experiment \| Also see the General Confidence Interval Activity]]

	<hr>		<hr>

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

@@ Line 4: / Line 4: @@
 === Point Estimates of Population Variance and Standard Deviation===
-The most unbiased point estimate for the population variance <math>\sigma^2</math> is the [[AP_Statistics_Curriculum_2007_EDA_Var | sample-variance (s<sup>2</sup>)]] and the point estimate for the population standard deviation <math>\sigma</math> is the [[AP_Statistics_Curriculum_2007_EDA_Var | sample standard deviation (s)]].
+The most unbiased point estimate for the population variance <math>\sigma^2</math> is the [[AP_Statistics_Curriculum_2007_EDA_Var | Sample-Variance (s<sup>2</sup>)]] and the point estimate for the population standard deviation <math>\sigma</math> is the [[AP_Statistics_Curriculum_2007_EDA_Var | Sample Standard Deviation (s)]].
-We use a [http://en.wikipedia.org/wiki/Chi_square_distribution Chi-square distribution] to construct confidence intervals for the variance and standard distribution. If the process or phenomenon we study generates a Normal random variable, then computing the following random variable (for a sample of size <math>n>1</math>) has a Chi-square distribution
+We use a [http://en.wikipedia.org/wiki/Chi_square_distribution Chi-Square Distribution] to construct confidence intervals for the variance and standard distribution. If the process or phenomenon we study generates a Normal random variable, then computing the following random variable (for a sample of size <math>n>1</math>) has a Chi-Square Distribution
 : <math>\chi_o^2 = {(n-1)s^2 \over \sigma^2}</math>
-===Chi-square Distribution Properties===
+===Chi-Square Distribution Properties===
 * All chi-squares values <math>\chi_o^2 \geq 0</math>.
 * The chi-square distribution is a family of curves, each determined by the degrees of freedom (n-1). See the interactive [http://socr.ucla.edu/htmls/SOCR_Distributions.html Chi-Square distribution].
 * To form a confidence interval for the variance (<math>\sigma^2</math>), use the <math>\chi^2(df=n-1)</math> distribution with degrees of freedom equal to one less than the sample size.
-* The area under each curve of the Chi-square distribution equals one.
+* The area under each curve of the Chi-Square Distribution equals one.
-* All Chi-square distributions are positively skewed.
+* All Chi-Square Distributions are positively skewed.
 <center>[[Image:SOCR_EBook_Dinov_Estim_Var_020408_Fig1.jpg|500px]]</center>
-===Interval Estimates of population Variance and Standard Deviation===
+===Interval Estimates of Population Variance and Standard Deviation===
-Notice that the Chi-square distribution is not symmetric (positively skewed) and therefore, there are two critical values for each level of confidence.  The value <math>\chi_L^2</math> represents the left-tail critical value and <math>\chi_R^2</math> represents the right-tail critical value.  For various degrees of freedom and areas, you can compute all critical values either using the [http://socr.ucla.edu/htmls/SOCR_Distributions.html SOCR Distributions] or using the [http://socr.ucla.edu/Applets.dir/Normal_T_Chi2_F_Tables.htm SOCR Chi-square distribution calculator].
+Notice that the Chi-Square Distribution is '''not''' symmetric (positively skewed) and therefore, there are two critical values for each level of confidence.  The value <math>\chi_L^2</math> represents the left-tail critical value and <math>\chi_R^2</math> represents the right-tail critical value.  For various degrees of freedom and areas, you can compute all critical values either using the [http://socr.ucla.edu/htmls/SOCR_Distributions.html SOCR Distributions] or using the [http://socr.ucla.edu/Applets.dir/Normal_T_Chi2_F_Tables.htm SOCR Chi-square Distribution Calculator].
 * Example: Find the critical values, <math>\chi_L^2</math> and <math>\chi_R^2</math>, for a 95% confidence interval when the sample size is 25. Use the following Protocol:
@@ Line 25: / Line 25: @@
 <center>[[Image:SOCR_EBook_Dinov_Estim_Var_020408_Fig2.jpg|500px]]</center>
-====Confidence interval for <math>\sigma^2</math>====
+====Confidence Interval for <math>\sigma^2</math>====
 : <math>{(n-1)s^2 \over \chi_R^2} \leq \sigma^2 \leq {(n-1)s^2 \over \chi_L^2}</math>
-====Confidence interval for <math>\sigma</math>====
+====Confidence Interval for <math>\sigma</math>====
 : <math>\sqrt{(n-1)s^2 \over \chi_R^2} \leq \sigma \leq \sqrt{(n-1)s^2 \over \chi_L^2}</math>
-===Hands-on activities===
+===Hands-on Activities===
 * Construct the confidence intervals for <math>\sigma^2</math> and <math>\sigma</math> assuming the observations below represent a random sample from the liquid content (in fluid ounces) of 16 beverage cans and can be considered as Normally distributed. Use a 90% level of confidence.
 <center>
@@ Line 53: / Line 53: @@
 : <math>0.0564=\sqrt{15\times 0.0053 \over 24.9958} \leq \sigma \leq \sqrt{15\times 0.0053 \over 7.261}=0.10464</math>
-===More examples===
+===More Examples===
 * You randomly select and measure the contents of 15 bottles of cough syrup.  The results (in fluid ounces) are shown.  Use a 95% level of confidence to construct a confidence interval for the standard deviation (<math>\sigma</math>) assuming the contents of these cough syrup bottles is Normally distributed. Does this CI(<math>\sigma</math>) suggest that the variation in the bottles is at an acceptable level if the '''population standard deviation''' of the bottle’s contents should be less than 0.025 fluid ounce?

@@ Line 48: / Line 48: @@
 * CI(<math>\sigma^2</math>)
-: <math>{15\times 0.0053 \over 24.9958} \leq \sigma^2 \leq {15\times 0.0052 \over 7.261}</math>
+: <math>0.00318={15\times 0.0053 \over 24.9958} \leq \sigma^2 \leq {15\times 0.0053 \over 7.261}=0.01095</math>
 * CI(<math>\sigma</math>)
-: <math>\sqrt{15\times 0.0053 \over 24.9958} \leq \sigma \leq \sqrt{15\times 0.0052 \over 7.261}</math>
+: <math>0.0564=\sqrt{15\times 0.0053 \over 24.9958} \leq \sigma \leq \sqrt{15\times 0.0053 \over 7.261}=0.10464</math>
 ===More examples===

@@ Line 43: / Line 43: @@
 <center>[[Image:SOCR_EBook_Dinov_Estim_Var_020408_Fig3.jpg|500px]]</center>
-* Identify the degrees of freedom (<math>df=n-1=15</math>) and the level of confidence (<math>{\alpha/2}=0.05</math>), as we are looking for a <math>(1-\alpha)100% CI(\sigma^2)</math>).
+* Identify the degrees of freedom (<math>df=n-1=15</math>) and the level of confidence (<math>{\alpha/2}=0.05</math>), as we are looking for a <math>(1-\alpha)100% CI(\sigma^2)</math>.
 * Find the left and right critical values, <math>\chi_L^2=7.261</math> and <math>\chi_R^2=24.9958</math> using [http://socr.ucla.edu/htmls/SOCR_Distributions.html SOCR Chi-Square Distribution], as in the image below.
 <center>[[Image:SOCR_EBook_Dinov_Estim_Var_020408_Fig4.jpg|500px]]</center>