AP Statistics Curriculum 2007 Estim S Mean - Revision history

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

2020-03-03T21:00:54Z

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

IvoDinov: /* Interval Estimation of a Population Mean */

2011-05-18T17:06:36Z

‎Interval Estimation of a Population Mean

IvoDinov: /* Example */

2011-02-16T16:42:18Z

‎Example

IvoDinov: /* Interval Estimation of a Population Mean */

2011-02-16T16:40:30Z

‎Interval Estimation of a Population Mean

Jenny at 17:15, 28 June 2010

2010-06-28T17:15:31Z

IvoDinov: added a link to the Problems set

2009-10-26T05:08:24Z

added a link to the Problems set

IvoDinov: /* Assumptions */ typo

2008-11-14T06:40:25Z

‎Assumptions: typo

IvoDinov: added an assumptions section

2008-11-14T06:39:33Z

added an assumptions section

IvoDinov: added T-distribution FIgure 4

2008-06-06T04:38:48Z

added T-distribution FIgure 4

IvoDinov: /* Example */

2008-06-06T04:37:09Z

‎Example

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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_S_Mean}}	+	"{{translate\|pageName=http://wiki.socr.umich.edu/index.php?title=AP_Statistics_Curriculum_2007_Estim_S_Mean}}

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 : <math>CI(\alpha): \overline{x} \pm t_{\alpha\over 2} E.</math>
-* The '''Margin of Error''' E is defined as
+* The '''Error''' term, E, is defined as
 <math>E = \begin{cases}{\sigma\over\sqrt{n}},& \texttt{for-known}-\sigma,\\
 {SE},& \texttt{for-unknown}-\sigma.\end{cases}</math>

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 :99% confidence (0.99), <math>{\alpha \over 2}=0.005</math>, [http://socr.ucla.edu/Applets.dir/T-table.html t<sub>(df=9)</sub> = 3.250]
-Notice that for a fixed <math>\alpha</math>, the ''t-critical values'' (for any degree-of-freedom) exceeds the [[AP_Statistics_Curriculum_2007#Estimating_a_Population_Mean:_Large_Samples |corresponding normal z-critical values]], which are used in the large-sample interval estimation.
+Notice that for a fixed <math>\alpha</math>, the ''t-critical values'' (for any degree-of-freedom) exceeds the [[AP_Statistics_Curriculum_2007#Estimating_a_Population_Mean:_Large_Samples |corresponding normal z-critical values]], which are used in the large-sample interval estimation. The [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html SOCR Student's T distribution calculator] may be used to obtain accurate critical and probability values for the T distribution.
 <center>[[Image:SOCR_EBook_Dinov_Estimates_L_Mean_060608_Fig4.jpg|500px]]</center>

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 ===Interval Estimation of a Population Mean===
-For small samples, interval estimations of the population means (or '''Confidence Intervals''') are constructed as follows. Choose a confidence level <math>(1-\alpha)100%</math>, where <math>\alpha</math> is small (e.g., 0.1, 0.05, 0.025, 0.01, 0.001, etc.). Then a <math>(1-\alpha)100%</math> confidence interval for <math>\mu</math> is defined in terms of the T-distribution:
+For small samples, interval estimations of the population means (or '''Confidence Intervals''') are constructed as follows. Choose a confidence level <math>(1-\alpha)100%</math>, where <math>\alpha</math> is small (e.g., 0.1, 0.05, 0.025, 0.01, 0.001, etc.). Then a <math>(1-\alpha)100%</math> confidence interval for <math>\mu</math> is defined in terms of the [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html T-distribution]:
 : <math>CI(\alpha): \overline{x} \pm t_{\alpha\over 2} E.</math>

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 === Point Estimation of a Population Mean===
-The population mean may also be estimated by the sample average using a small sample. That is the sample average <math>\overline{X_n}={1\over n}\sum_{i=1}^n{X_i}</math>, constructed from a random sample of the process {<math>X_1, X_2, X_3, \cdots , X_n</math>}, is an [http://en.wikipedia.org/wiki/Estimator_bias unbiased] estimate of the population mean <math>\mu</math>, if it exists! Note that the [[AP_Statistics_Curriculum_2007_EDA_Center | sample average may be susceptible to outliers]].
+The population mean may also be estimated by the sample average using a small sample. That is the sample average <math>\overline{X_n}={1\over n}\sum_{i=1}^n{X_i}</math>, constructed from a random sample of the process {<math>X_1, X_2, X_3, \cdots , X_n</math>}, which is an [http://en.wikipedia.org/wiki/Estimator_bias unbiased] estimate of the population mean <math>\mu</math>, if it exists! Note that the [[AP_Statistics_Curriculum_2007_EDA_Center | sample average may be susceptible to outliers]].
 ===Interval Estimation of a Population Mean===
-For small samples, interval estimation of the population means (or '''Confidence Intervals''') are constructed as follows. Choose a confidence level <math>(1-\alpha)100%</math>, where <math>\alpha</math> is small (e.g., 0.1, 0.05, 0.025, 0.01, 0.001, etc.). Then a <math>(1-\alpha)100%</math> confidence interval for <math>\mu</math> is defined in terms of the T-distribution:
+For small samples, interval estimations of the population means (or '''Confidence Intervals''') are constructed as follows. Choose a confidence level <math>(1-\alpha)100%</math>, where <math>\alpha</math> is small (e.g., 0.1, 0.05, 0.025, 0.01, 0.001, etc.). Then a <math>(1-\alpha)100%</math> confidence interval for <math>\mu</math> is defined in terms of the T-distribution:
 : <math>CI(\alpha): \overline{x} \pm t_{\alpha\over 2} E.</math>
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 ===Assumptions===
-To use the small-sample method for constructing a confidence interval for the population mean, we must have evidence that the data we observed and are using for the point and interval estimates come from a distribution which is (approximately) ''Normal''. If this assumption is violated, the interval estimate we compute may be significantly misrepresenting the real confidence interval!
+To use the small-sample method for constructing a confidence interval for the population mean, we must have evidence that the data we observed and used for the point and interval estimates come from a distribution which is (approximately) ''Normal''. If this assumption is violated, the interval estimate we compute may be significantly misrepresenting the real confidence interval!
 ===Example===

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

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	* <math>t_{\alpha\over 2}</math> is the [[AP_Statistics_Curriculum_2007_StudentsT \|Critical Value for the T(df=sample-size -1) distribution at <math>{\alpha\over 2}</math>]].		* <math>t_{\alpha\over 2}</math> is the [[AP_Statistics_Curriculum_2007_StudentsT \|Critical Value for the T(df=sample-size -1) distribution at <math>{\alpha\over 2}</math>]].
		+
		+	===Assumptions===
		+	To use the small-sample method for constructing a confidence interval for the population mean, we must have evidence that the data we observed and are using for the point and interval estimate comes from a distribution which is (approximately) Normal. If this assumption is violated, the interval estimate we compute may be significantly misrepresenting the real confidence interval!

	===Example===		===Example===

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	Notice that for a fixed <math>\alpha</math>, the ''t-critical values'' (for any degree-of-freedom) exceeds the [[AP_Statistics_Curriculum_2007#Estimating_a_Population_Mean:_Large_Samples \|corresponding normal z-critical values]], which are used in the large-sample interval estimation.		Notice that for a fixed <math>\alpha</math>, the ''t-critical values'' (for any degree-of-freedom) exceeds the [[AP_Statistics_Curriculum_2007#Estimating_a_Population_Mean:_Large_Samples \|corresponding normal z-critical values]], which are used in the large-sample interval estimation.
		+
		+	<center>[[Image:SOCR_EBook_Dinov_Estimates_L_Mean_060608_Fig4.jpg\|500px]]</center>

	====Known Variance====		====Known Variance====

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 Before you find an interval estimate, you should first determine how confident you want to be that your interval estimate contains the population mean.
-* [[AP_Statistics_Curriculum_2007_Normal_Critical | Recall these critical values for Standard Normal Distribution]]:
+* [[AP_Statistics_Curriculum_2007_StudentsT | Recall these critical values for Student's T Distribution]]:
 :80% confidence (0.80), <math>{\alpha \over 2}=0.1</math>, [http://socr.ucla.edu/Applets.dir/T-table.html t<sub>(df=9)</sub> = 1.383]
 :90% confidence (0.90), <math>{\alpha \over 2}=0.05</math>, [http://socr.ucla.edu/Applets.dir/T-table.html t<sub>(df=9)</sub> = 1.833]