AP Statistics Curriculum 2007 Estim L Mean - Revision history

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

2020-03-03T17:09:13Z

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

IvoDinov: /* Example */ typo

2011-05-20T17:31:10Z

‎Example: typo

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

2011-05-18T17:05:26Z

‎Interval Estimation of a Population Mean

IvoDinov: added a CI definition subsection

2011-02-14T18:38:40Z

added a CI definition subsection

IvoDinov: /* Unknown Variance */ added a link to CI Applet

2011-02-14T16:50:36Z

‎Unknown Variance: added a link to CI Applet

IvoDinov: /* Example */ added transpose of table

2011-02-14T16:44:33Z

‎Example: added transpose of table

Jenny: /* Example */

2010-06-28T20:29:25Z

‎Example

Jenny at 17:12, 28 June 2010

2010-06-28T17:12:52Z

IvoDinov: added a link to the Problems set

2009-10-26T05:07:33Z

added a link to the Problems set

IvoDinov: /* Assumptions */ typo

2008-11-14T06:47:19Z

‎Assumptions: typo

← Older revision		Revision as of 17:09, 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_L_Mean}}	+	"{{translate\|pageName=http://wiki.socr.umich.edu/index.php?title=AP_Statistics_Curriculum_2007_Estim_L_Mean}}

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 </center>
-A ''confidence interval estimate'' of <math>\mu</math> is a range of values used to estimate a population parameter (interval estimates are normally used more than point estimates because it is very unlikely that the sample mean would match exactly with the population mean). The interval estimate uses a margin of error about the point estimate.  For example if you have a point estimate of 12. 25 with a margin of error of 1.75, then the interval estimate would be (10.5 to 14).
+A ''confidence interval estimate'' of <math>\mu</math> is a range of values used to estimate a population parameter (interval estimates are normally used more than point estimates because it is very unlikely that the sample mean would match exactly with the population mean). The interval estimate uses a margin of error about the point estimate.  For example if you have a point estimate of 12.25 with a margin of error of 1.75, then the interval estimate would be (10.5 to 14).
 Before you find an interval estimate, you should first determine how confident you want to be that your interval estimate contains the population mean.

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 : <math>CI(\alpha): \overline{x} \pm z_{\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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 === Point Estimation of a Population Mean===
 For any process, the population mean may be estimated by a (large) sample average. 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===

← Older revision		Revision as of 16:50, 14 February 2011
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	Notice the increase of the CI's (directly related to the decrease of <math>\alpha</math>) reflecting our choice for higher confidence.		Notice the increase of the CI's (directly related to the decrease of <math>\alpha</math>) reflecting our choice for higher confidence.
		+
		+	You can use the [http://www.socr.ucla.edu/htmls/ana/ConfidenceInterval_Analysis.html SOCR CI Analysis Applet] to compute these interval estimates.

	===Hands-on Activities===		===Hands-on Activities===

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 ===Example===
-Market researchers use the ''number of sentences per advertisement'' as a measure of readability for magazine advertisements.  A random sample of the number of sentences found in 30 magazine advertisements is listed.  Use this sample to find ''point estimate'' for the population mean <math>\mu</math>.
+Market researchers use the ''number of sentences per advertisement'' as a measure of readability for magazine advertisements.  A random sample of the number of sentences found in 30 magazine advertisements is listed.  Use this sample to find ''point estimate'' for the population mean <math>\mu</math>. (See the [[AP_Statistics_Curriculum_2007_Estim_L_Mean_Table1 |data in (transposed) column format]])
 <center>
 {| class="wikitable" style="text-align:center; width:75%" border="1"

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 === Point Estimation of a Population Mean===
-For any process, the population mean may be estimated by a (large) sample average. 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]].
+For any process, the population mean may be estimated by a (large) sample average. 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===
 [[Image:SOCR_EBook_Dinov_Estimates_L_Mean_121208_Fig3.png|200px|thumbnail|right]]
-For large 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> will be
+For large 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> will be
 : <math>CI(\alpha): \overline{x} \pm z_{\alpha\over 2} E,</math>
@@ Line 16: / Line 16: @@
 {SE},& \texttt{for-unknown}-\sigma.\end{cases}</math>
-* The '''Standard Error''' of the estimate <math>\overline {x}</math> is obtained by replacing the unknown population standard deviation by the sample standard deviation:
+* The '''Standard Error''' of the estimated <math>\overline {x}</math> is obtained by replacing the unknown population standard deviation by the sample standard deviation:
 <math>SE(\overline {x}) = {1\over \sqrt{n}} \sqrt{\sum_{i=1}^n{(x_i-\overline{x})^2\over n-1}}</math>

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

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 ===Assumptions===
-The [[AP_Statistics_Curriculum_2007_Limits_CLT | Central Limit Theorem]] guarantees that for large-samples, the method above provides a valid recipe for constructing a confidence interval for the population mean, no matter what the distribution of the observed data may be. Of course, for significantly non-Normal distributions, we may need to increase the sample size to guarantee that the distribution of the sample average is approximately Normal!
+The [[AP_Statistics_Curriculum_2007_Limits_CLT | Central Limit Theorem]] guarantees that for large-samples, the method above provides a valid recipe for constructing a confidence interval for the population mean, no matter what the distribution of the observed data may be. Of course, for significantly non-Normal distributions, we may need to increase the sample size to guarantee that the sampling distribution of the mean is approximately Normal!
 ===Example===