AP Statistics Curriculum 2007 Hypothesis S Mean
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=== Testing a Claim about a Mean: Small Samples===  === Testing a Claim about a Mean: Small Samples===  
  
  
  +  The [[AP_Statistics_Curriculum_2007_Hypothesis_L_Mean  previous section discussed inference on the population mean for large smaples]]. Now, we show how to do hypothesis testing about the mean for small samplesizes.  
  +  
  *  +  ===[[AP_Statistics_Curriculum_2007_Estim_L_Mean  Background]]=== 
+  * Recall that for a random sample {<math>X_1, X_2, X_3, \cdots , X_n</math>} of the process, the population mean may be estimated by the sample average, <math>\overline{X_n}={1\over n}\sum_{i=1}^n{X_i}</math>.  
  ===  +  * For a given small <math>\alpha</math> (e.g., 0.1, 0.05, 0.025, 0.01, 0.001, etc.), the <math>(1\alpha)100%</math> Confidence interval for the mean is constructed by 
  +  : <math>CI(\alpha): \overline{x} \pm t_{df=n1,\alpha\over 2} {{1\over \sqrt{n}} \sqrt{\sum_{i=1}^n{(x_i\overline{x})^2\over n1}}}</math>  
+  : and <math>t_{df=n1, \alpha\over 2}</math> is the [[AP_Statistics_Curriculum_2007_Normal_Critical  critical value]] for a [[AP_Statistics_Curriculum_2007_StudentsT Tdistribution]] of df=(sample size  1) at <math>{\alpha\over 2}</math>.  
  *  +  === Hypothesis Testing about a Mean: Small Samples=== 
+  * Null Hypothesis: <math>H_o: \mu=\mu_o</math> (e.g., 0)  
+  * Alternative Research Hypotheses:  
+  ** One sided (unidirectional): <math>H_1: \mu >\mu_o</math>, or <math>H_o: \mu<\mu_o</math>  
+  ** Double sided: <math>H_1: \mu \not= \mu_o</math>  
  ===  +  ====Normal Process with Known Variance==== 
  *  +  * If the population is Normally distributed and we know the population variance, then the [http://en.wikipedia.org/wiki/Hypothesis_testing#Common_test_statistics Test statistics] is: 
+  : <math>Z_o = {\overline{x}  \mu_o \over \sigma} \sim N(0,1)</math>.  
  ===  +  ====(Approximately) Nornal Process with Unknown Variance==== 
  +  * If the population is approximately Normally distributed and we do not know the population variance, then the[http://en.wikipedia.org/wiki/Hypothesis_testing#Common_test_statistics Test statistics] is:  
+  : <math>T_o = {\overline{x}  \mu_o \over SE(\overline{x})} = {\overline{x}  \mu_o \over {{1\over \sqrt{n}} \sqrt{\sum_{i=1}^n{(x_i\overline{x})^2\over n1}}})} \sim N(0,1)</math>.  
  +  ===Example===  
  +  Let's use again the smallsample example of the [[AP_Statistics_Curriculum_2007_Estim_L_Mean  ''number of sentences per advertisement'']], where we measure of readability for magazine advertisements. A random sample of the number of sentences found in 10 magazine advertisements is listed below. Suppose we want to test at <math>\alpha=0.01</math> a null hypothesis: <math>H_o: \mu=12</math> against a onesided research alternative hypothesis: <math>H_1: \mu > 12</math>. Recall that we had the following [[AP_Statistics_Curriculum_2007_Estim_S_Mean#Example sample statistics: '''samplemean=22.1, samplevariance=737.88 and sampleSD=27.16390579''']] for these data.  
  ===  +  <center> 
  +  { class="wikitable" style="textalign:center; width:75%" border="1"  
+    
+   16  9  14  11 17  12 99  18  13 12  
+  }  
+  </center>  
  *  +  As the population variance is not given, we have to use the [[AP_Statistics_Curriculum_2007_StudentsT Tstatistics]]: <math>T_o = {\overline{x}  \mu_o \over SE(\overline{x})} \sim T(df=9)</math> 
+  : <math>T_o = {\overline{x}  \mu_o \over SE(\overline{x})} = {22.1  12 \over {{1\over \sqrt{10}} \sqrt{\sum_{i=1}^{10}{(x_i22.1)^2\over 9}}})}=....</math>.  
+  : <math>pvalue=P(T_{(df=9)}>T_o=....)=....</math> for this (onesided) test. Therefore, we '''can not reject''' the null hypothesis at <math>\alpha=0.01</math>! The left and right white areas at the tails of the T(df=9) distribution depict graphically the probability of interest, which represents the strenght of the evidence (in the data) against the Null hypothesis. In this case, the cummulative tail area is 0.094, which is larger than the initially set [[AP_Statistics_Curriculum_2007_Hypothesis_Basics  Type I]] error <math>\alpha = 0.05</math> and we can not reject the nul hypothesis.  
+  <center>[[Image:SOCR_EBook_Dinov_Hypothesis_020508_Fig3.jpg600px]]</center>  
+  
+  * You can see use the [http://socr.ucla.edu/htmls/SOCR_Analyses.html SOCR Analyses (OneSample TTest)] to carry out these calculations as shown in the figure below.  
+  <center>[[Image:SOCR_EBook_Dinov_Hypothesis_020508_Fig2.jpg600px]]</center>  
+  
+  * This [[SOCR_EduMaterials_AnalysisActivities_OneT  SOCR One Smaple Ttest Activity]] provides additional handson demonstrations of onesample hypothesis testing.  
+  
+  ===Examples===  
+  
+  ====Cavendish Mean Density of the Earth====  
+  A number of famous early experiments of measuring physical constants have later been shown to be biased. In the 1700's [http://en.wikipedia.org/wiki/Henry_Cavendish Henry Cavendish] measured the [http://www.jstor.org/view/02610523/ap000022/00a00200/0 Mean density of the Earth]. Formulate and test null and research hypotheses about these data regarding the now know exact meandensity value = 5.517. These sample statistics may be helpful  
+  : n = 23, sample mean = 5.483, sample SD = 0.1904  
+  <center>  
+  { class="wikitable" style="textalign:center; width:75%" border="1"  
+    
+   5.36  5.29  5.58  5.65  5.57  5.53  5.62  5.29  5.44  5.34  5.79  5.10  5.27  5.39  5.42  5.47  5.63  5.34  5.46  5.30  5.75  5.68  5.85  
+  }  
+  </center>  
<hr>  <hr>  
  ===  +  
  *  +  ===Hypothesis Testing Summary=== 
+  Important parts of Hypothesis test conclusions:  
+  * Decision (significance or no significance)  
+  * Parameter of interest  
+  * Variable of interest  
+  * Population under study  
+  * (optional but preferred) Pvalue  
+  
+  === Parallels between Hypothesis Testing and Confidence Intervals===  
+  These are different methods for coping with the uncertainty about the true value of a parameter caused by the sampling variation in estimates.  
+  
+  * [[AP_Statistics_Curriculum_2007_Estim_L_Mean  Confidence intervals]]: A fixed level of confidence is chosen. We determine a range of possible values for the parameter that are consistent with the data (at the chosen confidence level).  
+  
+  * Hypothesis (Significance) testing: Only one possible value for the parameter, called the hypothesized value, is tested. We determine the strength of the evidence (confidence) provided by the data against the proposition that the hypothesized value is the true value.  
<hr>  <hr> 
Revision as of 20:52, 6 February 2008
Contents

General AdvancePlacement (AP) Statistics Curriculum  Testing a Claim about a Mean: Small Samples
Testing a Claim about a Mean: Small Samples
The previous section discussed inference on the population mean for large smaples. Now, we show how to do hypothesis testing about the mean for small samplesizes.
Background
 Recall that for a random sample {} of the process, the population mean may be estimated by the sample average, .
 For a given small α (e.g., 0.1, 0.05, 0.025, 0.01, 0.001, etc.), the (1 − α)100% Confidence interval for the mean is constructed by
 and is the critical value for a Tdistribution of df=(sample size  1) at .
Hypothesis Testing about a Mean: Small Samples
 Null Hypothesis: H_{o}:μ = μ_{o} (e.g., 0)
 Alternative Research Hypotheses:
 One sided (unidirectional): H_{1}:μ > μ_{o}, or H_{o}:μ < μ_{o}
 Double sided:
Normal Process with Known Variance
 If the population is Normally distributed and we know the population variance, then the Test statistics is:
 .
(Approximately) Nornal Process with Unknown Variance
 If the population is approximately Normally distributed and we do not know the population variance, then theTest statistics is:
 .
Example
Let's use again the smallsample example of the number of sentences per advertisement, where we measure of readability for magazine advertisements. A random sample of the number of sentences found in 10 magazine advertisements is listed below. Suppose we want to test at α = 0.01 a null hypothesis: H_{o}:μ = 12 against a onesided research alternative hypothesis: H_{1}:μ > 12. Recall that we had the following sample statistics: samplemean=22.1, samplevariance=737.88 and sampleSD=27.16390579 for these data.
16  9  14  11  17  12  99  18  13  12 
As the population variance is not given, we have to use the Tstatistics:
 .
 p − value = P(T_{(df = 9)} > T_{o} = ....) = .... for this (onesided) test. Therefore, we can not reject the null hypothesis at α = 0.01! The left and right white areas at the tails of the T(df=9) distribution depict graphically the probability of interest, which represents the strenght of the evidence (in the data) against the Null hypothesis. In this case, the cummulative tail area is 0.094, which is larger than the initially set Type I error α = 0.05 and we can not reject the nul hypothesis.
 You can see use the SOCR Analyses (OneSample TTest) to carry out these calculations as shown in the figure below.
 This SOCR One Smaple Ttest Activity provides additional handson demonstrations of onesample hypothesis testing.
Examples
Cavendish Mean Density of the Earth
A number of famous early experiments of measuring physical constants have later been shown to be biased. In the 1700's Henry Cavendish measured the Mean density of the Earth. Formulate and test null and research hypotheses about these data regarding the now know exact meandensity value = 5.517. These sample statistics may be helpful
 n = 23, sample mean = 5.483, sample SD = 0.1904
5.36  5.29  5.58  5.65  5.57  5.53  5.62  5.29  5.44  5.34  5.79  5.10  5.27  5.39  5.42  5.47  5.63  5.34  5.46  5.30  5.75  5.68  5.85 
Hypothesis Testing Summary
Important parts of Hypothesis test conclusions:
 Decision (significance or no significance)
 Parameter of interest
 Variable of interest
 Population under study
 (optional but preferred) Pvalue
Parallels between Hypothesis Testing and Confidence Intervals
These are different methods for coping with the uncertainty about the true value of a parameter caused by the sampling variation in estimates.
 Confidence intervals: A fixed level of confidence is chosen. We determine a range of possible values for the parameter that are consistent with the data (at the chosen confidence level).
 Hypothesis (Significance) testing: Only one possible value for the parameter, called the hypothesized value, is tested. We determine the strength of the evidence (confidence) provided by the data against the proposition that the hypothesized value is the true value.
 SOCR Home page: http://www.socr.ucla.edu
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