AP Statistics Curriculum 2007 Hypothesis L Mean
From Socr
(→ Background) 
m (→Hypothesis Testing About a Mean: Large Samples) 

(24 intermediate revisions not shown)  
Line 1:  Line 1:  
  ==[[AP_Statistics_Curriculum_2007  General AdvancePlacement (AP) Statistics Curriculum]]  Testing a Claim  +  ==[[AP_Statistics_Curriculum_2007  General AdvancePlacement (AP) Statistics Curriculum]]  Testing a Claim About a Mean: Large Samples== 
  We already saw [[AP_Statistics_Curriculum_2007_Estim_L_Mean  how to construct point and interval estimates for the population mean in the large sample case]]. Now, we show how to do hypothesis testing  +  We already saw [[AP_Statistics_Curriculum_2007_Estim_L_Mean  how to construct point and interval estimates for the population mean in the large sample case]]. Now, we show how to do hypothesis testing of the mean for large samplesizes. 
===[[AP_Statistics_Curriculum_2007_Estim_L_Mean  Background]]===  ===[[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  +  * 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  * 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  
Line 13:  Line 13:  
: and <math>z_{\alpha\over 2}</math> is the [[AP_Statistics_Curriculum_2007_Normal_Critical  critical value]] for a [[AP_Statistics_Curriculum_2007_Normal_Std Standard Normal]] distribution at <math>{\alpha\over 2}</math>.  : and <math>z_{\alpha\over 2}</math> is the [[AP_Statistics_Curriculum_2007_Normal_Critical  critical value]] for a [[AP_Statistics_Curriculum_2007_Normal_Std Standard Normal]] distribution at <math>{\alpha\over 2}</math>.  
  === Hypothesis Testing  +  === Hypothesis Testing About a Mean: Large Samples=== 
  * Null Hypothesis: <math>H_o: \mu=\mu_o</math> (e.g., 0)  +  * Null Hypothesis: <math>H_o: \mu=\mu_o</math> (e.g., <math>\mu_o=0</math>) 
* Alternative Research Hypotheses:  * Alternative Research Hypotheses:  
  ** One sided (unidirectional): <math>H_1: \mu >\mu_o</math>, or <math>  +  ** One sided (unidirectional): <math>H_1: \mu >\mu_o</math>, or <math>H_1: \mu<\mu_o</math> 
  ** Double sided:  +  ** Double sided: \(H_1: \mu \not= \mu_o\) 
====Known Variance====  ====Known Variance====  
  * [http://en.wikipedia.org/wiki/Hypothesis_testing#Common_test_statistics Test  +  * [http://en.wikipedia.org/wiki/Hypothesis_testing#Common_test_statistics Test Statistics]: 
  : <math>Z_o = {\overline{x}  \mu_o \over \sigma} \sim N(0,1)</math>.  +  : <math>Z_o = {\overline{x}  \mu_o \over {\sigma \over \sqrt{n}}} \sim N(0,1)</math>. 
====Unknown Variance====  ====Unknown Variance====  
  * [http://en.wikipedia.org/wiki/Hypothesis_testing#Common_test_statistics Test  +  * [http://en.wikipedia.org/wiki/Hypothesis_testing#Common_test_statistics Test Statistics]: 
  : <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  +  : <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 T_{(df=n1)}</math>. 
===Example===  ===Example===  
  Let's [[AP_Statistics_Curriculum_2007_Estim_L_Mean  revisit the ''number of sentences per advertisement'' example]], where we measure  +  Let's [[AP_Statistics_Curriculum_2007_Estim_L_Mean#Example  revisit the ''number of sentences per advertisement'' example]], where we measure the readability for magazine advertisements. A random sample of the number of sentences found in 30 magazine advertisements is listed below. Suppose: 
+  : We want to test at <math>\alpha=0.05</math>  
+  : Null hypothesis: <math>H_o: \mu=20</math>  
+  : Against a doublesided research alternative hypothesis: <math>H_1: \mu \not= 20</math>.  
+  
<center>  <center>  
{ class="wikitable" style="textalign:center; width:75%" border="1"  { class="wikitable" style="textalign:center; width:75%" border="1"  
Line 40:  Line 44:  
: <math>s=\hat{\sigma}=16.54</math>  : <math>s=\hat{\sigma}=16.54</math>  
  As the population variance is not given, we have to use the [[AP_Statistics_Curriculum_2007_StudentsT T  +  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=29)</math> 
  : <math>T_o = {\overline{x}  \mu_o \over SE(\overline{x})} = {14.77  20 \over {{1\over \sqrt{30}} \sqrt{\sum_{i=1}^{30}{(x_i14.77)^2\over 29}}})}  +  : <math>T_o = {\overline{x}  \mu_o \over SE(\overline{x})} = {14.77  20 \over {{1\over \sqrt{30}} \sqrt{\sum_{i=1}^{30}{(x_i14.77)^2\over 29}}})}=1.733</math>. 
+  : <math>P(T_{(df=29)} < T_o=1.733)=0.047</math>, thus  
+  : the <math>pvalue=2\times 0.047= 0.094</math> for this (doublesided) test.  
  ==  +  Therefore, we '''can not reject''' the null hypothesis at <math>\alpha=0.05</math>! The left and right white areas at the tails of the T(df=29) distribution depict graphically the probability of interest, which represents the strength of the evidence (in the data) against the Null hypothesis. In this case, the cumulative 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> so we can not reject the null hypothesis. 
  +  <center>[[Image:SOCR_EBook_Dinov_Hypothesis_020508_Fig3.jpg600px]]</center>  
  *  +  
  <center>[[Image:  +  * You can 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 Sample 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 has 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/stable/pdfplus/107617.pdf Mean density of the Earth]. Formulate and test null and research hypotheses about these data regarding the now known exact meandensity value = 5.517. These sample statistics may be helpful (you can also find the [[AP_Statistics_Curriculum_2007_Hypothesis_L_Mean_Tbl1_trtransposed table here]]):  
+  : 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>  
+  
+  ====US Federal Budget Deficit====  
+  Use the [[SOCR_Data_US_BudgetsDeficits_1849_2016  US Federal Budget Deficit data (18492016)]] to formulate and test several Null hypotheses on whether the US Federal Budget Deficit is trivial (<math>\mu_o=0</math>) in different time frames (e.g., 18492000 or 19002016). Start with some [[AP_Statistics_Curriculum_2007_EDA_Plots exploratory data analyses]] to plot the data as shown in the [[SOCR_Data_US_BudgetsDeficits_1849_2016data page]]. Then you can use the [http://www.socr.ucla.edu/htmls/ana/OneSampleTTest_Analysis.html SOCR One Sample TTest] and the [http://www.socr.ucla.edu/htmls/ana/ConfidenceInterval_Analysis.html SOCR Confidence Interval] applets. What are your conclusions?  
<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.  
+  
+  ===[[EBook_Problems_Hypothesis_L_MeanProblems]]===  
<hr>  <hr> 
Current revision as of 15:48, 22 May 2013
Contents

General AdvancePlacement (AP) Statistics Curriculum  Testing a Claim About a Mean: Large Samples
We already saw how to construct point and interval estimates for the population mean in the large sample case. Now, we show how to do hypothesis testing of the mean for large 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
 where the margin of error E is defined as
 and is the critical value for a Standard Normal distribution at .
Hypothesis Testing About a Mean: Large Samples
 Null Hypothesis: H_{o}:μ = μ_{o} (e.g., μ_{o} = 0)
 Alternative Research Hypotheses:
 One sided (unidirectional): H_{1}:μ > μ_{o}, or H_{1}:μ < μ_{o}
 Double sided: \(H_1: \mu \not= \mu_o\)
Known Variance
 .
Unknown Variance
 .
Example
Let's revisit the number of sentences per advertisement example, where we measure the readability for magazine advertisements. A random sample of the number of sentences found in 30 magazine advertisements is listed below. Suppose:
 We want to test at α = 0.05
 Null hypothesis: H_{o}:μ = 20
 Against a doublesided research alternative hypothesis: .
16  9  14  11  17  12  99  18  13  12  5  9  17  6  11  17  18  20  6  14  7  11  12  5  18  6  4  13  11  12 
We had the following 2 sample statistics computed earlier
As the population variance is not given, we have to use the TStatistics:
 .
 P(T_{(df = 29)} < T_{o} = − 1.733) = 0.047, thus
 the for this (doublesided) test.
Therefore, we can not reject the null hypothesis at α = 0.05! The left and right white areas at the tails of the T(df=29) distribution depict graphically the probability of interest, which represents the strength of the evidence (in the data) against the Null hypothesis. In this case, the cumulative tail area is 0.094, which is larger than the initially set Type I error α = 0.05 so we can not reject the null hypothesis.
 You can use the SOCR Analyses (OneSample TTest) to carry out these calculations as shown in the figure below.
 This SOCR One Sample 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 has 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 known exact meandensity value = 5.517. These sample statistics may be helpful (you can also find the transposed table here):
 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 
US Federal Budget Deficit
Use the US Federal Budget Deficit data (18492016) to formulate and test several Null hypotheses on whether the US Federal Budget Deficit is trivial (μ_{o} = 0) in different time frames (e.g., 18492000 or 19002016). Start with some exploratory data analyses to plot the data as shown in the data page. Then you can use the SOCR One Sample TTest and the SOCR Confidence Interval applets. What are your conclusions?
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.
Problems
 SOCR Home page: http://www.socr.ucla.edu
Translate this page: