AP Statistics Curriculum 2007 Infer 2Means Dep - Revision history

WikiSysop at 14:51, 5 March 2014

2014-03-05T14:51:59Z

IvoDinov: /* Paired vs. Independent Testing */

2013-06-17T20:05:11Z

‎Paired vs. Independent Testing

IvoDinov: /* Paired vs. Independent Testing */

2013-06-17T20:04:04Z

‎Paired vs. Independent Testing

IvoDinov: /* Paired vs. Independent Testing */

2013-06-17T20:02:29Z

‎Paired vs. Independent Testing

IvoDinov: /* Paired vs. Independent Testing */

2013-06-17T19:57:55Z

‎Paired vs. Independent Testing

IvoDinov: /* Test Statistics */

2013-06-17T19:55:52Z

‎Test Statistics

IvoDinov: /* Test Statistics */

2012-03-06T05:45:39Z

‎Test Statistics

Jenny: /* Effects of Ignoring the Pairing */

2010-06-28T20:46:33Z

‎Effects of Ignoring the Pairing

Jenny: /* Analysis Protocol for Paired Designs */

2010-06-28T20:44:44Z

‎Analysis Protocol for Paired Designs

Jenny: /* Inferences About Two Means: Dependent Samples */

2010-06-28T20:44:09Z

‎Inferences About Two Means: Dependent Samples

@@ Line 128: / Line 128: @@
 <center>[[Image:SOCR_EBook_Dinov_Infer_2Means_Dep_020908_Fig4.jpg|600px]]</center>
-* You can also use the [http://socr.ucla.edu/htmls/SOCR_Analyses.html SOCR Analyses (One-Sample T-Test)] to carry out these calculations as shown in the figure below.
+* You can also use the [http://socr.umich.edu/html/ana/ SOCR Analyses (One-Sample T-Test)] to carry out these calculations as shown in the figure below.
 <center>[[Image:SOCR_EBook_Dinov_Infer_2Means_Dep_020908_Fig5.jpg|600px]]</center>
@@ Line 164: / Line 164: @@
 <hr>
-* SOCR Home page: http://www.socr.ucla.edu
+* SOCR Home page: http://www.socr.umich.edu
-{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_Infer_2Means_Dep}}
+{{translate|pageName=http://wiki.socr.umich.edu/index.php?title=AP_Statistics_Curriculum_2007_Infer_2Means_Dep}}

@@ Line 154: / Line 154: @@
 : \(T_o = {\overline{x}-\overline{y} - \mu_o \over SE(\overline{x}+\overline{y})} \sim T(df=17)\)
 : \(T_o = {\overline{x}-\overline{y} - \mu_o \over SE(\overline{x}+\overline{y})} = {0.682 -0.637- 0 \over \sqrt{SE^2(\overline{x})+SE^2(\overline{y})}}= \) \({0.682 -0.637\over \sqrt{{0.0742^2\over 10}+ {0.0709^2\over 10}}}={0.682 -0.637\over 0.0325}=1.38\)
-: \(p-value=P(T>1.38)= 0.100449\) and we would have failed to reject the null-hypothesis ('''incorrect!''')
+: \(p-value=P(T>1.38)= 0.100449\) and we would have failed to reject the null-hypothesis ('''[[AP_Statistics_Curriculum_2007_Infer_2Means_Dep#Inference|incorrect!]]''')
 Similarly, had we incorrectly used the [[AP_Statistics_Curriculum_2007_Infer_2Means_Indep |independent design]] and constructed a corresponding Confidence interval, we would obtain an incorrect inference:

@@ Line 152: / Line 152: @@
 ====Paired vs. Independent Testing====
 Suppose we accidentally analyzed the groups independently (using the [[AP_Statistics_Curriculum_2007_Infer_2Means_Indep |independent T-test]]) rather than using this paired test (this would be an incorrect way of analyzing this ''before-after'' data). How would this change our results and findings?
-: <math>T_o = {\overline{x}-\overline{y} - \mu_o \over SE(\overline{x}+\overline{y})} \sim T(df=17)</math>
+: \(T_o = {\overline{x}-\overline{y} - \mu_o \over SE(\overline{x}+\overline{y})} \sim T(df=17)\)
-: <math>T_o = {\overline{x}-\overline{y} - \mu_o \over SE(\overline{x}+\overline{y})} = {0.682 -0.637- 0 \over \sqrt{SE^2(\overline{x})+SE^2(\overline{y})}}= {0.682 -0.637\over \sqrt{{0.0742^2\over 10}+ {0.0709^2\over 10}}}={0.682 -0.637\over 0.0325}=1.38</math>
+: \(T_o = {\overline{x}-\overline{y} - \mu_o \over SE(\overline{x}+\overline{y})} = {0.682 -0.637- 0 \over \sqrt{SE^2(\overline{x})+SE^2(\overline{y})}}= \) \({0.682 -0.637\over \sqrt{{0.0742^2\over 10}+ {0.0709^2\over 10}}}={0.682 -0.637\over 0.0325}=1.38\)
-: <math>p-value=P(T>1.38)= 0.100449</math> and we would have failed to reject the null-hypothesis ('''incorrect!''')
+: \(p-value=P(T>1.38)= 0.100449\) and we would have failed to reject the null-hypothesis ('''incorrect!''')
 Similarly, had we incorrectly used the [[AP_Statistics_Curriculum_2007_Infer_2Means_Indep |independent design]] and constructed a corresponding Confidence interval, we would obtain an incorrect inference:
-: <math>CI:  {\overline{x}-\overline{y} - \mu_o \pm t_{(df=17, \alpha/2)} \times SE(\overline{x}+\overline{y})} = 0.045 \pm 1.740\times 0.0325 = [-0.0116 ; 0.1016]</math>
+: \(CI:  {\overline{x}-\overline{y} - \mu_o \pm t_{(df=17, \alpha/2)} \times SE(\overline{x}+\overline{y})} = \) \(0.045 \pm 1.740\times 0.0325 = [-0.0116 ; 0.1016].\)
 <hr>

@@ Line 153: / Line 153: @@
 Suppose we accidentally analyzed the groups independently (using the [[AP_Statistics_Curriculum_2007_Infer_2Means_Indep |independent T-test]]) rather than using this paired test (this would be an incorrect way of analyzing this ''before-after'' data). How would this change our results and findings?
 : <math>T_o = {\overline{x}-\overline{y} - \mu_o \over SE(\overline{x}+\overline{y})} \sim T(df=17)</math>
-: <math>T_o = {\overline{x}-\overline{y} - \mu_o \over SE(\overline{x}+\overline{y})} = {0.682 -0.687- 0 \over \sqrt{SE^2(\overline{x})+SE^2(\overline{y})}}= {0.682 -0.687\over \sqrt{{0.0742^2\over 10}+ {0.0709^2\over 10}}}={0.682 -0.687\over 0.0325}=1.38</math>
+: <math>T_o = {\overline{x}-\overline{y} - \mu_o \over SE(\overline{x}+\overline{y})} = {0.682 -0.637- 0 \over \sqrt{SE^2(\overline{x})+SE^2(\overline{y})}}= {0.682 -0.637\over \sqrt{{0.0742^2\over 10}+ {0.0709^2\over 10}}}={0.682 -0.637\over 0.0325}=1.38</math>
 : <math>p-value=P(T>1.38)= 0.100449</math> and we would have failed to reject the null-hypothesis ('''incorrect!''')
 Similarly, had we incorrectly used the [[AP_Statistics_Curriculum_2007_Infer_2Means_Indep |independent design]] and constructed a corresponding Confidence interval, we would obtain an incorrect inference:
-: <math>CI:  {\overline{x}-\overline{y} - \mu_o \pm t_{(df=17, \alpha/2)} \times SE(\overline{x}+\overline{y})} = 0.682 -0.687 \pm 1.740\times 0.0325 = [-0.0116 ; 0.1016]</math>
+: <math>CI:  {\overline{x}-\overline{y} - \mu_o \pm t_{(df=17, \alpha/2)} \times SE(\overline{x}+\overline{y})} = 0.045 \pm 1.740\times 0.0325 = [-0.0116 ; 0.1016]</math>
 <hr>

@@ Line 157: / Line 157: @@
 Similarly, had we incorrectly used the [[AP_Statistics_Curriculum_2007_Infer_2Means_Indep |independent design]] and constructed a corresponding Confidence interval, we would obtain an incorrect inference:
-: <math>CI:  {\overline{x}-\overline{y} - \mu_o \pm t_{(df=17, \alpha/2)} \times SE(\overline{x_1}+\overline{x_1})} = 0.682 -0.687 \pm 1.740\times 0.0325 = [-0.0116 ; 0.1016]</math>
+: <math>CI:  {\overline{x}-\overline{y} - \mu_o \pm t_{(df=17, \alpha/2)} \times SE(\overline{x}+\overline{y})} = 0.682 -0.687 \pm 1.740\times 0.0325 = [-0.0116 ; 0.1016]</math>
 <hr>

@@ Line 44: / Line 44: @@
 ====Test Statistics====
 * If the two populations that the {<math>X_i</math>} and {<math>Y_i</math>} samples were drawn from are approximately Normal, then the [http://en.wikipedia.org/wiki/Hypothesis_testing#Common_test_statistics Test Statistics] is:
-: <math>T_o = {\overline{d} - \mu_o \over SE(\overline{d})} = {\overline{x} - \mu_o \over {{1\over \sqrt{n}} \sqrt{\sum_{i=1}^n{(d_i-\overline{d})^2\over n-1}}}} \sim T_{(df=n-1)}</math>.
+: <math>T_o = {\overline{d} - \mu_o \over SE(\overline{d})} = {\overline{d} - \mu_o \over {{1\over \sqrt{n}} \sqrt{\sum_{i=1}^n{(d_i-\overline{d})^2\over n-1}}}} \sim T_{(df=n-1)}</math>.
 ====Effects of Ignoring the Pairing====

@@ Line 47: / Line 47: @@
 ====Effects of Ignoring the Pairing====
-The SE estimate will be '''smaller''' for correctly paired data. If we look at the data within each sample we notice variation from one subject to the next. This information gets incorporated into the SE for the independent t-test via <math>s_1</math> and <math>s_2</math>. The original reason we paired was to try to control for some of this inter-subject variation, which is not of interest in the paired design. Notice that the inter-subject variation has no influence on the SE for the paired test, because only the differences were used in the calculation. The price of pairing is smaller degrees of freedom of the T-test. However, this can be compensated with a smaller SE if we had paired correctly.
+The SE estimate will be '''smaller''' for correctly paired data. If we look  within each sample at the data, we notice variation from one subject to the next. This information gets incorporated into the SE for the independent t-test via <math>s_1</math> and <math>s_2</math>. The original reason we paired was to try to control for some of this inter-subject variation, which is not of interest in the paired design. Notice that the inter-subject variation has no influence on the SE for the paired test, because only the differences were used in the calculation. The price of pairing is smaller degrees of freedom of the T-test. However, this can be compensated with a smaller SE if we had paired correctly.
 '''Pairing''' is used to ''reduce'' bias and ''increase'' precision in our inference. By '''matching/blocking''' we can control variation due to extraneous variables.

@@ Line 24: / Line 24: @@
 Now we can clearly see that the group effect (group differences) is directly represented in the {<math>d_i</math>} sequence. The [[AP_Statistics_Curriculum_2007_Hypothesis_S_Mean#.28Approximately.29_Nornal_Process_with_Unknown_Variance | one-sample T test]] is the proper strategy to analyze the difference sample {<math>d_i</math>}, if the <math>X_i</math> and <math>Y_i</math> samples come from [[AP_Statistics_Curriculum_2007#Chapter_V:_Normal_Probability_Distribution |Normal distributions]].
-Because we are focusing on the differences, we can use the same reasoning as we did in the [[AP_Statistics_Curriculum_2007_Hypothesis_S_Mean#.28Approximately.29_Nornal_Process_with_Unknown_Variance |single sample case]] to calculate the standard error (i.e., the standard deviation of the sampling distribution of <math>\overline{d}</math>) of <math>\overline{d}={1\over n}\sum_{i=1}^n{d_i}</math>.
+Since we are focusing on the differences, we can use the same reasoning as we did in the [[AP_Statistics_Curriculum_2007_Hypothesis_S_Mean#.28Approximately.29_Nornal_Process_with_Unknown_Variance |single sample case]] to calculate the standard error (i.e., the standard deviation of the sampling distribution of <math>\overline{d}</math>) of <math>\overline{d}={1\over n}\sum_{i=1}^n{d_i}</math>.
 Thus, the standard error of <math>\overline{d}</math> is given by <math>{{1\over \sqrt{n}} \sqrt{\sum_{i=1}^n{(d_i-\overline{d})^2\over n-1}}}</math>, where <math>d_i=X_i-Y_i, \forall 1\leq i\leq n</math>.

@@ Line 4: / Line 4: @@
 === Inferences About Two Means: Dependent Samples===
-In all study designs, it is always critical to clearly identify whether samples we compare come from dependent or independent populations. There is a general formulation for the significance testing when the samples are independent. The fact that there may be uncountable many different types of dependencies prevents us from having a similar analysis protocol for ''all'' dependent sample cases. However, in one specific case - paired samples - we have a theory to generalize the significance testing analysis protocol. Two populations (or samples) are ''dependent because of pairing'' (or paired) if they are linked in some way, usually by a direct relationship. For example, measure the weight of subjects before and after a six month diet.
+In all study designs, it is always critical to clearly identify whether samples we compare come from dependent or independent populations. There is a general formulation for the significance testing when the samples are independent. The fact that there may be uncountable many different types of dependencies that prevents us from having a similar analysis protocol for ''all'' dependent sample cases. However, in one specific case - paired samples - we have a theory to generalize the significance testing analysis protocol. Two populations (or samples) are ''dependent because of pairing'' (or paired) if they are linked in some way, usually by a direct relationship. For example, measure the weight of subjects before and after a six month diet.
 ===Paired Designs===