AP Statistics Curriculum 2007 Infer 2Means Indep - Revision history

Dinov: /* Detailed specifications for independent-sample design studies */

2014-11-16T16:20:02Z

‎Detailed specifications for independent-sample design studies

Dinov at 21:48, 8 October 2014

2014-10-08T21:48:02Z

Dinov at 21:47, 8 October 2014

2014-10-08T21:47:34Z

IvoDinov: /* Detailed specifications for independent-sample design studies */

2011-03-02T20:43:10Z

‎Detailed specifications for independent-sample design studies

IvoDinov: added a new section on Detailed specifications for independent-sample design studies

2011-03-02T20:42:21Z

added a new section on Detailed specifications for independent-sample design studies

IvoDinov: /* Large Samples */ typo

2011-03-01T00:51:26Z

‎Large Samples: typo

IvoDinov: /* Small Samples */ typo

2011-02-28T18:21:50Z

‎Small Samples: typo

Jenny at 20:47, 28 June 2010

2010-06-28T20:47:56Z

Jenny at 17:28, 28 June 2010

2010-06-28T17:28:57Z

IvoDinov: /* Inference */ typo

2010-03-05T16:44:43Z

‎Inference: typo

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 : ''S'' is the standard error of the difference between two means
 : df = degrees of freedom
-: <math>n_1</math> and <math>n_2</math> are the number of observations in each group.
+: $n_1$ and $n_2$ are the number of observations in each group.
 <center>
 {| class="wikitable" style="text-align:center; width:55%" border="1"

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 <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_Indep}}
+{{translate|pageName=http://wiki.socr.umich.edu/index.php?title=AP_Statistics_Curriculum_2007_Infer_2Means_Indep}}

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 |}
 </center>
 ===[[EBook_Problems_Infer_2Means_Indep|Problems]]===

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 <math>\ S_{X_1X_2} = \sqrt{\frac{1}{2}(S_{X_1}^2+S_{X_2}^2)}</math> is the [http://en.wikipedia.org/wiki/Pooled_standard_deviation pooled standard deviation] and df =2<math>n_1</math>−2
 ||
-<math>T = \frac{\bar {X}_1 - \bar{X}_2}{S_{X_1X_2} \cdot \sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}</math>, where <math> S_{X_1X_2} = \sqrt{\frac{(n_1-1)S_{X_1}^2+(n_2-1)S_{X_2}^2}{n_1+n_2-2}}</math>, <math>S_{X_1X_2}</math> is an estimator of the common standard deviation of the two samples, df=''n''<sub>1</sub>&nbsp;+&nbsp;''n''<sub>2</sub>&nbsp;&minus;&nbsp;2
+<math>T = \frac{\bar {X}_1 - \bar{X}_2}{S_{X_1X_2} \cdot \sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}</math>, where <math> S_{X_1X_2} = \sqrt{\frac{(n_1-1)S_{X_1}^2+(n_2-1)S_{X_2}^2}{n_1+n_2-2}}</math> is an estimator of the common standard deviation of the two samples and df=''n''<sub>1</sub>&nbsp;+&nbsp;''n''<sub>2</sub>&nbsp;&minus;&nbsp;2
 |-
 | Non-equivalent (different) || colspan="2"|

← Older revision		Revision as of 20:42, 2 March 2011
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	<hr>		<hr>
		+
		+	===Detailed specifications for independent-sample design studies===
		+	There are several different situations that arise in studies involving independent samples inference. These cases are separated by whether the sample sizes are equal and the sample variances (approximately) '''equivalent''' (i.e., <math>\frac{\sigma_1}{2}\le \sigma_2 \le 2\sigma_1</math>). The table below illustrates the differences in the statistical inference in each of these situations. This table uses the following notation:
		+	: Indexes 1 and 2 denote group one and group two, respectively
		+	: ''T'' is the test statistics
		+	: ''S'' is the standard error of the difference between two means
		+	: df = degrees of freedom
		+	: <math>n_1</math> and <math>n_2</math> are the number of observations in each group.
		+	<center>
		+	{\| class="wikitable" style="text-align:center; width:55%" border="1"
		+	\|-
		+	! colspan="2" rowspan="2"\|Design \|\| colspan="2"\|Sample Size
		+	\|-
		+	! Equal \|\| Unequal (different)
		+	\|-
		+	\| rowspan="2"\|Population Variance \|\| Equivalent \|\|
		+	<math>T = \frac{\bar {X}_1 - \bar{X}_2}{S_{X_1X_2} \cdot \sqrt{\frac{2}{n}}}</math>
		+	where
		+	<math>\ S_{X_1X_2} = \sqrt{\frac{1}{2}(S_{X_1}^2+S_{X_2}^2)}</math> is the [http://en.wikipedia.org/wiki/Pooled_standard_deviation pooled standard deviation] and df =2<math>n_1</math>−2
		+	\|\|
		+	<math>T = \frac{\bar {X}_1 - \bar{X}_2}{S_{X_1X_2} \cdot \sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}</math>, where <math> S_{X_1X_2} = \sqrt{\frac{(n_1-1)S_{X_1}^2+(n_2-1)S_{X_2}^2}{n_1+n_2-2}}</math>, <math>S_{X_1X_2}</math> is an estimator of the common standard deviation of the two samples, df=''n''<sub>1</sub> + ''n''<sub>2</sub> − 2
		+	\|-
		+	\| Non-equivalent (different) \|\| colspan="2"\|
		+	<math>T = {\overline{X}_1 - \overline{X}_2 \over s_{\overline{X}_1 - \overline{X}_2}}</math>, where <math>s_{\overline{X}_1 - \overline{X}_2} = \sqrt{{s_1^2 \over n_1} + {s_2^2 \over n_2}}
		+	</math> (not a pooled variance), <math> \mathrm{df} = \frac{(s_1^2/n_1 + s_2^2/n_2)^2}{(s_1^2/n_1)^2/(n_1-1) + (s_2^2/n_2)^2/(n_2-1)}</math>, which is called the [http://en.wikipedia.org/wiki/Welch%E2%80%93Satterthwaite_equation Welch–Satterthwaite equation]
		+	\|}
		+	</center>

	===[[EBook_Problems_Infer_2Means_Indep\|Problems]]===		===[[EBook_Problems_Infer_2Means_Indep\|Problems]]===

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 * '''Confidence Intervals''': <math>(1-\alpha)100%</math> confidence interval for <math>\mu_1-\mu_2</math> will be
-: <math>CI(\alpha): \overline{x}-\overline{y} \pm z_{\alpha\over 2} SE(\overline{x}-\overline{y})= \overline{x}-\overline{y} \pm z_{\alpha\over 2}  \sqrt{{1\over {n_1}} {\sum_{i=1}^{n_1}{(x_i-\overline{x})^2\over n_1-1}} + {1\over {n_2}} {\sum_{i=1}^{n_2}{(y_i-\overline{y})^2\over n_2-1}}}</math>. Note that the <math>SE(\overline{x} -\overline{y})=\sqrt{SE(\overline{x})+SE(\overline{y})}</math>, as the samples are independent. Also, <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>.
+: <math>CI(\alpha): \overline{x}-\overline{y} \pm z_{\alpha\over 2} SE(\overline{x}-\overline{y})= \overline{x}-\overline{y} \pm z_{\alpha\over 2}  \sqrt{{1\over {n_1}} {\sum_{i=1}^{n_1}{(x_i-\overline{x})^2\over n_1-1}} + {1\over {n_2}} {\sum_{i=1}^{n_2}{(y_i-\overline{y})^2\over n_2-1}}}</math>. Note that the <math>SE(\overline{x} -\overline{y})=\sqrt{SE^2(\overline{x})+SE^2(\overline{y})}</math>, as the samples are independent. Also, <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>.
 ====Small Samples====

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 * '''Confidence Intervals''': <math>(1-\alpha)100%</math> confidence interval for <math>\mu_1-\mu_2</math> will be
-: <math>CI(\alpha): \overline{x}-\overline{y} \pm t_{df, {\alpha\over 2}} SE(\overline{x}-\overline{y})= \overline{x}-\overline{y} \pm t_{df, {\alpha\over 2}}  \sqrt{{1\over {n_1}} {\sum_{i=1}^{n_1}{(x_i-\overline{x})^2\over n_1-1}} + {1\over {n_2}} {\sum_{i=1}^{n_2}{(y_i-\overline{y})^2\over n_2-1}}}</math>. Note that the <math>SE(\overline{x} -\overline{y})=\sqrt{SE(\overline{x})+SE(\overline{y})}</math>, as the samples are independent.
+: <math>CI(\alpha): \overline{x}-\overline{y} \pm t_{df, {\alpha\over 2}} SE(\overline{x}-\overline{y})= \overline{x}-\overline{y} \pm t_{df, {\alpha\over 2}}  \sqrt{{1\over {n_1}} {\sum_{i=1}^{n_1}{(x_i-\overline{x})^2\over n_1-1}} + {1\over {n_2}} {\sum_{i=1}^{n_2}{(y_i-\overline{y})^2\over n_2-1}}}</math>. Note that the <math>SE(\overline{x} -\overline{y})=\sqrt{SE^2(\overline{x})+SE^2(\overline{y})}</math>, as the samples are independent.
 : The ''degrees of freedom'' is: <math>df={( SE^2(\overline{x})+SE^2(\overline{y}))^2 \over {SE^4(\overline{x}) \over n_1-1} + {SE^4(\overline{y}) \over n_2-1} } \approx n_1+n_2-2.</math> Always round down the degrees of freedom to the next smaller integer. Also, <math>t_{df, {\alpha\over 2}}</math> is the [[AP_Statistics_Curriculum_2007_Normal_Critical | critical value]] for a [[AP_Statistics_Curriculum_2007_StudentsT |Student's T]] distribution at <math>{\alpha\over 2}</math>.

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 Both the confidence intervals and the hypothesis testing methods in the independent-sample design require Normality of both samples. If the sample sizes are large (say >50), Normality is not as critical, as the [[AP_Statistics_Curriculum_2007_Limits_CLT | CLT]] implies the sampling distributions of the means are approximately Normal. If these parametric assumptions are invalid we must use a [[AP_Statistics_Curriculum_2007_NonParam_2MedianIndep | non-parametric (distribution free test)]], even if the latter is less powerful.
-The plots below indicate that Normal assumptions are not unreasonable for these data, and hence we may be justified in using the two independent sample T-test in this case.
+The plots below indicate that Normal assumptions are not unreasonable for these data, and hence we may be justified in using the two independent sample T-tests in this case.
 * [[AP_Statistics_Curriculum_2007_Normal_Prob#Assessing_Normality | QQ-Normal plot]] of the first sample:

@@ Line 33: / Line 33: @@
 ===Example===
-Nine observations of surface soil [http://en.wikipedia.org/wiki/PH pH] were made at two different locations.  Does the data suggest that the true mean soil [http://en.wikipedia.org/wiki/PH pH] values differ for the two locations?  Formulate testable hypothesis and make inference about the effect of the treatment at <math>\alpha=0.05</math>. Check any necessary assumptions for the validity of your test.
+Nine observations of surface soil [http://en.wikipedia.org/wiki/PH pH] were made at two different locations.  Does the data suggest that the true mean soil [http://en.wikipedia.org/wiki/PH pH] values differs for the two locations?  Formulate testable hypothesis and make inference about the effect of the treatment at <math>\alpha=0.05</math>. Check any necessary assumptions for the validity of your test.
 ====Data in row format====
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 : <math>t_o = {7.9322-7.460-0 \over  0.081}=5.829</math>.
-: <math>p-value=P(T_{(df=11)}>T_o=5.829)=0.00003</math> for this (two-sided) test. Therefore, we '''can reject''' the null hypothesis at <math>\alpha=0.05</math>! The left white area at the tails of the T(df=11) distribution depict graphically the probability of interest, which represents the strength of the evidence (in the data) against the Null hypothesis. In this case, this area is 0.00003, which is much smaller than the initially set [[AP_Statistics_Curriculum_2007_Hypothesis_Basics | Type I]] error <math>\alpha = 0.05</math> and we reject the null hypothesis.
+: <math>p-value=P(T_{(df=11)}>T_o=5.829)=0.00003</math> for this (two-sided) test. Therefore, we '''can reject''' the null hypothesis at <math>\alpha=0.05</math>! The left white area at the tails of the T(df=11) distribution depicts graphically the probability of interest, which represents the strength of the evidence (in the data) against the Null hypothesis. In this case, this area is 0.00003, which is much smaller than the initially set [[AP_Statistics_Curriculum_2007_Hypothesis_Basics | Type I]] error <math>\alpha = 0.05</math> and we reject the null hypothesis.
 <center>[[Image:SOCR_EBook_Dinov_Infer_2Means_Indep_020908_Fig4.jpg|600px]]</center>