AP Statistics Curriculum 2007 Infer 2Means Indep

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Revision as of 04:55, 10 February 2008

General Advance-Placement (AP) Statistics Curriculum - Inferences about Two Means: Independent Samples

In the previous section we discussed the inference on two paired random samples. Now, we show how to do inference on two independent samples.

Indepenent Samples Designs

Independent samples designs refer to design of experiments or observations where all measurements are individually independent from each other within their groups and the groups are independent. The groups may be drawn from different populations with different distribution characteristics.

Background

Recall that for a random sample { $X_1, X_2, X_3, \cdots , X_n$ } of the process, the population mean may be estimated by the sample average, $\overline{X_n}={1\over n}\sum_{i=1}^n{X_i}$ .

The standard error of $\overline{x}$ is given by ${{1\over \sqrt{n}} \sqrt{\sum_{i=1}^n{(x_i-\overline{x})^2\over n-1}}}$ .

Analysis Protocol for Independent Designs

To study independent samples we would like to examine the differences between two group means. Suppose { $X_1^1, X_2^1, X_3^1, \cdots , X_n^1$ } and { $Y_1^2, Y_2^2, Y_3^2, \cdots , Y_n^2$ } represent the two independent samples. Then we want to study the differences of the two group means relative to the internal sample variations. If the two samples were drawn from populations that had different centers, then we would expect that the two sample averages will be distinct.

Large Samples

Significance Testing: We have a standard null-hypothesis $H o :μ X - μ Y = μ o$ (e.g., $μ o = 0$ ). Then the test statistics is:

$Z_o = {\overline{x}-\overline{y}-\mu_o \over SE(\overline{x}-\overline{y})} \sim N(0,1)$ .

$z_o= {\overline{x}-\overline{y} \over \sqrt{{1\over \sqrt{n_1}} \sqrt{\sum_{i=1}^{n_1}{(x_i-\overline{x})^2\over n_1-1}} + {1\over \sqrt{n_2}} \sqrt{\sum_{i=1}^{n_2}{(y_i-\overline{y})^2\over n_2-1}}}}$

Confidence Intervals: $(1 - α)100%$ confidence interval for $μ 1 - μ 2$ will be

$CI(\alpha): \overline{x}-\overline{y} \pm z_{\alpha\over 2} SE(\overline{x}-\overline{y})= \overline{x}-\overline{y} \pm z_{\alpha\over 2} \sqrt{{1\over \sqrt{n_1}} \sqrt{\sum_{i=1}^{n_1}{(x_i-\overline{x})^2\over n_1-1}} + {1\over \sqrt{n_2}} \sqrt{\sum_{i=1}^{n_2}{(y_i-\overline{y})^2\over n_2-1}}}$ . Note that the $SE(\overline{x} -\overline{x})=\sqrt{SE(\overline{x})+SE(\overline{y})}$ , as the samples are independent. Also, $z_{\alpha\over 2}$ is the critical value for a Standard Normal distribution at ${\alpha\over 2}$ .

Small Samples

References

SOCR Home page: http://www.socr.ucla.edu

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@@ Line 1: / Line 1: @@
-==[[AP_Statistics_Curriculum_2007 | General Advance-Placement (AP) Statistics Curriculum]] - Inferences about Two Means: Independent and Large Samples==
+==[[AP_Statistics_Curriculum_2007 | General Advance-Placement (AP) Statistics Curriculum]] - Inferences about Two Means: Independent Samples==
-=== Inferences about Two Means: Independent and Large Samples===
+In the [[AP_Statistics_Curriculum_2007_Infer_2Means_Dep | previous section we discussed the inference on two paired random samples]]. Now, we show how to do inference on two independent samples.
-Example on how to attach images to Wiki documents in included below (this needs to be replaced by an appropriate figure for this section)!
-<center>[[Image:AP_Statistics_Curriculum_2007_IntroVar_Dinov_061407_Fig1.png|500px]]</center>
-===Approach===
+===Indepenent Samples Designs===
-Models & strategies for solving the problem, data understanding & inference.
+Independent samples designs refer to design of experiments or observations where all measurements are individually independent from each other within their groups and the groups are independent. The groups may be drawn from different populations with different distribution characteristics.
-* TBD
+===[[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>.
-===Model Validation===
+* The standard error of <math>\overline{x}</math> is given by <math>{{1\over \sqrt{n}} \sqrt{\sum_{i=1}^n{(x_i-\overline{x})^2\over n-1}}}</math>.
-Checking/affirming underlying assumptions.
-* TBD
+===Analysis Protocol for Independent Designs===
+To study independent samples we would like to examine the differences between two group means. Suppose {<math>X_1^1, X_2^1, X_3^1, \cdots , X_n^1</math>} and {<math>Y_1^2, Y_2^2, Y_3^2, \cdots , Y_n^2</math>} represent the two independent samples. Then we want to study the differences of the two group means relative to the internal sample variations. If the two samples were drawn from populations that had different [[AP_Statistics_Curriculum_2007_EDA_Center |centers]], then we would expect that the two sample averages will be distinct.
-===Computational Resources: Internet-based SOCR Tools===
+====Large Samples====
-* TBD
+*Significance Testing: We have a standard null-hypothesis <math>H_o: \mu_X -\mu_Y = \mu_o</math> (e.g., <math>\mu_o=0</math>). Then the test statistics is:
+: <math>Z_o = {\overline{x}-\overline{y}-\mu_o \over SE(\overline{x}-\overline{y})} \sim N(0,1)</math>.
+: <math>z_o= {\overline{x}-\overline{y} \over  \sqrt{{1\over \sqrt{n_1}} \sqrt{\sum_{i=1}^{n_1}{(x_i-\overline{x})^2\over n_1-1}} + {1\over \sqrt{n_2}} \sqrt{\sum_{i=1}^{n_2}{(y_i-\overline{y})^2\over n_2-1}}}}</math>
-===Examples===
+* Confidence Intervals: <math>(1-\alpha)100%</math> confidence interval for <math>\mu_1-\mu_2</math> will be
-Computer simulations and real observed data.
+: <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 \sqrt{n_1}} \sqrt{\sum_{i=1}^{n_1}{(x_i-\overline{x})^2\over n_1-1}} + {1\over \sqrt{n_2}} \sqrt{\sum_{i=1}^{n_2}{(y_i-\overline{y})^2\over n_2-1}}}</math>. Note that the <math>SE(\overline{x} -\overline{x})=\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>.
+====Small Samples====
-* TBD
-===Hands-on activities===
-Step-by-step practice problems.
-* TBD
 <hr>
 ===References===
-* TBD
 <hr>

AP Statistics Curriculum 2007 Infer 2Means Indep

From Socr

Revision as of 04:55, 10 February 2008

Contents

General Advance-Placement (AP) Statistics Curriculum - Inferences about Two Means: Independent Samples

Indepenent Samples Designs

Background

Analysis Protocol for Independent Designs

Large Samples

Small Samples

References

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