AP Statistics Curriculum 2007 IntroVar

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==This is an Outline of a General Advance-Placement (AP) Statistics Curriculum==
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[[AP_Statistics_Curriculum_2007 | General Advance-Placement (AP) Statistics Curriculum]] - Introduction to Statistics
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===Outline===
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==The Nature of Data & Variation==
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Each topic discussed in the SOCR AP Curricumum should contain the following subsections:
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No matter how controlled are the environment, the protocol or the design, virtually any repeated measurement, observation, experiment, trial, study or survey is bounded to generate data that varies because of intrinsic (internal to the system) or extrinsic (due to the ambient environment) effects.
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* '''Motivation/Problem''': A real data set and fundamental challenge.
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* '''Approach''': Models & strategies for solving the problem, data understanding & inference.
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* '''Model Validation''': Checking/affirming underlying assumptions.
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* '''Computational Resources''': Internet-based SOCR Tools (including offline resources, e.g., tables).
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* '''Examples''': computer simulations and real observed data.
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* '''Hands-on activities''': Step-by-step practice problems.
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===Introduction to Statistics===
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For example, the UCLA's [[AP_Statistics_Curriculum_2007_IntroVar#References | study of Alzheimer’s disease*]] analyzed the data of 31 Mild Cognitive Impairment (MCI) and 34 probable Alzheimer’s disease (AD) patients. The investigators made every attempt to control as many variables as possible. Yet, the demographic information they collected from the outcomes of the subjects contained unavoidable variation. The same study found variation in the MMSE cognitive scores even in the same subject. The table below shows the demographic characteristics for the subjects and patients included in this study, where the following notation is used M: male; F: female; W: white; AA: African American; A: Asian:
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====[[AP_Statistics_Curriculum_2007_IntroVar | The Nature of Data & Variation]]====
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No mater how controlled the environment, the protocol or the design, virtually any repeated measurement, observation, experiment, trial, study or survey is bound to generate data that varies because of intrinsic (internal to the system) or extrinsic (due to the ambient environment) effects.
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====Uses and Abuses of Statistics ====
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<center>
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====Design of Experiments ====
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{| class="wikitable" style="text-align:center; width:75%" border="1"
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====Statistics with Calculators and Computers====
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| '''Variable''' || '''Alzheimer’s disease''' || '''MCI''' || '''Test statistics''' || '''Test score''' || '''P-value'''
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===Describing, Exploring, and Comparing Data===
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|-
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====Summarizing data with Frequency Tables ====
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| '''Age (years)''' || 76.2 (8.3) range 52–89 || 73.7 (7.4) range 57–84 || Student’s T  || <math>t_o = 1.284</math> || ''p=0.21''
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====Pictures of Data ====
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|-
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====Measures of Central Tendency ====
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| '''Gender (M:F)''' || 15:19 || 15:16 || Proportion || <math>z_o = -0.345</math>  || ''p=0.733''
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====Measures of Variation ====
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|-
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====Measures of Position ====
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| '''Education (years)''' || 14.0 (2.1) range 12–19 || 16.23 (2.7) range 12–20 || Wilcoxon rank sum || <math>w_o = 773.0</math> || ''p<0.001''
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====Exploratory Data Analysis ====
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|-
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| '''Race (W:AA:A)''' || 29:1:4 || 26:2:3 || <math>\chi_{(df=2)}^2</math> || <math>\chi_{(df=2)}^2=1.18</math> || 0.55
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===Probability===
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|-
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====Fundamentals====
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| '''MMSE''' || 20.9 (6.3) range 4–29 || 28.2 (1.6) range 23–30 || Wilcoxon rank-sum || <math>w_o= 977.5</math>  || ''p<0.001''
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====Addition Rule ====
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|}
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====Multiplication Rule ====
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</center>
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====Probabilities through Simulations ====
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====Counting ====
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===Probability Distributions===
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====Random Variables ====
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====Bernoulli & Binomial Experiments ====
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====Geometric, HyperGeometric & Negative Binomial====
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====Mean, Variance, and Standard Deviation for the Binomial Distribution ====
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====Poisson Distribution====
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===Normal Probability Distributions===
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====The Standard Normal Distribution ====
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====Nonstandard Normal Distribution: Finding Probabilities ====
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====Nonstandard Normal Distributions: Finding Scores ====
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===Relations Between Distributions===
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==Approach==
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====The Central Limit Theorem ====
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Models and strategies for solving  problems and understanding data and inferences.
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====Law of Large Numbers====
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====Normal Distribution as Approximation to Binomial Distribution ====
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* Once we accept that all natural phenomena are inherently variant and there are no completely deterministic processes, we need to look for models and techniques that allow us to study such acquired data in the presence of variation, uncertainty and chance.
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====Poisson Approximation to Binomial Distribution ====
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* '''Statistics''' is the data science that investigates natural processes and allows us to quantify variation to make population inferences based on limited observations.
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====Binomial Approximation to HyperGeometric====
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====Normal Approximation to Poisson====
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==Model Validation==
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Checking/affirming underlying assumptions.
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* Each model or technique for data exploration, analysis and understanding relies on a set of assumptions, which always need to be validated before the model or analysis tool is employed to study real data (observations or measurements that are perceived or detected by the investigator).
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* Such prior model conjectures or presumptions could take the form of mathematical constraints about the properties of the underlying process, restrictions on the study design or demands on the data acquisition protocol.
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* Common assumptions include (statistical) independence of the measurements, specific limitations on the shape of the observed distribution, restrictions on the parameters of the processes being studied, etc.
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==Computational Resources: Internet-based SOCR Tools==
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* The [[SOCR]] resource contains a variety of educational materials, demonstration applets and learning resources that illustrate data generation, experimentation, exploratory and statistical data analysis.
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* [[SOCR_EduMaterials_Activities_RNG | (Numeric Pseudo-Random) Data Generation]]
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* [[SOCR_EduMaterials_ExperimentsActivities | Interactive SOCR Experimentation]] with computer generated models of natural phenomena
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**  [[SOCR_EduMaterials_Activities_DieCoinExperiment | Bivariate Die-Coin Experiment]]
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* [[SOCR_EduMaterials_Activities_Histogram_Graphs | Exploratory Data Analysis]]
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* [[SOCR_EduMaterials_AnalysisActivities_ANOVA_1 | Statistical Data Analysis]]
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==Datasets==
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There are [[SOCR_Data | a number of large, natural, useful and demonstrative datasets]] that are provided as part of this statistics [[EBook]]. Many of these data collections are intentionally selected to be large and complex. This choice is driven by the need of emphasizing the symbiosis between driving challenges, statistical concepts, mathematical derivations and the use of technology to solve relevant research problems.
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==Examples==
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Computer simulations and real observed data.
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* For example, [[SOCR_EduMaterials_Activities_Histogram_Graphs | exploratory data analysis using data histograms]]. This SOCR activity illustrates the generation and interpretation of the histogram of quantitative data.
   
   
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===Estimates and Sample Sizes===
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==Hands-on activities==
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====Estimating a Population Mean: Large Samples ====
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Step-by-step practice problems.
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====Estimating a Population Mean: Small Samples ====
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====Estimating a Population Proportion ====
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* [[SOCR_EduMaterials_Activities_Histogram_Graphs | Histograms and Frequency Graphs Activity]]
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====Estimating a Population Variance====
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* [[SOCR_EduMaterials_Activities_CardsCoinsSampling | Bivariate Cards and Coins Meta-Activity]]
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===Hypothesis Testing===
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==[[EBook_Problems_EDA_IntroVar|Problems]]==
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====Fundamentals of Hypothesis Testing ====
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====Testing a Claim about a Mean: Large Samples ====
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<hr>
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====Testing a Claim about a Mean: Small Samples ====
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==References==
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====Testing a Claim about a Proportion ====
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* Apostolova LG, Dinov ID, Dutton RA, Hayashi KM, Toga AW, Cummings JL, Thompson PM. [http://brain.oxfordjournals.org/cgi/reprint/awl274v1.pdf 3D comparison of hippocampal atrophy in amnestic mild cognitive impairment and Alzheimer's disease.] Brain. 2006 Nov; 129(Pt 11):2867-73.
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====Testing a Claim about a Standard Deviation or Variance====
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===Inferences from Two Samples===
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====Inferences about Two Means: Dependent Samples ====
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====Inferences about Two Means: Independent and Large Samples ====
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====Comparing Two Variances ====
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====Inferences about Two Means: Independent and Small Samples====
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====Inferences about Two Proportions ====
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===Correlation and Regression===
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====Correlation ====
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====Regression ====
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====Variation and Prediction Intervals ====
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====Multiple Regression ====
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===Multinomial Experiments and Contingency Tables===
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====Multinomial Experiments: Goodness-of-Fit ====
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====Contingency Tables: Independence and Homogeneity====
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===Statistical Process Control===
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====Control Charts for Variation and Mean ====
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====Control Charts for Attributes====
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<hr>
<hr>

Current revision as of 17:40, 31 March 2013

General Advance-Placement (AP) Statistics Curriculum - Introduction to Statistics

Contents

The Nature of Data & Variation

No matter how controlled are the environment, the protocol or the design, virtually any repeated measurement, observation, experiment, trial, study or survey is bounded to generate data that varies because of intrinsic (internal to the system) or extrinsic (due to the ambient environment) effects.

For example, the UCLA's study of Alzheimer’s disease* analyzed the data of 31 Mild Cognitive Impairment (MCI) and 34 probable Alzheimer’s disease (AD) patients. The investigators made every attempt to control as many variables as possible. Yet, the demographic information they collected from the outcomes of the subjects contained unavoidable variation. The same study found variation in the MMSE cognitive scores even in the same subject. The table below shows the demographic characteristics for the subjects and patients included in this study, where the following notation is used M: male; F: female; W: white; AA: African American; A: Asian:

Variable Alzheimer’s disease MCI Test statistics Test score P-value
Age (years) 76.2 (8.3) range 52–89 73.7 (7.4) range 57–84 Student’s T to = 1.284 p=0.21
Gender (M:F) 15:19 15:16 Proportion zo = − 0.345 p=0.733
Education (years) 14.0 (2.1) range 12–19 16.23 (2.7) range 12–20 Wilcoxon rank sum wo = 773.0 p<0.001
Race (W:AA:A) 29:1:4 26:2:3 \chi_{(df=2)}^2 \chi_{(df=2)}^2=1.18 0.55
MMSE 20.9 (6.3) range 4–29 28.2 (1.6) range 23–30 Wilcoxon rank-sum wo = 977.5 p<0.001

Approach

Models and strategies for solving problems and understanding data and inferences.

  • Once we accept that all natural phenomena are inherently variant and there are no completely deterministic processes, we need to look for models and techniques that allow us to study such acquired data in the presence of variation, uncertainty and chance.
  • Statistics is the data science that investigates natural processes and allows us to quantify variation to make population inferences based on limited observations.

Model Validation

Checking/affirming underlying assumptions.

  • Each model or technique for data exploration, analysis and understanding relies on a set of assumptions, which always need to be validated before the model or analysis tool is employed to study real data (observations or measurements that are perceived or detected by the investigator).
  • Such prior model conjectures or presumptions could take the form of mathematical constraints about the properties of the underlying process, restrictions on the study design or demands on the data acquisition protocol.
  • Common assumptions include (statistical) independence of the measurements, specific limitations on the shape of the observed distribution, restrictions on the parameters of the processes being studied, etc.

Computational Resources: Internet-based SOCR Tools

Datasets

There are a number of large, natural, useful and demonstrative datasets that are provided as part of this statistics EBook. Many of these data collections are intentionally selected to be large and complex. This choice is driven by the need of emphasizing the symbiosis between driving challenges, statistical concepts, mathematical derivations and the use of technology to solve relevant research problems.

Examples

Computer simulations and real observed data.

Hands-on activities

Step-by-step practice problems.

Problems


References




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