Difference between revisions of "AP Statistics Curriculum 2007"
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===[[AP_Statistics_Curriculum_2007_Estim_Var |Estimating a Population Variance]]===
===[[AP_Statistics_Curriculum_2007_Estim_Var |Estimating a Population Variance]]===
==Chapter VIII: Hypothesis Testing==
==Chapter VIII: Hypothesis Testing==
Revision as of 14:53, 4 February 2008
This is a General Advanced-Placement (AP) Statistics Curriculum E-Book
- 1 Preface
- 2 Chapter I: Introduction to Statistics
- 3 Chapter II: Describing, Exploring, and Comparing Data
- 4 Chapter III: Probability
- 5 Chapter IV: Probability Distributions
- 6 Chapter V: Normal Probability Distribution
- 7 Chapter VI: Relations Between Distributions
- 8 Chapter VII: Point and Interval Estimates
- 9 Chapter VIII: Hypothesis Testing
- 10 Chapter IX: Inferences from Two Samples
- 11 Chapter X: Correlation and Regression
- 12 Chapter XI: Non-Parametric Inference
- 12.1 Differences of Means of Two Paired Samples
- 12.2 Differences of Means of Two Independent Samples
- 12.3 Differences of Medians of Two Paired Samples
- 12.4 Differences of Medians of Two Independent Samples
- 12.5 Differences of Proportions of Two Independent Samples
- 12.6 Differences of Means of Several Independent Samples
- 12.7 Differences of Variances of Two Independent Samples
- 13 Chapter XII: Multinomial Experiments and Contingency Tables
- 14 Chapter XIII: Statistical Process Control
- 15 Chapter XIV: Survival/Failure Analysis
- 16 Chapter XV: Multivariate Statistical Analyses
- 17 Chapter XVI: Time Series Analysis
This is an Internet-based E-Book for advanced-placement (AP) statistics educational curriculum. The E-Book is initially developed by the UCLA Statistics Online Computational Resource (SOCR), however, any statistics instructor, researcher or educator is encouraged to contribute to this effort and improve the content of these learning materials.
Follow the instructions in this page to expand, revise or improve the materials in this E-Book.
Chapter I: Introduction to Statistics
The Nature of Data & Variation
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. How many natural processes or phenomena in real life can we describe that have an exact mathematical closed-form description and are completely deterministic? How do we model the rest of the processes that are unpredictable and have random characteristics?
Uses and Abuses of Statistics
Statistics is the science of variation, randomness and chance. As such, statistics is different from other sciences, where the processes being studied obey exact deterministic mathematical laws. Statistics provides quantitative inference represented as long-time probability values, confidence or prediction intervals, odds, chances, etc., which may ultimately be subjected to varying interpretations. The phrase Uses and Abuses of Statistics refers to the notion that in some cases statistical results may be used as evidence to seemingly opposite theses. However, most of the time, common principles of logic allow us to disambiguate the obtained statistical inference.
Design of Experiments
Design of experiments is the blueprint for planning a study or experiment, performing the data collection protocol and controlling the study parameters for accuracy and consistency. Data, or information, is typically collected in regard to a specific process or phenomenon being studied to investigate the effects of some controlled variables (independent variables or predictors) on other observed measurements (responses or dependent variables). Both types of variables are associated with specific observational units (living beings, components, objects, materials, etc.)
Statistics with Tools (Calculators and Computers)
All methods for data analysis, understanding or visualization are based on models that often have compact analytical representations (e.g., formulas, symbolic equations, etc.) Models are used to study processes theoretically. Empirical validations of the utility of models are achieved by plugging in data and actually testing the models. This validation step may be done manually, by computing the model prediction or model inference from recorded measurements. This however is possible by hand only for small number of observations (<10). In practice, we write (or use existent) algorithms and computer programs that automate these calculations for better efficiency, accuracy and consistency in applying models to larger datasets.
Chapter II: Describing, Exploring, and Comparing Data
Types of Data
There are two important concepts in any data analysis - population and sample. Each of these may generate data of two major types - quantitative or qualitative measurements.
Summarizing data with Frequency Tables
There are two important ways to describe a data set (sample from a population) - Graphs or Tables.
Pictures of Data
There are many different ways to display and graphically visualize data. These graphical techniques facilitate the understanding of the dataset and enable the selection of an appropriate statistical methodology for the analysis of the data.
Measures of Central Tendency
There are three main features of populations (or sample data) that are always critical in understanding and interpreting their distributions - Center, Spread and Shape. The main measures of centrality are mean, median and mode(s).
Measures of Variation
There are many measures of (population or sample) spread, e.g., the range, the variance, the standard deviation, mean absolute deviation, etc. These are used to assess the dispersion or variation in the population.
Measures of Shape
The shape of a distribution can usually be determined by just looking at a histogram of a (representative) sample from that population frequency plots, dot plots or stem and leaf displays may be helpful.
Variables can be summarized using statistics - functions of data samples.
Graphs & Exploratory Data Analysis
Graphical visualization and interrogation of data are critical components of any reliable method for statistical modeling, analysis and interpretation of data.
Chapter III: Probability
Probability is important in many studies and disciplines because measurements, observations and findings are often influenced by variation. In addition, probability theory provides the theoretical groundwork for statistical inference.
Some fundamental concepts of probability theory include random events, sampling, types of probabilities, event manipulations and axioms of probability.
Rules for Computing Probabilities
There are many important rule for computing probabilities of composite events. These include conditional probability, statistical independence, multiplication and addition rules, the law of total probability and the Bayesian rule.
Probabilities Through Simulations
Many experimental setting require probability computations of complex events. Such calculations may be carried out exactly, using theoretical models, or approximately, using estimation or simulations.
There are many useful counting principles (including permutations and combinations) to compute the number of ways that certain arrangements of objects can be formed. This allows counting-based estimation of probabilities of complex events.
Chapter IV: Probability Distributions
There are two basic types of processes that we observe in nature - discrete and continuous. We begine by discussing several important discrete random processes, their distributions, expectations, variances and applications. In the next chapter, we will discuss their continuous counterparts.
To simplify the calculations of probabilities, we will define the concept of a random variable which will allows ut to study uniformly various processes, using the same mathamatical and computational techniques.
Expectation (Mean) and Variance
The expectation and the variance for any discrete random variable or process are important measures of centrality and dispersion.
Bernoulli & Binomial Experiments
The Bernoulli and Binomial processes provide the simplest models for discrete random experiments.
Geometric, Hypergeometric & Negative Binomial
The Geometric, Hypergeometric and Negative Binomial distributions provide computational models for calculating probabilities for a large number of experiment and random variables. This section presents the theoretical foundations and the applications of each of these discrete distributions.
The Poisson distribution models many different discrete processes where the probability of the observed phenomenon is constant in time or space. Poisson distribution may be used as approximation to Binomial.
Chapter V: Normal Probability Distribution
The normal distribution is perhaps the most important model for studying quantitative phenomena in the natural and behavioral sciences - this is due to the Central Limit Theorem. Many numerical measurements (e.g., weight, time, etc.) can be well approximated by the normal distribution.
The Standard Normal Distribution
The standard Normal distribution is the simplest version (zero-mean, unit-standard-deviation) of the (general) Normal dsitribtuion. Yet, it is perheps the most frequently used version because many tables and computational resources are explicitely available for cclculating probabilities.
Nonstandard Normal Distribution: Finding Probabilities
In practice, the mechanisms underlying natural phenomena may be unknown, yet the use of the normal model can be theoretically justified in many situations to compute critical and probability values for various processes.
Nonstandard Normal Distribution: Finding Scores (critical values)
In addition to being able to compute probability (p) values, we often need to estimate the critical values of the Normal distribution for a given p-value.
Chapter VI: Relations Between Distributions
In this chapter we explore the relations between different distributions. This knowledge will help us in two ways: First, some inter-distribution relations will enable us to compute difficult probabilities using reasonable approximations; Second, it would help us identify appropriate probability models, graphical and statistical analysis tools for data interpretation.
The Central Limit Theorem
The exploration of the relation between different distributions begines with the study of the sampling distribution of the sample average. This will demonstrate the universally important role of normal distribution.
Law of Large Numbers
Suppose the relative frequency of occurrence of one event whose probability to be observed at each experiment is p. If we repeat the same experiment over and over, the ratio of the observed frequency of that event to the total number of repetitions converges towards p as the number of experiments increases. Why is that and why is this important?
Normal Distribution as Approximation to Binomial Distribution
Normal distribution provides a valuable approximation to Binomial when the sample sizes are large and the probability of success and failure are not close to zero.
Poisson Approximation to Binomial Distribution
Poisson provides an approximation to Binomial distribution when the sample sizes are large and the probability of success or failure is close to zero.
Binomial Approximation to HyperGeometric
Binomial distribution is much simpler to compute, compared to Hypergeometric, and can be used as an approximation when the poo\pulation sizes are large (relative to the sample size) and the probability of success are not close to zero.
Normal Approximation to Poisson
The Poisson can be approximated fairly well by Normal distribution when λ is large.
Chapter VII: Point and Interval Estimates
Estimation of population parameters is critical in many applications. Estimation is most frequently carried in terms of point-estimates or interval (range) estimates for population papameters that are of interest.
Estimating a Population Mean: Large Samples
This section discusses how to find point and interval estimates when the sample-sizes are large.
Estimating a Population Mean: Small Samples
Next, we discuss point and interval estimates when the sample-sizes are small. Naturally, the point estimates are less precise and the interval estimates produce wider intervals, compared to the case of large-samples.
Student's T distribution
The Student's t-distribution arises in the problem of estimating the mean of a normally distributed population when the sample size is small and the population variance is unknown.
Estimating a Population Proportion
Normal distribution is appropriate model for proportions, when the sample size is large enough. In this section we demonstrate how to obtain point and interval estimates for population proportion.
Estimating a Population Variance
In many processes and experiments, controlling the amount of variance is of critical importance. Thus the ability to assess variation, using point and interval estimates, facilitates our ability to make inference, revise manufacturing protocols, improve clinical trials, etc.
Chapter VIII: Hypothesis Testing
Fundamentals of Hypothesis Testing
Testing a Claim about a Mean: Large Samples
Testing a Claim about a Mean: Small Samples
Testing a Claim about a Proportion
Testing a Claim about a Standard Deviation or Variance
Chapter IX: Inferences from Two Samples
Inferences about Two Means: Dependent Samples
Inferences about Two Means: Independent and Large Samples
Comparing Two Variances
Inferences about Two Means: Independent and Small Samples
Inferences about Two Proportions
Chapter X: Correlation and Regression
Variation and Prediction Intervals
Chapter XI: Non-Parametric Inference
Differences of Means of Two Paired Samples
Differences of Means of Two Independent Samples
Differences of Medians of Two Paired Samples
Differences of Medians of Two Independent Samples
Differences of Proportions of Two Independent Samples
Differences of Means of Several Independent Samples
Differences of Variances of Two Independent Samples
Chapter XII: Multinomial Experiments and Contingency Tables
Multinomial Experiments: Goodness-of-Fit
Contingency Tables: Independence and Homogeneity
Chapter XIII: Statistical Process Control
Control Charts for Variation and Mean
Control Charts for Attributes
Chapter XIV: Survival/Failure Analysis
Chapter XV: Multivariate Statistical Analyses
Multivariate Analysis of Variance
Multiple Linear Regression
Multivariate Analysis of Covariance
Chapter XVI: Time Series Analysis
- SOCR Home page: http://www.socr.ucla.edu
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