EBook Problems

Probability and Statistics EBook Practice Problems

The problems provided below may be useful for practicing the concepts, methods and analysis protocols, and for self-evaluation of learning of the materials presented in the EBook.

I. Introduction to Statistics

The Nature of Data and Variation

Although natural phenomena in real life are unpredictable, the designs of experiments are 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 visualizing 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 inputting data and executing tests of the models. This validation step may be done manually, by computing the model prediction or model inference from recorded measurements. This process may be possible by hand, but only for small numbers 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.

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 looking at a histogram of a (representative) sample from that population; Frequency Plots, Dot Plots or Stem and Leaf Displays may be helpful.

Statistics

Variables can be summarized using statistics - functions of data samples.

Graphs and Exploratory Data Analysis

Graphical visualization and interrogation of data are critical components of any reliable method for statistical modeling, analysis and interpretation of data.

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.

Fundamentals

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 rules 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.

Counting

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.

IV. Probability Distributions

There are two basic types of processes that we observe in nature - Discrete and Continuous. We begin by discussing several important discrete random processes, emphasizing the different distributions, expectations, variances and applications. In the next chapter, we will discuss their continuous counterparts. The complete list of all SOCR Distributions is available here.

Random Variables

To simplify the calculations of probabilities, we will define the concept of a random variable which will allow us to study uniformly various processes with the same mathematical 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. This section also presents the definitions of some common population- or sample-based moments.

Bernoulli and Binomial Experiments

The Bernoulli and Binomial processes provide the simplest models for discrete random experiments.

Multinomial Experiments

Multinomial processes extend the Binomial experiments for the situation of multiple possible outcomes.

Geometric, Hypergeometric and 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.

Poisson Distribution

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 an approximation to the Binomial distribution.

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 Distribution. Yet, it is perhaps the most frequently used version because many tables and computational resources are explicitly available for calculating 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.

Multivariate Normal Distribution

The multivariate normal distribution (also known as multivariate Gaussian distribution) is a generalization of the univariate (one-dimensional) normal distribution to higher dimensions (2D, 3D, etc.) The multivariate normal distribution is useful in studies of correlated real-valued random variables.

VI. Relations Between Distributions

In this chapter, we will explore the relations between different distributions. This knowledge will help us to compute difficult probabilities using reasonable approximations and identify appropriate probability models, graphical and statistical analysis tools for data interpretation. The complete list of all SOCR Distributions is available here and the SOCR Distributome applet provides an interactive graphical interface for exploring the relations between different distributions.

The Central Limit Theorem

The exploration of the relation between different distributions begins 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 successes and failures 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 successes or failures 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 population sizes are large (relative to the sample size) and the probability of successes is not close to zero.

Normal Approximation to Poisson

The Poisson can be approximated fairly well by Normal Distribution when λ is large.

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 parameters that are of interest.

Method of Moments and Maximum Likelihood Estimation

There are many ways to obtain point (value) estimates of various population parameters of interest, using observed data from the specific process we study. The method of moments and the maximum likelihood estimation are among the most popular ones frequently used in practice.

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.

VIII. Hypothesis Testing

Hypothesis Testing is a statistical technique for decision making regarding populations or processes based on experimental data. It quantitatively answers the possibility that chance alone might be responsible for the observed discrepancy between a theoretical model and the empirical observations.

Fundamentals of Hypothesis Testing

In this section, we define the core terminology necessary to discuss Hypothesis Testing (Null and Alternative Hypotheses, Type I and II errors, Sensitivity, Specificity, Statistical Power, etc.)

Testing a Claim about a Mean: Large Samples

As we already saw how to construct point and interval estimates for the population mean in the large sample case, we now show how to do hypothesis testing in the same situation.

Testing a Claim about a Mean: Small Samples

We continue with the discussion on inference for the population mean for small samples.

Testing a Claim about a Proportion

When the sample size is large, the sampling distribution of the sample proportion \(\hat{p}\) is approximately Normal, by CLT. This helps us formulate hypothesis testing protocols and compute the appropriate statistics and p-values to assess significance.

Testing a Claim about a Standard Deviation or Variance

The significance testing for the variation or the standard deviation of a process, a natural phenomenon or an experiment is of paramount importance in many fields. This chapter provides the details for formulating testable hypotheses, computation, and inference on assessing variation.

IX. Inferences from Two Samples

In this chapter, we continue our pursuit and study of significance testing in the case of having two populations. This expands the possible applications of one-sample hypothesis testing we saw in the previous chapter.

Inferences About Two Means: Dependent Samples

We need to clearly identify whether samples we compare are Dependent or Independent in all study designs. In this section, we discuss one specific dependent-samples case - Paired Samples.

Inferences About Two Means: Independent Samples

Independent Samples designs refer to experiments or observations where all measurements are individually independent from each other within their groups and the groups are independent. In this section, we discuss inference based on independent samples.

Comparing Two Variances

In this section, we compare variances (or standard deviations) of two populations using randomly sampled data.

Inferences about Two Proportions

This section presents the significance testing and inference on equality of proportions from two independent populations.

X. Correlation and regression

Many scientific applications involve the analysis of relationships between two or more variables involved in a process of interest. We begin with the simplest of all situations where Bivariate Data (X and Y) are measured for a process and we are interested on determining the association, relation or an appropriate model for these observations (e.g., fitting a straight line to the pairs of (X,Y) data).

Correlation

The Correlation between X and Y represents the first bivariate model of association which may be used to make predictions.

Regression

We are now ready to discuss the modeling of linear relations between two variables using Regression Analysis. This section demonstrates this methodology for the SOCR California Earthquake dataset.

Variation and Prediction Intervals

In this section, we discuss point and interval estimates about the slope of linear models.

Multiple Regression

Now, we are interested in determining linear regressions and multilinear models of the relationships between one dependent variable Y and many independent variables \(X_i\).

XI. Analysis of Variance (ANOVA)

One-Way ANOVA

We now expand our inference methods to study and compare k independent samples. In this case, we will be decomposing the entire variation in the data into independent components.

Two-Way ANOVA

Now we focus on decomposing the variance of a dataset into (independent/orthogonal) components when we have two (grouping) factors. This procedure called Two-Way Analysis of Variance.

XII. Non-Parametric Inference

To be valid, many statistical methods impose (parametric) requirements about the format, parameters and distributions of the data to be analyzed. For instance, the Independent T-Test requires the distributions of the two samples to be Normal, whereas Non-Parametric (distribution-free) statistical methods are often useful in practice, and are less-powerful.

Differences of Medians (Centers) of Two Paired Samples

The Sign Test and the Wilcoxon Signed Rank Test are the simplest non-parametric tests which are also alternatives to the One-Sample and Paired T-Test. These tests are applicable for paired designs where the data is not required to be normally distributed.

Differences of Medians (Centers) of Two Independent Samples

The Wilcoxon-Mann-Whitney (WMW) Test (also known as Mann-Whitney U Test, Mann-Whitney-Wilcoxon Test, or Wilcoxon rank-sum Test) is a non-parametric test for assessing whether two samples come from the same distribution.

Differences of Proportions of Two Samples

Depending upon whether the samples are dependent or independent, we use different statistical tests.

Differences of Means of Several Independent Samples

We now extend the multi-sample inference which we discussed in the ANOVA section, to the situation where the ANOVA assumptions are invalid.

Differences of Variances of Independent Samples (Variance Homogeneity)

There are several tests for variance equality in k samples. These tests are commonly known as tests for Homogeneity of Variances.

XIII. Multinomial Experiments and Contingency Tables

Multinomial Experiments: Goodness-of-Fit

The Chi-Square Test is used to test if a data sample comes from a population with specific characteristics.

Contingency Tables: Independence and Homogeneity

The Chi-Square Test may also be used to test for independence (or association) between two variables.

References

• SOCR
• EBook
• UCLA Statistics Moodle

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