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STUDY GUIDE FOR MODERN MATHEMATICAL STATISTICS MAT 446 WITH DAMIEN PITMAN Introduction & Instructions Content Goals. We will work towards an understanding of how to quantify uncertainty and how to make decisions in the face of uncertainty. First, we will lean how to make statements about how likely an event is based on certain assumptions. This forms the basis for the theory of probability. We will then learn to summarize and visualize a collection of such statements that are all based on the same assumptions or else based on observations made in an experiment. This forms the basis for the theory of descriptive statistics. With the basic theories of probability and descriptive statistics we will learn to make inferences and conclusions about unknown parameters of a population. This forms the basis for the theory of statistical inference. General Goals. This course should reinforce much of the mathematics that you have learned thus far in your life. This means that you are expected to use what you have previously learned. This course should also develop your problem solving skills. This means that I will go out of my way to challenge you to consider nontrivial questions. This course should also be applicable to the world beyond the classroom. This means that I will try to relate what we learn form the textbook to ideas that you may not be familiar with from coursework in the math department. This could mean that you have to do some independent learning, which brings me to the most important goal of all. This course should develop your skills as an independent learner. In my opinion, a good college course should make you feel like you took the bull by the horns and wrestled your way to a greater understanding of life. Personal Goals. Well, what are they? 1 STUDY GUIDE FOR MODERN MATHEMATICAL STATISTICS 2 Course Structure : Reading the textbook is part of your homework and this document is your guide for that reading as well as the homework problems that you are to turn in. : A typical class day will consist of two or three distinct segments. For the first several minutes of class, we will have a sort of study hall, during which time you should ask questions individually and work on homework from sections that have already been discussed in class. After this, we will have a discussion and work examples from the section that was assigned reading for the day. Lastly, there could be a five minute quiz that will test your knowledge or understanding of a vocabulary word. The word could be any vocabulary word from the assigned reading for that class day or any previous class day. The discussion and examples will rely on a basic understanding of the assigned reading, which is why I reserve the option to quiz you over the vocabulary at any time. : I recommend making a two-column reference sheet with every vocabulary words in one column and the definition in the other. : Do your best to complete all homework. Keep in mind that the goals for homework are 1) to develop a practical understanding of the concepts from the reading, and 2) to learn and practice the technical skills that will enable you to apply your understanding of these concepts. : While reading or working problems, write out any questions you have for me. : Homework content is given in this document, but the timeline will be given in class. STUDY GUIDE FOR MODERN MATHEMATICAL STATISTICS 3 Chapter 1: Overview and Descriptive Statistics Homework Problems. 2(c), 16, 26, 34, 38, 46, 50, 52, 62 Additional Exercises. 5, 9, 11, 33, 39, 45, 71 1. Populations and Samples Vocabulary. data, population, census, sample, variable, univariate, bivariate, multivariate, descriptive statistics, inferential statistics, concrete population, conceptual population, simple random sample, stratified random sample, sampling frame 2. Pictorial and Tabular Methods in Descriptive Statistics Vocabulary. sample size, stem-and-leaf display, dotplot, frequency, relative frequency, histogram for counting data, classes, class boundary rule, histogram for measurement data: equal class widths, histogram for measurement data: unequal class widths, total area of each rectangle in an histogram with unequal class widths, total area of all rectangles in a histogram with unequal class widths, unimodal or bimodal histograms, symmetric, positively or negatively skewed, qualitative data, multivariate data 3. Measures of Location Vocabulary. sample mean x, population mean µ, sample median x̃, population median µ̃, quartiles (first, second, and third), trimmed mean, sample proportion 4. Measures of Variability Vocabulary. range, deviations from the mean (signed, absolute, and squared), sample variance s2 , sample standard deviation s, population variance σ2 , population standard deviation σ, degrees of freedom, Sxx , lower fourth, upper fourth, fourth spread f s , boxplot, outlier (mild and extreme), boxplot showing outliers, comparative boxplot Results. computational formula for s2 , outliers formulas Chapter 2: Probability Homework Problems. 2, 4, 10, 12, 22, 24, 26, 28, 30, 32, 34, 44, 48, 50, 58, 62, 66, 70, 78, 80 Additional Exercises. 11, 33, 84, 88-90, 104, 105 1. Sample Spaces and Events Vocabulary. experiment, sample space, event (simple and compound), union, intersection, complement, disjoint or mutually exclusive events STUDY GUIDE FOR MODERN MATHEMATICAL STATISTICS 4 2. Axioms, Interpretations, and Properties of Probability Vocabulary. axioms of probability, null set, equally likely events Results. probability of the null set and implication for Axiom 3, sum of a geometric series, complementary probability, sum rule for probabilities with or statements (A ∪ B) 3. Counting Techniques Vocabulary. product rule for k-tuples, permutation Pk,n , factorial m!, combination (nk) (or Ck,n ) Results. the computational formulas for each of the vocabulary words from this section 4. Conditional Probability Vocabulary. conditional probability of A given that B has occurred Results. multiplication rule for probabilities with and statements (A ∩ B), law of total probability, Bayes’ theorem 5. Independence Vocabulary. independent events, dependent events, mutually independent events Chapter 3: Discrete Random Variables and Probability Distributions Homework Problems. 4, 7(a-d), 12, 14, 16, 22, 28, 30, 34, 60, 62, 70, 116, 119 Additional Exercises. 1, 11, 65, 113, 115 1. Random Variables Vocabulary. random variable (rv), Bernoulli rv, discrete rv, continuous rv, 2. Probability Distributions for Discrete Random Variables Vocabulary. probability distribution or probability mass function (pmf), probability histogram, parameter, family of probability distributions, cumulative distribution function (cdf), step function, F ( a−) Reading Exercises. (1) Is it true that the cdf of any discrete rv is a step function? Why? 3. Expected Values of Discrete Random Variables Vocabulary. expected value or mean value of a random variable, expected value of a function, variance of X σX2 = σ2 , standard deviation (SD) of X σx = σ Results. computational formula for σ2 STUDY GUIDE FOR MODERN MATHEMATICAL STATISTICS 5 5. The Binomial Probability Distribution Vocabulary. trials, dichotomous, binomial experiment, without replacement, binomial random variable Results. b( x; n, p) =?, Bin( x; n, p) =? Chapter 4: Continuous Random Variables and Probability Distributions Homework Problems. 2, 4, 12, 18, 20, 26, 40, 42, 44, 48, 54, 62 Additional Exercises. 1, 3, 5, 11, 27, 39, 41, 43 1. Probability Density Functions and Cumulative Distribution Functions Vocabulary. probability density function (pdf) f ( x ), uniform distribution, cumulative distribution function (cdf) F ( x ), (100p)th percentile η ( p) Results. P( a ≤ X ≤ b) = F (b) − F ( a), F 0 ( x ) = f ( x ) 2. Expected Values and Moment Generating Functions Vocabulary. expected or mean value, E[h( X )], variance of X, σX2 = V ( X ) Results. computational formula V ( X ) = E( X 2 ) − [ E( X )]2 3. The Normal Distribution Vocabulary. normal distribution, standard normal distribution, normal random variable, standard normal random variable Z, standard normal cdf Φ(z), z critical values zα , standardized variable Results. standardized normal probabilities and percentiles, binomial approximation by the normal Chapter 5: Joint Probability Distributions Homework Problems. 2, 12, 18, 21, 26 Additional Exercises. 1, 3, 17, 19 1. Jointly Distributed random Variables Vocabulary. joint probability mass function, joint probability table, marginal probability mass function, joint probability density function, marginal probability density function, independent random variables 2. Expected Values, Covariance, and Correlation Vocabulary. covariance, correlation coefficient ρ STUDY GUIDE FOR MODERN MATHEMATICAL STATISTICS 6 Results. E(h( X, Y )) = . . ., Cov( X, Y ) = E( XY ) − µ X · µY , relationship between independence and ρ = 0 Chapter 6: Statistics and Sampling Distributions Homework Problems. 2, 6, 12, 18, 24, 32, 42, 46, 47, 50(a,b) Additional Exercises. 1, 11, 15, 27, 53 1. Statistics and Their Distributions Vocabulary. statistic, sample mean X, sample standard deviation S, sample total T0 , sampling distribution, random sample, independent and identically distributed (i.i.d.), simulation experiment Reading Questions. What is the difference between X and x? Does it make sense to say that statistics with capital letters are random variables and those with lower case letters are values of the relevant random variable? Does it make sense to say that capital letters correspond to unobserved values and lower case letters correspond to observed values? 2. The Distribution of the Sample Mean Results. E( X ) = µ X , V ( X ) = σ2 /n, E( T0 ) = nµ, V ( T0 ) = nσ2 , sampling distribution of a normal rv, Central Limit Theorem (CLT), Law of Large Numbers (LLN) 3. The Mean, Variance, (and MGF) for Several Variables Vocabulary. linear combination (of rv’s) Results. mean and variance of linear combinations (of rv’s), mean and variance of the difference between two rv’s 4. Distributions Based on a Normal Random Sample Vocabulary. chi-squared distribution χ2ν , degrees of freedom ν, t distribution, F distribution √ Results. T = ( X − µ)/(S/ n) Chapter 7: Point Estimation Homework Problems. 4(a-c), 14 Additional Exercises. 1, 11 STUDY GUIDE FOR MODERN MATHEMATICAL STATISTICS 7 1. General Concepts and Criteria Vocabulary. point estimate, parameter, statistic, point estimator, bias, unbiased estimator, principle of unbiased estimation, principle of minimum variance unbiased estimation, MVUE, standard error, estimated standard error, Chapter 8: Statistical Intervals Based on a Single Sample Homework Problems. 2, 6, 12, 20, 30, 34, 44, 46 Additional Exercises. 1, 5, 13, 21, 23, 29, 31, 35, 45, 47 1. Basic Properties of Confidence Intervals Vocabulary. 100(1 − α)% confidence interval for the mean µ, bound on the error of estimation (a.k.a. the error bound for the mean EBM or error bound for the proportion EBP), confidence level, α as error probability 2. Large-Sample Confidence Intervals for a Population Mean and Proportion Vocabulary. meaning of large sample confidence interval and necessary assumptions to use it 3. Intervals Based on a Normal Population Distribution Vocabulary. t distribution, normality assumption for t distribution, properties of t distribution, prediction interval Chapter 9: Tests of Hypotheses Based on a Single Sample Homework. 2, 6, 10, 22, 26, 38, 46, 48 Additional Exercises. 5, 7, 15, 19, 21, 35, 41, 45, 46, 51 1. Hypotheses and Test Procedures Vocabulary. statistical hypothesis, null hypothesis, alternative hypothesis, null value, test statistic, rejection region, type I error, type II error, significance α 2. Tests about a Population Mean Vocabulary. assumption of normality, known σ, large sample test, unknown σ 3. Tests Concerning a Population Proportion Vocabulary. np ≥ 10 assumption, small sample test for a proportion 4. P-Values Vocabulary. P-value, decision rule based on P-value