A Concise Course in Advanced Level Statistics with worked examples Export Edition

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Description

Contents

Reviews

Language

English

ISBN

9781382018012

Front Cover

IFC

Imprint

Contents

Preface

1 Representation and summary of data

Discrete data

Continuous data

Stem and leaf diagrams (stemplots)

Ways of grouping data

Histograms

Frequency polygons

Frequency curves

Circular diagrams or pie charts

The mean

Variability of data

The standard deviation, s, and the variance, s²

Combining sets of data

Scaling sets of data

Using a method of coding to find the mean and standard deviation

Cumulative frequency

Cumulative percentage frequency diagrams

Median, quartiles and percentiles

Skewness

The normal distribution

Box and whisker diagrams (box plots)

Summary

2 Regression and correlation

Scatter diagrams

Regression function

Linear correlation and regression lines

The product-moment correlation coefficient, r

Spearman’s coefficient of rank correlation, r

Summary

3 Probability

Experimental probability

Probability when outcomes are equally likely

Subjective probabilities

Probability notation and probability laws

Illustrating two or more events using Venn diagrams

Probability rule for combined events

Exclusive (or mutually exclusive) events

Exhaustive events

Conditional probability

Independent events

Probability trees

Bayes’ Theorem

Some useful methods

Arrangements

Permutations of r objects from n objects

Combinations of r objects from n objects

Summary

4 Probability distributions I – discrete variables

Probability distributions

Expectation of X, E(X)

Expectation of any function of X, E(g(X))

Variance, Var(X) or V(X)

The Cumulative distribution function, F(x)

Two independent random variables

Distribution of X₁ + X₂ +⋯+ X

Comparing the distributions of X₁ + X₂ and 2X

Summary

5 Special discrete probability distributions

The uniform distribution

The geometric distribution

Expectation and variance of the geometric distribution

The binomial distribution

Expectation and variance of the binomial distribution

The Poisson distribution

Using the Poisson distribution as an approximation to the binomial distribution

The sum of independent Poisson variables

Summary

6 Probability distributions II – continuous variables

Continuous random variables

Probability density function (p.d.f.)

Expectation of X, E(X)

Expectation of any function of X

Variance of X, Var(X)

The mode

Cumulative distribution function F(x)

Obtaining the p.d.f., f(x), from the cumulative distribution function

The continous uniform (or rectangular) distribution

Expectation and variance of the uniform distribution

The cumulative distribution function, F(x), for a uniform distribution

Summary

7 The normal distribution

Finding probabilities

The standard normal variable, Z

Using standard normal tables

Using standard normal tables for any normal variable, X

Using the standard normal tables in reverse to find z when Φ(z) is known

Using the tables in reverse for any normal variable, X

Value of μ or σ or both

The normal approximation to the binomial distribution

Continuity corrections

Deciding when to use a normal approximation and when to use a Poisson approximation for a binomial distribution

The normal approximation to the Poisson distribution

Summary

8 Linear combinations of normal variables

The sum of independent normal variables

The difference of independent normal variables

Multiples of independent normal variables

Summary

9 Sampling and estimation

Sampling

Surveys

Sampling methods

Simulating random samples from given distributions

Sample statistics

The distribution of the sample mean

Central limit theorem

The distribution of the sample proportion, p

Unbiased estimates of population parameters

Point estimates

Interval estimates

The t-distribution

Confidence intervals for the population proportion, p

Summary

10 Hypothesis tests: discrete distributions

Hypothesis test for a binomial proportion, p (small sample size)

Procedure for carrying out a hypothesis test

One-tailed and two-tailed tests

Summary of stages of a hypothesis (significance) test

Type I and Type II errors

Significance test for a Poisson mean λ

Summary of stages of a significance test

Summary of Type I and Type II errors

11 Hypothesis testing (z-tests and t-tests)

Hypothesis testing

One-tailed and two-tailed tests

Critical z-values

Summary of critical values and rejection criteria

Stages in the hypothesis test

Hypothesis test 1: testing μ (the mean of a population)

Type I and Type II errors

Hypothesis test 2: testing a binomial proportion p when n is large

Hypothesis test 3: testing μ₁ − μ₂, the difference between means of two normal populations

Summary

12 The χ² significance test

The χ² significance test

Performing a χ² goodness-of-fit test

Summary of the procedure for performing a χ² goodness-of-fit test

Test 1 – goodness-of-fit test for a uniform distribution

Test 2 – goodness-of-fit test for a distribution in a given ratio

Test 3 – goodness-of-fit test for a binomial distribution

Test 4 – goodness-of-fit test for a Poisson distribution

Test 5 – goodness-of-fit test for a normal distribution

Summary of the number of degrees of freedom for a goodness-of-fit test

The χ² significance test for independence

Summary

13 Significance tests for correlation coefficients

Significance tests for correlation coefficients

Test for the product-moment correlation coefficient, r

Spearman’s coefficient of rank correlation, r

Summary

ICT statistics supplement

Appendix

Cumulative binomial probabilities

Cumulative Poisson probabilities

The standard normal distribution function

Critical values for the normal distribution

Critical values for the t-distribution

Critical values for the χ² distribution

Critical values for correlation coefficients

Random numbers

Answers

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