Understanding Statistics

Preface, 1 Variables, Population, and Samples, 1.1 Statistics is concerned with describing and explaining how things vary., 1.2 Inferences about populations based on sample information are subject to error., 1.3 Other forms of random samples include stratified and cluster sampling., 2 Basic Ideals of Statistical Inference, 2.1 There are two basic tasks of statistical inference., 2.2 To evaluate degree of certainty, first consider all possible samples., 2.3 To find degree of certainty, find the sampling distribution of the statistic., 2.4 A handy way of representing a distribution is to draw a histogram., 2.5 The larger the sample, the less spread out is the sampling distribution., 2.6 So what?, 3 Describing Data for a Single Variable, 3.1 There are many ways of summarizing a set of data, 3.2 Often we are interested in the entire population distribution., 3.3 Some distributions can be approximated by mathematical functions., 3.4 A histogram and frequency distribution are very useful descriptive tools., 3.5 In a histogram, area represents relative frequency., 3.6 The mean, median, and mode are useful for describing central tendency., 3.7 The range and interquartile range are measures of variability., 3.8 The standard deviation is the most commonly used measure of variability., 3.9 The standard deviation is computed differently for samples versus populations., 4 Some Distributions Used in Statistical Inference, 4.1 Knowing the sampling distribution of a statistic allows us to draw inferences from sample data., 4.2 The standard normal distribution is used to find areas under any normal curve., 4.3 The binomial distribution is used for variables that count the number of yeses., 4.4 To calculate binomial probabilities, we need to find the probability of each possible outcome., 4.5 We next find the number of relevant outcomes and multiply it by the probability of each relevant outcome., 4.6 Binomial probabilities can be computed from a general formula and are also available in tables., 4.7 A binomial distribution with large n and moderate p is approximately normal., 5 Interval Estimation, 5.1 The standard error of a statistic is the standard deviation of its sampling distribution., 5.2 The CLT can be applied to draw inferences about the population mean., 5.3 If we know σ we can use z scores to form a confidence interval for the mean., 6 Hypothesis Testing, 6.1 The CLT can also be applied to perform hypothesis tests concerning averages., 6.2 Rejecting a true null hypothesis is called a type I error., 6.3 P values come in two flavors: one-tailed and two-tailed., 6.4 Failing to reject a false null hypothesis is called a type II error., 6.5 Do not confuse statistical significance with practical significance., 7 Drawing Inference About a Population Mean, 7.1 The normal distribution can be used to test hypotheses about μ when σ is known., 7.2 If we must estimate the standard error from the data, we lose some certainty., 7.3 The family of t distributions is used to draw inferences when a is known., 7.4 The t distribution with n – 1 df is used to draw inferences about a mean., 8 Further Topics in Inference About Single Populations, 8.1 Proportions can be treated as a special kind of mean., 8.2 The sign test requires few assumptions., 8.3 The Wilcoxon signed rank test works with ranks instead of the original units., 9 Drawing Inferences About Group Differences, 9.1 Many scientific hypotheses can be stated in terms of group differences., 9.2 For two paired samples, the problem reduces to the one-sample case., 9.3 The two-sample t test is used for problems involving independent samples., 9.4 The Wilcoxon rank sum test is another method for comparing two groups., 10 One-Way Analysis of Variance, 10.1 Analysis of variance is a general method used in analyzing group differences., 10.2 One-way ANOVA can be used to test for differences among more than two groups., 10.3 The Kruskal-Wallis test is a nonparametric analog of one-way ANOVA., 11 Describing Relationships Between Two Variables, 11.1 A scatterplot shows the shape of a relationship between two variables., 11.2 Easy-r is a simple measure of the strength of a monotonic relationship., 11.3 The correlation coefficient tells how well a straight line describes the plot., 11.4 A form of t test can be used to draw inferences about correlation coefficients., 11.5 Spearman’s rho and Kendall’s tau are nonparametric measures of correlation., 11.6 A nonzero r does not imply causality, nor does a zero r imply no correlation., 12 Introduction to Regression Methods, 12.1 Often we wish to form a model to describe how one variable responds to others., 12.2 Stochastic models may include one independent variable, or several., 12.3 Regression methods are used to estimate the parameters of a stochastic model., 12.4 The coefficient of determination tells how well the model fits the data., 12.5 The F test is used to test the model as a whole for statistical significance., 12.6 Individual regression coefficients can be tested using a form of t test., 13 Further Topics in Regression, 13.1 Stepwise methods can be used to select independent variables for a model., 13.2 Beware of the misuses of stepwise regression!, 13.3 Plotting the residuals can help identify violations of assumptions., 13.4 Violations of assumptions can be treated by a variety of methods., 14 Further Topics in Analysis of Variance, 14.1 Analysis of variance is used to compare group means., 14.2 Interaction means that the effect of one factor depends on another factor., 14.3 The F test is used to test for significant main effects and interactions., 15 Analyzing Categorical Data, 15.1 In many situations our data are not measurements, but simply counts., 15.2 Expected counts are what we would observe on the average if the null is true., 15.3 The chi-square test evaluates the difference between observed and expected frequencies., References, Appendix A: The Coronary Care Data Set, Appendix B: The Electricity Consumer Questionnaire Data Set, Appendix C: Selected Advanced Topics, C.1 Checking for normality and homogeneity, C.2 Dealing with lack of homogeneity and normality, C.3 Comparing individual groups in ANOVA, C.4 Fisher’s z transformation for testing correlation, C.5 Derivation of least-squares formulas for simple regression coefficients, C.6 Formula for standard error of b1 in simple regression, C.7 Formulas for mean squares and df in regression, Appendix D: Statistical Tables, D .l Standard normal distribution, D.2 Binomial probabilities, D.3 Student’s t distribution, D.4 Critical values for the Wilcoxon signed rank test, D.5 Critical values for the Wilcoxon rank sum test, D.6 The F distribution, D.7 The chi-square distribution, D.8 Critical values of r, the correlation coefficient, Index, Statpal Manual, Statpal Manual Index