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Bayes factor, 400 Bayesian: hypothesis tests, 399 interval estimation, 397 point estimation, 394 predictive distributions, 401 Bayes s theorem, 17 Beta distribution, 133 Binomial coefficients, 21 Binomial distribution, 118 moment generating function of, 121 normal approximation to, 126, 129 Poisson approximation to, 128 Bivariate normal distribution, 140 Buffon s needle problem, 64 Calculation of probabilities, 12 Cauchy distribution, 132, 133 characteristic function of, 132 moment generating function of, 132 relation to uniform distribution, 133 Central limit theorem, 129 for binomial random variables, 129 proof of, 130 Central tendency, measures of, 94 Change of variables, 51, 63 Characteristic function, 90, 97 of Cauchy distribution, 132 Chebyshev s inequality, 93 Chi-square distribution: moment generating function of, 134 relationship to F and t distributions, 139 relationship to normal distribution, 134 tests involving, 233, 242 Chi-square test of goodness of fit, 242, 246, 252 Coefficient of: contingency, 250 correlation, 91 determination, 301 linear correlation, 289 Combinations, 20 Combinatorial analysis, 17 probability using, 22 Conditional: density, 58 distribution, 58 expectation, 93 moments, 93 probability, 14 variance, 93 Confidence interval estimates for: differences of means, 203 differences of proportions, 202 means in large samples, 200 means in small samples, 202 mean when population variance is unknown, 174 proportions, 202, 207 standard deviation, 208 variances, 204 variance ratios, 205 Conjugate prior distributions, 393 Contingency, coefficient of, 250 Contingency tables, 246 Continuous distribution function, 46 Convolutions, 56 Correlation: coefficient, 91, 140 generalized, 292 linear, 289 multiple, 293 probability interpretation of, 295 product-moment formula for, 290 rank, 293 sampling theory of, 298 table, 305 Counting, 17 Covariance, 91, 184 Cyclic pattern, in runs test, 361 Determination, coefficient of, 301 Discrete distribution function, 45 Discrete random variable, 44 Dispersion, measures of, 96 Distribution: Bayesian predictive, 401 beta, 133 binomial, 118 bivariate normal, 140 Cauchy, 132 chi-square, 134 conditional, 58 conjugate prior, 393 continuous, 46 of differences and sums, 169 discrete, 4 Fisher s F, 138 frequency, 175 gamma, 133 hypergeometric, 131 improper prior, 392 joint, 47 marginal, 48 multinomial, 131 normal, 122 of means, 163 of proportions, 166 of ratios of variances, 174 Poisson, 128 prior and posterior, 388 relationships among, F, chi-square, and t, 139 sampling, 163, 166, 169, 171, 174 Student s t, 136 uniform, 132 of variances, 171 Weibull, 141 Distribution functions: continuous, 46 discrete, 45 Marginal, 48, 49 Efficient estimates, 199 Estimates: confidence interval, 200, 202 efficient, 199 maximum likelihood, 206 unbiased, 199 Events, 10 independent, 14 mutually exclusive, 13 Expectation of random variables, 85 conditional, 93 F distribution, 138 relationship to chi-square and t distributions, 139 tests involving, 233 Fitting of data by theoretical distributions, 239 Fourier coefficients, 97 Fourier series, 97 Frequency: distributions, 175 histograms, 176 polygons, 176 Gamma distribution, 133 Generalized correlation coefficient, 292 Geometric probability, applications to, 60 Goodness of fit, 242, 246 Graeco-Latin squares, 335 Hypergeometric distribution, 131 Hypotheses tests (see Tests) Improper distributions, 392 Independent events, 14 Independent random variables, 47, 59 Joint density functions, 48 Joint distributions, 47 Kruskal-Wallis H test, 283 Kurtosis, 96, 97, 98 Latin squares, 334 Law of large numbers, 93 for Bernoulli trials, 122 Least squares: line, 275 parabola, 284 Linear correlation coefficient (see Correlation) Linear relationship, 292 Log-log graph paper, 283.

That is, we don t assume that the di erence between the rst and second student is the same as the di erence between the 100th and 101st student Any formula that converts from one ordinal scale measure to another ordinal scale measure for the same entity must preserve the ordering In an interval scale measure, the amount of the di erence is constant An example is temperature There are two instances of temperature measures that are commonly used, Fahrenheit and Celsius The formula for converting from a Celsius scale measure to a Fahrenheit scale measure is 9=5 x 32 With any two interval scale measures for the same attribute, the formula for conversion must be of the form ax b In a ratio scale measure, the amount of the di erence is constant and there is a well-understood zero that any scale measure would use.

Fig. 2-2

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