\documentclass[a4paper,12pt]{report} \usepackage{rotating,graphicx,amsbsy,amsfonts,epsfig,amsmath,enumerate,amssymb,psfrag,lscape,epsf,color,fancybox} %\usepackage{stevenhead} %\linespread{1.6} \linespread{1.2} %\textheight23cm %\textwidth16cm \oddsidemargin0.5cm \evensidemargin0.5cm \leftmargin0cm \rightmargin0cm \topmargin0mm %\headheight15mm %\headwidth16cm %\headsep7mm \begin{document} \LARGE \noindent \ovalbox{\textbf{Formulae and statistical tables}} \normalsize \section*{Sample statistics} \begin{tabular}{l c} Lower quartile: & $Q_{1} = \frac{(n+1)}{4} {\rm th}\ \textrm{smallest observation.} $ \\ & \\ Median: & $Q_{2} = \frac{(n+1)}{2} {\rm th}\ \textrm{smallest observation.} $ \\ & \\ Upper quartile: & $Q_{3} = \frac{3(n+1)}{4} {\rm th}\ \textrm{smallest observation.}$ \\ & \\ Sample mean: & $\bar{x} = \displaystyle \frac{1}{n} \sum_{i=1}^{n} x_{i}$ \\ & \\ Sample variance:& $s^{2} = \displaystyle \frac{1}{n-1} \sum_{i=1}^{n}(x_{i} - \bar{x})^{2}$ \\ \end{tabular} \vspace{1cm} \section*{Combining probabilities} \subsection*{Addition} \[ P(A\ {\rm or}\ B) = P(A) + P(B) - P(A\ {\rm and}\ B) \] If $ A $ and $ B $ are mutually exclusive then \[ P(A\ {\rm or}\ B) = P(A) + P(B) \] \subsection*{Multiplication} \[ P(A\ {\rm and}\ B) = P(A) P(B \mid A)\] If $ A $ and $ B $ are independent then \[ P(A\ {\rm and}\ B) = P(A) P(B) \] \section*{Probability distributions} \begin{tabular}{|l| c| c c|} \hline \textbf{Distribution} &PMF/PDF &Mean &Variance \\ \hline Binomial, $\displaystyle X \sim \textrm{Bin}(n,p)$&${n\choose x}p^{x}(1-p)^{n-x}$ &$np$ &$np(1-p)$ \\ & & & \\ Poisson, $\displaystyle X\sim \textrm{Po}(\lambda)$&$\frac{e^{-\lambda}\lambda^{x}}{x!}$&$\lambda$ &$\lambda$ \\ & & & \\ \hline Exponential, $\displaystyle X \sim \textrm{Exp}(\lambda)$&$\lambda e^{-\lambda x}$ &$1/\lambda$ &$1/\lambda^{2}$\\ & & & \\ Normal, $\displaystyle X \sim N(\mu, \sigma^{2})$ &$\frac{1}{\sigma\sqrt{2 \pi}}\textrm{exp}\left\{-\frac{(x-\mu)^{2}}{2 \sigma^{2}} \right\}$ &$\mu$ &$\sigma^{2}$ \\ & & & \\ Standard Normal, $\displaystyle Z=\frac{X-\mu}{\sigma}$&$\frac{1}{\sqrt{2 \pi}}e^{-\frac{1}{2}x^{2}}$& 0 &1 \\ & & & \\ \hline \end{tabular} \section*{Confidence intervals and hypothesis tests} \noindent \textbf{Confidence interval for the population mean $\boldsymbol{\mu}$} \vspace{0.3cm} \begin{tabular}{l c} $\sigma$ known: &$\bar{x} \pm z_{\alpha} \times \sigma/\sqrt{n}$ \\ $\sigma$ unknown: &$\bar{x} \pm t_{\nu,\alpha} \times s/\sqrt{n}$ \\ \end{tabular} \vspace{0.7cm} \noindent \begin{tabular}{| l | c| c|} \hline \textbf{Hypothesis test} &Test statistic &Other comments\\ \hline \textit{Tests for one mean} & & \\ $\sigma$ known &$z=\frac{\left|\bar{x}-\mu\right|}{\sigma/\sqrt{n}}$ &\\ & & \\ $\sigma$ unknown &$t=\frac{\left|\bar{x}-\mu\right|}{s/\sqrt{n}}$ &$\nu=n-1$\\ \hline \textit{Tests for two means} & & \\ $\sigma_{1},\sigma_{2}$ known &$z=\frac{\left|\bar{x}_{1}-\bar{x}_{2}\right|}{\sqrt{\frac{\sigma_{1}^{2}}{n_{1}}+\frac{\sigma_{2}^{2}}{n_{2}}}}$ & \\ & & \\ $\sigma_{1}, \sigma_{2}$ unknown &$t=\frac{\left|\bar{x}_{1}-\bar{x}_{2}\right|}{s\times \sqrt{\frac{1}{n_{1}}+\frac{1}{n_{2}}}}$ &$s=\sqrt{\frac{\left(n_{1}-1\right)s_{1}^{2}+\left(n_{2}-1\right)s_{2}^{2}}{n_{1}+n_{2}-2}}$ \\ & &$\nu=n_{1}+n_{2}-2$ \\ \hline \textit{Goodness--of--fit tests} &$X^{2}=\sum\frac{\left(O-E\right)^{2}}{E}$ &$\nu=(\textrm{no. of categories after pooling})$ \\ & &$-(\textrm{no. of estimated parameters})-1$ \\ \hline \end{tabular} \section*{Correlation and linear regression} \noindent The sample correlation coefficient is given by \begin{eqnarray*} r &=& \frac{S_{XY}}{\sqrt{S_{XX}\times S_{YY}}}, \end{eqnarray*} where \begin{eqnarray*} S_{XY} &=& \left(\sum xy \right)-n\bar{x}\bar{y}, \\ S_{XX} &=& \left(\sum x^{2}\right)-n \bar{x}^{2}, \qquad \textrm{and} \\ S_{YY} &=& \left(\sum y^{2}\right)-n \bar{y}^{2}. \end{eqnarray*} \noindent The simple linear regression equation is given by \begin{eqnarray*} Y &=& \alpha + \beta X + \epsilon, \end{eqnarray*} where $\left\{\epsilon_{i}\right\}$ are independent $N\left(0,\sigma^{2}\right)$ random variables and $\alpha$ and $\beta$ can be estimated using \begin{eqnarray*} \hat{\beta} &=& \frac{S_{XY}}{S_{XX}} \qquad \textrm{and} \\ \hat{\alpha} &=& \bar{y} - \hat{\beta}\bar{x}. \end{eqnarray*} \newpage \subsection*{Statistical tables} \begin{table}[!h] {\scriptsize \centering \begin{tabular}{|c c| c c c c c|} \hline & &\multicolumn{5}{c|}{$p$} \\ &One--tailed test &10\% &5\% &2.5\% &1\% &0.5\% \\ &Two--tailed test &20\% &10\% &5\% &2\% &1\% \\ \hline &1 &3.078 &6.314 &12.706 &31.821 &63.657 \\ &2 &1.886 &2.920 &4.303 &6.965 &9.925 \\ &3 &1.638 &2.353 &3.182 &4.541 &5.841 \\ &4 &1.533 &2.132 &2.776 &3.747 &4.604 \\ &5 &1.476 &2.015 &2.571 &3.365 &4.032 \\ &6 &1.440 &1.943 &2.447 &3.143 &3.707 \\ &7 &1.415 &1.895 &2.365 &2.998 &3.449 \\ &8 &1.397 &1.860 &2.306 &2.896 &3.355 \\ &9 &1.383 &1.833 &2.262 &2.821 &3.250 \\ &10 &1.372 &1.812 &2.228 &2.764 &3.169 \\ &11 &1.363 &1.796 &2.201 &2.718 &3.106 \\ &12 &1.356 &1.782 &2.179 &2.681 &3.055 \\ &13 &1.350 &1.771 &2.160 &2.650 &3.012 \\ &14 &1.345 &1.761 &2.145 &2.624 &2.977 \\ $\nu$&15 &1.341 &1.753 &2.131 &2.602 &2.947 \\ &16 &1.337 &1.746 &2.120 &2.583 &2.921 \\ &17 &1.333 &1.740 &2.110 &2.567 &2.898 \\ &18 &1.330 &1.734 &2.101 &2.552 &2.878 \\ &19 &1.328 &1.729 &2.093 &2.539 &2.861 \\ &20 &1.325 &1.725 &2.086 &2.528 &2.845 \\ &21 &1.323 &1.721 &2.080 &2.518 &2.831 \\ &22 &1.321 &1.717 &2.074 &2.508 &2.819 \\ &23 &1.319 &1.714 &2.069 &2.500 &2.807 \\ &24 &1.318 &1.711 &2.064 &2.492 &2.797 \\ &25 &1.316 &1.708 &2.060 &2.485 &2.787 \\ &26 &1.315 &1.706 &2.056 &2.479 &2.779 \\ &27 &1.314 &1.703 &2.052 &2.473 &2.771 \\ &28 &1.313 &1.701 &2.048 &2.467 &2.763 \\ &29 &1.311 &1.699 &2.045 &2.462 &2.756 \\ &\vdots &\vdots &\vdots &\vdots &\vdots &\vdots \\ &$\infty$ &1.282 &1.645 &1.960 &2.326 &2.576 \\ \hline \end{tabular} \caption{This table contains values of $t$ for which Pr($T>t$) = $p$, where $T \sim t_{\nu}$. In a two--tailed test, the tabulated values correspond to Pr($|T|>t$) = $p$} } \end{table} \begin{table}[!h] {\scriptsize \centering \begin{tabular}{|c c| c c c c c|} \hline & & & &$p$ & & \\ & &50\% &10\% &5\% &1\% &0.1\% \\ \hline &1 &0.45 &2.17 &3.84 &6.63 &10.83 \\ &2 &1.39 &4.61 &5.99 &9.21 &13.82 \\ &3 &2.37 &6.25 &7.82 &11.34 &16.27 \\ &4 &3.36 &7.78 &9.49 &13.28 &18.47 \\ &5 &4.34 &9.24 &11.07 &15.09 &20.52 \\ &6 &5.35 &10.64 &12.59 &16.81 &22.46 \\ &7 &6.35 &12.02 &14.07 &18.48 &24.32 \\ $\nu$&8 &7.34 &13.36 &15.51 &20.09 &26.13 \\ &9 &8.34 &14.68 &16.92 &21.67 &27.88 \\ &10 &9.34 &15.99 &18.31 &23.21 &29.59 \\ &12 &11.34 &18.55 &21.03 &26.22 &32.91 \\ &15 &14.34 &22.31 &25.00 &30.58 &37.70 \\ &20 &19.34 &28.41 &31.41 &37.57 &45.32 \\ &25 &24.34 &34.38 &37.65 &44.31 &52.62 \\ &30 &29.34 &40.26 &43.77 &50.89 &59.70 \\ \hline \end{tabular} \caption{This table contains values of $x$ for which $\textrm{Pr}(X^{2}>x)=p$, where $X^{2} \sim \chi^{2}_{\nu}$} } \end{table} \begin{center} \begin{table} \centering {\scriptsize \begin{tabular}{|c|cccccccccc|} \hline $z$ & -0.09 & -0.08 & -0.07 & -0.06 & -0.05 & -0.04 & -0.03 & -0.02 & -0.01 & 0.00 \\ \hline -2.9 & 0.0014 & 0.0014 & 0.0015 & 0.0015 & 0.0016 & 0.0016 & 0.0017 & 0.0018 & 0.0018 & 0.0019 \\ -2.8 & 0.0019 & 0.0020 & 0.0021 & 0.0021 & 0.0022 & 0.0023 & 0.0023 & 0.0024 & 0.0025 & 0.0026 \\ -2.7 & 0.0026 & 0.0027 & 0.0028 & 0.0029 & 0.0030 & 0.0031 & 0.0032 & 0.0033 & 0.0034 & 0.0035 \\ -2.6 & 0.0036 & 0.0037 & 0.0038 & 0.0039 & 0.0040 & 0.0041 & 0.0043 & 0.0044 & 0.0045 & 0.0047 \\ -2.5 & 0.0048 & 0.0049 & 0.0051 & 0.0052 & 0.0054 & 0.0055 & 0.0057 & 0.0059 & 0.0060 & 0.0062 \\ -2.4 & 0.0064 & 0.0066 & 0.0068 & 0.0069 & 0.0071 & 0.0073 & 0.0075 & 0.0078 & 0.0080 & 0.0082 \\ -2.3 & 0.0084 & 0.0087 & 0.0089 & 0.0091 & 0.0094 & 0.0096 & 0.0099 & 0.0102 & 0.0104 & 0.0107 \\ -2.2 & 0.0110 & 0.0113 & 0.0116 & 0.0119 & 0.0122 & 0.0125 & 0.0129 & 0.0132 & 0.0136 & 0.0139 \\ -2.1 & 0.0143 & 0.0146 & 0.0150 & 0.0154 & 0.0158 & 0.0162 & 0.0166 & 0.0170 & 0.0174 & 0.0179 \\ -2.0 & 0.0183 & 0.0188 & 0.0192 & 0.0197 & 0.0202 & 0.0207 & 0.0212 & 0.0217 & 0.0222 & 0.0228 \\ -1.9 & 0.0233 & 0.0239 & 0.0244 & 0.0250 & 0.0256 & 0.0262 & 0.0268 & 0.0274 & 0.0281 & 0.0287 \\ -1.8 & 0.0294 & 0.0301 & 0.0307 & 0.0314 & 0.0322 & 0.0329 & 0.0336 & 0.0344 & 0.0351 & 0.0359 \\ -1.7 & 0.0367 & 0.0375 & 0.0384 & 0.0392 & 0.0401 & 0.0409 & 0.0418 & 0.0427 & 0.0436 & 0.0446 \\ -1.6 & 0.0455 & 0.0465 & 0.0475 & 0.0485 & 0.0495 & 0.0505 & 0.0516 & 0.0526 & 0.0537 & 0.0548 \\ -1.5 & 0.0559 & 0.0571 & 0.0582 & 0.0594 & 0.0606 & 0.0618 & 0.0630 & 0.0643 & 0.0655 & 0.0668 \\ -1.4 & 0.0681 & 0.0694 & 0.0708 & 0.0721 & 0.0735 & 0.0749 & 0.0764 & 0.0778 & 0.0793 & 0.0808 \\ -1.3 & 0.0823 & 0.0838 & 0.0853 & 0.0869 & 0.0885 & 0.0901 & 0.0918 & 0.0934 & 0.0951 & 0.0968 \\ -1.2 & 0.0985 & 0.1003 & 0.1020 & 0.1038 & 0.1056 & 0.1075 & 0.1093 & 0.1112 & 0.1131 & 0.1151 \\ -1.1 & 0.1170 & 0.1190 & 0.1210 & 0.1230 & 0.1251 & 0.1271 & 0.1292 & 0.1314 & 0.1335 & 0.1357 \\ -1.0 & 0.1379 & 0.1401 & 0.1423 & 0.1446 & 0.1469 & 0.1492 & 0.1515 & 0.1539 & 0.1562 & 0.1587 \\ -0.9 & 0.1611 & 0.1635 & 0.1660 & 0.1685 & 0.1711 & 0.1736 & 0.1762 & 0.1788 & 0.1814 & 0.1841 \\ -0.8 & 0.1867 & 0.1894 & 0.1922 & 0.1949 & 0.1977 & 0.2005 & 0.2033 & 0.2061 & 0.2090 & 0.2119 \\ -0.7 & 0.2148 & 0.2177 & 0.2206 & 0.2236 & 0.2266 & 0.2296 & 0.2327 & 0.2358 & 0.2389 & 0.2420 \\ -0.6 & 0.2451 & 0.2483 & 0.2514 & 0.2546 & 0.2578 & 0.2611 & 0.2643 & 0.2676 & 0.2709 & 0.2743 \\ -0.5 & 0.2776 & 0.2810 & 0.2843 & 0.2877 & 0.2912 & 0.2946 & 0.2981 & 0.3015 & 0.3050 & 0.3085 \\ -0.4 & 0.3121 & 0.3156 & 0.3192 & 0.3228 & 0.3264 & 0.3300 & 0.3336 & 0.3372 & 0.3409 & 0.3446 \\ -0.3 & 0.3483 & 0.3520 & 0.3557 & 0.3594 & 0.3632 & 0.3669 & 0.3707 & 0.3745 & 0.3783 & 0.3821 \\ -0.2 & 0.3859 & 0.3897 & 0.3936 & 0.3974 & 0.4013 & 0.4052 & 0.4090 & 0.4129 & 0.4168 & 0.4207 \\ -0.1 & 0.4247 & 0.4286 & 0.4325 & 0.4364 & 0.4404 & 0.4443 & 0.4483 & 0.4522 & 0.4562 & 0.4602 \\ 0.0 & 0.4641 & 0.4681 & 0.4721 & 0.4761 & 0.4801 & 0.4840 & 0.4880 & 0.4920 & 0.4960 & 0.5000 \\ \hline \end{tabular} \vspace{0.3cm} \begin{tabular}{|c|cccccccccc|} \hline $z$ & 0.00 & 0.01 & 0.02 & 0.03 & 0.04 & 0.05 & 0.06 & 0.07 & 0.08 & 0.09 \\ \hline 0.0 & 0.5000 & 0.5040 & 0.5080 & 0.5120 & 0.5160 & 0.5199 & 0.5239 & 0.5279 & 0.5319 & 0.5359 \\ 0.1 & 0.5398 & 0.5438 & 0.5478 & 0.5517 & 0.5557 & 0.5596 & 0.5636 & 0.5675 & 0.5714 & 0.5753 \\ 0.2 & 0.5793 & 0.5832 & 0.5871 & 0.5910 & 0.5948 & 0.5987 & 0.6026 & 0.6064 & 0.6103 & 0.6141 \\ 0.3 & 0.6179 & 0.6217 & 0.6255 & 0.6293 & 0.6331 & 0.6368 & 0.6406 & 0.6443 & 0.6480 & 0.6517 \\ 0.4 & 0.6554 & 0.6591 & 0.6628 & 0.6664 & 0.6700 & 0.6736 & 0.6772 & 0.6808 & 0.6844 & 0.6879 \\ 0.5 & 0.6915 & 0.6950 & 0.6985 & 0.7019 & 0.7054 & 0.7088 & 0.7123 & 0.7157 & 0.7190 & 0.7224 \\ 0.6 & 0.7257 & 0.7291 & 0.7324 & 0.7357 & 0.7389 & 0.7422 & 0.7454 & 0.7486 & 0.7517 & 0.7549 \\ 0.7 & 0.7580 & 0.7611 & 0.7642 & 0.7673 & 0.7704 & 0.7734 & 0.7764 & 0.7794 & 0.7823 & 0.7852 \\ 0.8 & 0.7881 & 0.7910 & 0.7939 & 0.7967 & 0.7995 & 0.8023 & 0.8051 & 0.8078 & 0.8106 & 0.8133 \\ 0.9 & 0.8159 & 0.8186 & 0.8212 & 0.8238 & 0.8264 & 0.8289 & 0.8315 & 0.8340 & 0.8365 & 0.8389 \\ 1.0 & 0.8413 & 0.8438 & 0.8461 & 0.8485 & 0.8508 & 0.8531 & 0.8554 & 0.8577 & 0.8599 & 0.8621 \\ 1.1 & 0.8643 & 0.8665 & 0.8686 & 0.8708 & 0.8729 & 0.8749 & 0.8770 & 0.8790 & 0.8810 & 0.8830 \\ 1.2 & 0.8849 & 0.8869 & 0.8888 & 0.8907 & 0.8925 & 0.8944 & 0.8962 & 0.8980 & 0.8997 & 0.9015 \\ 1.3 & 0.9032 & 0.9049 & 0.9066 & 0.9082 & 0.9099 & 0.9115 & 0.9131 & 0.9147 & 0.9162 & 0.9177 \\ 1.4 & 0.9192 & 0.9207 & 0.9222 & 0.9236 & 0.9251 & 0.9265 & 0.9279 & 0.9292 & 0.9306 & 0.9319 \\ 1.5 & 0.9332 & 0.9345 & 0.9357 & 0.9370 & 0.9382 & 0.9394 & 0.9406 & 0.9418 & 0.9429 & 0.9441 \\ 1.6 & 0.9452 & 0.9463 & 0.9474 & 0.9484 & 0.9495 & 0.9505 & 0.9515 & 0.9525 & 0.9535 & 0.9545 \\ 1.7 & 0.9554 & 0.9564 & 0.9573 & 0.9582 & 0.9591 & 0.9599 & 0.9608 & 0.9616 & 0.9625 & 0.9633 \\ 1.8 & 0.9641 & 0.9649 & 0.9656 & 0.9664 & 0.9671 & 0.9678 & 0.9686 & 0.9693 & 0.9699 & 0.9706 \\ 1.9 & 0.9713 & 0.9719 & 0.9726 & 0.9732 & 0.9738 & 0.9744 & 0.9750 & 0.9756 & 0.9761 & 0.9767 \\ 2.0 & 0.9772 & 0.9778 & 0.9783 & 0.9788 & 0.9793 & 0.9798 & 0.9803 & 0.9808 & 0.9812 & 0.9817 \\ 2.1 & 0.9821 & 0.9826 & 0.9830 & 0.9834 & 0.9838 & 0.9842 & 0.9846 & 0.9850 & 0.9854 & 0.9857 \\ 2.2 & 0.9861 & 0.9864 & 0.9868 & 0.9871 & 0.9875 & 0.9878 & 0.9881 & 0.9884 & 0.9887 & 0.9890 \\ 2.3 & 0.9893 & 0.9896 & 0.9898 & 0.9901 & 0.9904 & 0.9906 & 0.9909 & 0.9911 & 0.9913 & 0.9916 \\ 2.4 & 0.9918 & 0.9920 & 0.9922 & 0.9925 & 0.9927 & 0.9929 & 0.9931 & 0.9932 & 0.9934 & 0.9936 \\ 2.5 & 0.9938 & 0.9940 & 0.9941 & 0.9943 & 0.9945 & 0.9946 & 0.9948 & 0.9949 & 0.9951 & 0.9952 \\ 2.6 & 0.9953 & 0.9955 & 0.9956 & 0.9957 & 0.9959 & 0.9960 & 0.9961 & 0.9962 & 0.9963 & 0.9964 \\ 2.7 & 0.9965 & 0.9966 & 0.9967 & 0.9968 & 0.9969 & 0.9970 & 0.9971 & 0.9972 & 0.9973 & 0.9974 \\ 2.8 & 0.9974 & 0.9975 & 0.9976 & 0.9977 & 0.9977 & 0.9978 & 0.9979 & 0.9979 & 0.9980 & 0.9981 \\ 2.9 & 0.9981 & 0.9982 & 0.9982 & 0.9983 & 0.9984 & 0.9984 & 0.9985 & 0.9985 & 0.9986 & 0.9986 \\ \hline \end{tabular} \caption{This table contains values of $\textrm{Pr}(Z