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\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 $\textrm{Pr}(Z<z)$, where $Z\sim N(0,1)$}.
}
\end{table}

\end{center}
\newpage

\end{document}

