14.2.3: Formulas

Table B4

Basic probability rules
\(P(A \cap B)=P(A | B) \cdot P(B)\)			Multiplication rule
\(P(A \cup B)=P(A)+P(B)-P(A \cap B)\)			Addition rule
\(P(A \cap B)=P(A) \cdot P(B) \text { or } P(A | B)=P(A)\)			Independence test
Hypergeometric distribution formulae
\(n C x=\left(\begin{array}{c}{n} \\ {x}\end{array}\right)=\frac{n !}{x !(n-x) !}\)		Combinatorial equation
\(P(x)=\frac{\left(\begin{array}{c}{A} \\ {x}\end{array}\right)\left(\begin{array}{c}{N-A} \\ {n-x}\end{array}\right)}{\left(\begin{array}{c}{N} \\ {n}\end{array}\right)}\)		Probability equation
\(E(X)=\mu=n p\)		Mean
\(\sigma^{2}=\left(\frac{N-n}{N-1}\right) n p(q)\)		Variance
Binomial distribution formulae
\(P(x)=\frac{n !}{x !(n-x) !} p^{x}(q)^{n-x}\)		Probability density function
\(E(X)=\mu=n p\)		Arithmetic mean
\(\sigma^{2}=n p(q)\)		Variance
Geometric distribution formulae
\(P(X=x)=(1-p)^{x-1}(p)\)	Probability when \(x\) is the first success.	Probability when \(x\) is the number of failures before first success	\(P(X=x)=(1-p)^{x}(p)\)
\(\mu=\frac{1}{p}\)	Mean	Mean	\(\mu=\frac{1-p}{p}\)
\(\sigma^{2}=\frac{(1-p)}{p^{2}}\)	Variance	Variance	\(\sigma^{2}=\frac{(1-p)}{p^{2}}\)
Poisson distribution formulae
\(P(x)=\frac{e^{-\mu_{\mu} x}}{x !}\)		Probability equation
\(E(X)=\mu\)		Mean
\(\sigma^{2}=\mu\)		Variance
Uniform distribution formulae
\(f(x)=\frac{1}{b-a} \text { for } a \leq x \leq b\)		PDF
\(E(X)=\mu=\frac{a+b}{2}\)		Mean
\(\sigma^{2}=\frac{(b-a)^{2}}{12}\)		Variance
Exponential distribution formulae
\(P(X \leq x)=1-e^{-m x}\)		Cumulative probability
\(E(X)=\mu=\frac{1}{m} \text { or } m=\frac{1}{\mu}\)		Mean and decay factor
\(\sigma^{2}=\frac{1}{m^{2}}=\mu^{2}\)		Variance

Table B5

The following page of formulae requires the use of the "\(Z\)", "\(t\)", "\(\chi^2\)" or "\(F\)" tables.
\(Z=\frac{x-\mu}{\sigma}\)	Z-transformation for normal distribution
\(Z=\frac{x-n p^{\prime}}{\sqrt{n p^{\prime}\left(q^{\prime}\right)}}\)	Normal approximation to the binomial
Probability (ignores subscripts) Hypothesis testing	Confidence intervals [bracketed symbols equal margin of error] (subscripts denote locations on respective distribution tables)
\(Z_{c}=\frac{\overline{x}-\mu_{0}}{\frac{\sigma}{\sqrt{n}}}\)	Interval for the population mean when sigma is known \(\overline{x} \pm\left[Z_{(\alpha / 2)} \frac{\sigma}{\sqrt{n}}\right]\)
\(Z_{c}=\frac{\overline{x}-\mu_{0}}{\frac{s}{\sqrt{n}}}\)	Interval for the population mean when sigma is unknown but \(n>30\) \(\overline{x} \pm\left[Z_{(\alpha / 2)} \frac{s}{\sqrt{n}}\right]\)
\(t_{c}=\frac{\overline{x}-\mu_{0}}{\frac{s}{\sqrt{n}}}\)	Interval for the population mean when sigma is unknown but \(n<30\) \(\overline{x} \pm\left[t_{(n-1),(\alpha / 2)} \frac{s}{\sqrt{n}}\right]\)
\(Z_{c}=\frac{p^{\prime}-p_{0}}{\sqrt{\frac{p_{0} q_{0}}{n}}}\)	Interval for the population proportion \(p^{\prime} \pm\left[Z_{(\alpha / 2)} \sqrt{\frac{p^{\prime} q^{\prime}}{n}}\right]\)
\(t_{c}=\frac{\overline{d}-\delta_{0}}{s_{d}}\)	Interval for difference between two means with matched pairs \(\overline{d} \pm\left[t_{(n-1),(\alpha / 2)} \frac{s_{d}}{\sqrt{n}}\right]\) where \(s_d\) is the deviation of the differences
\(Z_{c}=\frac{\left(\overline{x_{1}}-\overline{x_{2}}\right)-\delta_{0}}{\sqrt{\frac{\sigma_{1}^{2}}{n_{1}}+\frac{\sigma_{2}^{2}}{n_{2}}}}\)	Interval for difference between two means when sigmas are known \(\left(\overline{x}_{1}-\overline{x}_{2}\right) \pm\left[Z_{(\alpha / 2)} \sqrt{\frac{\sigma_{1}^{2}}{n_{1}}+\frac{\sigma_{2}^{2}}{n_{2}}}\right]\)
\(t_{c}=\frac{\left(\overline{x}_{1}-\overline{x}_{2}\right)-\delta_{0}}{\sqrt{\left(\frac{\left(s_{1}\right)^{2}}{n_{1}}+\frac{\left(s_{2}\right)^{2}}{n_{2}}\right)}}\)	Interval for difference between two means with equal variances when sigmas are unknown \(\left(\overline{x}_{1}-\overline{x}_{2}\right) \pm\left[t_{d f,(\alpha / 2)} \sqrt{\left(\frac{\left(s_{1}\right)^{2}}{n_{1}}+\frac{\left(s_{2}\right)^{2}}{n_{2}}\right)}\right] \text { where } d f=\frac{\left(\frac{\left(s_{1}\right)^{2}}{n_{1}}+\frac{\left(s_{2}\right)^{2}}{n_{2}}\right)^{2}}{\left(\frac{1}{n_{1}-1}\right)\left(\frac{\left(s_{1}\right)^{2}}{n_{1}}\right)+\left(\frac{1}{n_{2}-1}\right)\left(\frac{\left(s_{2}\right)^{2}}{n_{2}}\right)}\)
\(Z_{c}=\frac{\left(p_{1}^{\prime}-p_{2}^{\prime}\right)-\delta_{0}}{\sqrt{\frac{p_{1}^{\prime}\left(q_{1}^{\prime}\right)}{n_{1}}+\frac{p_{2}^{\prime}\left(q_{2}^{\prime}\right)}{n_{2}}}}\)	Interval for difference between two population proportions \(\left(p_{1}^{\prime}-p_{2}^{\prime}\right) \pm\left[Z_{(\alpha / 2)} \sqrt{\frac{p_{1}^{\prime}\left(q_{1}^{\prime}\right)}{n_{1}}+\frac{p_{2}^{\prime}\left(q_{2}^{\prime}\right)}{n_{2}}}\right]\)
\(\chi_{c}^{2}=\frac{(n-1) s^{2}}{\sigma_{0}^{2}}\)	Tests for \(GOF\), Independence, and Homogeneity \(\chi_{c}^{2}=\sum \frac{(O-E)^{2}}{E}\)where \(O =\) observed values and \(E =\) expected values
\(F_{c}=\frac{s_{1}^{2}}{s_{2}^{2}}\)	Where \(s_{1}^{2}\) is the sample variance which is the larger of the two sample variances
The next 3 formule are for determining sample size with confidence intervals. (note: \(E\) represents the margin of error)
\(n=\frac{Z^{2}\left(\frac{a}{2}\right)^{\sigma^{2}}}{E^{2}}\) Use when sigma is known \(E=\overline{x}-\mu\)	\(n=\frac{Z^{2}\left(\frac{a}{2}\right)^{(0.25)}}{E^{2}}\) Use when \(p^{\prime}\) is unknown \(E=p^{\prime}-p\)	\(n=\frac{Z^{2}\left(\frac{a}{2}\right)^{\left[p^{\prime}\left(q^{\prime}\right)\right]}}{E^{2}}\) Use when p'p′ is uknown \(E=p^{\prime}-p\)

Table B6

Simple linear regression formulae for \(y=a+b(x)\)
\(r=\frac{\Sigma[(x-\overline{x})(y-\overline{y})]}{\sqrt{\Sigma(x-\overline{x})^{2} * \Sigma(y-\overline{y})^{2}}}=\frac{S_{x y}}{S_{x} S_{y}}=\sqrt{\frac{S S R}{S S T}}\)	Correlation coefficient
\(b=\frac{\Sigma[(x-\overline{x})(y-\overline{y})]}{\Sigma(x-\overline{x})^{2}}=\frac{S_{x y}}{S S_{x}}=r_{y, x}\left(\frac{s_{y}}{s_{x}}\right)\)	Coefficient \(b\) (slope)
\(a=\overline{y}-b(\overline{x})\)	\(y\)-intercept
\(s_{e}^{2}=\frac{\Sigma\left(y_{i}-\hat{y}_{i}\right)^{2}}{n-k}=\frac{\sum_{i=1}^{n} e_{i}^{2}}{n-k}\)	Estimate of the error variance
\(S_{b}=\frac{s_{e}^{2}}{\sqrt{\left(x_{i}-\overline{x}\right)^{2}}}=\frac{s_{e}^{2}}{(n-1) s_{x}^{2}}\)	Standard error for coefficient \(b\)
\(t_{c}=\frac{b-\beta_{0}}{s_b}\)	Hypothesis test for coefficient \(\beta\)
\(b \pm\left[t_{n-2, \alpha / 2} S_{b}\right]\)	Interval for coefficient \(\beta\)
\(\hat{y} \pm\left[t_{\alpha / 2} * s_{e}\left(\sqrt{\frac{1}{n}+\frac{\left(x_{p}-\overline{x}\right)^{2}}{s_{x}}}\right)\right]\)	Interval for expected value of \(y\)
\(\hat{y} \pm\left[t_{\alpha / 2} * s_{e}\left(\sqrt{1+\frac{1}{n}+\frac{\left(x_{p}-\overline{x}\right)^{2}}{s_{x}}}\right)\right]\)	Prediction interval for an individual \(y\)
ANOVA formulae
\(S S R=\sum_{i=1}^{n}\left(\hat{y}_{i}-\overline{y}\right)^{2}\)	Sum of squares regression
\(S S E=\sum_{i=1}^{n}\left(\hat{y}_{i}-\overline{y}_{i}\right)^{2}\)	Sum of squares error
\(S S T=\sum_{i=1}^{n}\left(y_{i}-\overline{y}\right)^{2}\)	Sum of squares total
\(R^{2}=\frac{S S R}{S S T}\)	Coefficient of determination

The following is the breakdown of a one-way ANOVA table for linear regression.
Source of variation	Sum of squares	Degrees of freedom	Mean squares	\(F\)-ratio
Regression	\(SSR\)	\(1\) or \(k−1\)	\(M S R=\frac{S S R}{d f_{R}}\)	\(F=\frac{M S R}{M S E}\)
Error	\(SSE\)	\(n-k\)	\(M S E=\frac{S S E}{d f_{E}}\)
Total	\(SST\)	\(n−1\)