Hypothesis Testing & SQC | t-test F-test ANOVA Charts

Module V: Statistical Techniques – III – All Important Formulas

Sample Mean & Variance

\bar{x} = \frac{\sum x_i}{n}, \qquad s^2 = \frac{1}{n-1}\sum (x_i - \bar{x})^2

Standard Error of Mean

\text{SE}(\bar{x}) = \frac{\sigma}{\sqrt{n}} \quad\text{(large sample / known }\sigma\text{)}

\text{SE}(\bar{x}) = \frac{s}{\sqrt{n}} \quad\text{(small sample / unknown }\sigma\text{)}

Confidence Limits (for mean)

\bar{x} \pm Z_{\alpha/2}\frac{\sigma}{\sqrt{n}} \quad\text{(large sample)}

\bar{x} \pm t_{\alpha/2,\,n-1}\frac{s}{\sqrt{n}} \quad\text{(small sample)}

Z = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\dfrac{\sigma_1^2}{n_1} + \dfrac{\sigma_2^2}{n_2}}}

One-sample t-test

t = \frac{\bar{x} - \mu}{s/\sqrt{n}} \sim t_{n-1}

Two-sample t-test (equal variance)

t = \frac{\bar{x}_1 - \bar{x}_2}{s_p\sqrt{\dfrac{1}{n_1}+\dfrac{1}{n_2}}}

where pooled variance

s_p^2 = \frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1+n_2-2}

Paired t-test

t = \frac{\bar{d}}{s_d/\sqrt{n}} \sim t_{n-1}

F = \frac{s_1^2}{s_2^2} \sim F_{(n_1-1,\,n_2-1)} \quad (s_1^2 > s_2^2)

Goodness of Fit / Independence

\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}

Test for Variance

\chi^2 = \frac{(n-1)s^2}{\sigma^2} \sim \chi^2_{n-1}

Total Sum of Squares

\text{SST} = \sum_{i=1}^{k}\sum_{j=1}^{n_i}(x_{ij} - \bar{x})^2

Between Sum of Squares

\text{SSB} = \sum_{i=1}^{k} n_i (\bar{x}_i - \bar{x})^2

Within Sum of Squares

\text{SSW} = \text{SST} - \text{SSB}

Mean Squares

\text{MSB} = \frac{\text{SSB}}{k-1}, \qquad \text{MSW} = \frac{\text{SSW}}{N-k}

F-Statistic

F = \frac{\text{MSB}}{\text{MSW}} \sim F_{(k-1,\,N-k)}

$\bar{X}$ -Chart (Process Mean)

\text{UCL} = \bar{\bar{X}} + A_2 \bar{R}, \qquad \text{CL} = \bar{\bar{X}}, \qquad \text{LCL} = \bar{\bar{X}} - A_2 \bar{R}

$R$ -Chart (Process Range)

\text{UCL} = D_4 \bar{R}, \qquad \text{CL} = \bar{R}, \qquad \text{LCL} = D_3 \bar{R}

( $A_2$ , $D_3$ , $D_4$ are control chart constants depending on subgroup size)

$p$ -Chart (Fraction Defective)

\text{UCL} = \bar{p} + 3\sqrt{\frac{\bar{p}(1-\bar{p})}{n}}, \qquad \text{CL} = \bar{p}, \qquad \text{LCL} = \bar{p} - 3\sqrt{\frac{\bar{p}(1-\bar{p})}{n}}

$np$ -Chart (Number Defective)

\text{UCL} = n\bar{p} + 3\sqrt{n\bar{p}(1-\bar{p})}, \qquad \text{CL} = n\bar{p}, \qquad \text{LCL} = n\bar{p} - 3\sqrt{n\bar{p}(1-\bar{p})}

$c$ -Chart (Number of Defects)

\text{UCL} = \bar{c} + 3\sqrt{\bar{c}}, \qquad \text{CL} = \bar{c}, \qquad \text{LCL} = \bar{c} - 3\sqrt{\bar{c}}