SESSION:12 F Distribution and One-Way ANOVAFormula Review

Session:12 F Distribution and One-Way ANOVA

Formula Review

Introductory Business Statistics | Leadership Development – Micro-Learning Session

Rice University 2020 | Michael Laverty, Colorado State University Global Chris Littel, North Carolina State University| https://openstax.org/details/books/introductory-business-statistics

12.1 Test of Two Variances

�_{0} : \frac{�_{1}^{2}}{�_{2}^{2}} = �_{0}

�_{�} : \frac{�_{1}^{2}}{�_{2}^{2}} \neq �_{0}

12.12

if δ₀=1 then

�_{0} : �_{1}^{2} = �_{2}^{2}

�_{�} : �_{1}^{2} \neq �_{2}

Test statistic is :

�_{�} = \frac{�_{1}^{2}}{�_{2}^{2}}

12.13

12.3 The F Distribution and the F-Ratio

$S S_{between} = \sum^{} [\frac{{(s_{j})}^{2}}{n_{j}}] - \frac{{(\sum^{} s_{j})}^{2}}{n}$

$� �_{between} = \sum^{} [\frac{{(�_{�})}^{2}}{�_{�}}] - \frac{{(\sum^{} �_{�})}^{2}}{�}$

$S S_{total} = \sum^{} x^{2} - \frac{{(\sum^{} x)}^{2}}{n}$

$� �_{total} = \sum^{} �^{2} - \frac{{(\sum^{} �)}^{2}}{�}$

$S S_{within} = S S_{total} - S S_{between}$

$� �_{within} = � �_{total} - � �_{between}$

df_between = df(num) = k – 1

df_within = df(denom) = n – k

MS_between = $\frac{S S_{between}}{d f_{between}}$

$\frac{� �_{between}}{� �_{between}}$

MS_within = $\frac{S S_{within}}{d f_{within}}$

$\frac{� �_{within}}{� �_{within}}$

F = $\frac{M S_{between}}{M S_{within}}$

$\frac{� �_{between}}{� �_{within}}$

k = the number of groups
n_j = the size of the j^th group
s_j = the sum of the values in the j^th group
n = the total number of all values (observations) combined
x = one value (one observation) from the data
$s_{\bar{x}}^{2}$

= the variance of the sample means
$s^{2}_{p o o l e d}$

= the mean of the sample variances (pooled variance)