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Limits to ANOVA
 population must be normally distributed
 samples are independent and random
 sd of sample = sd of larger population therefore pooled SE

ANOVA Hypothesis
 H_{0}: u_{1} = u_{2} = u_{3}
 H_{A}: at lease one u is different

n dot
total number of observations

y_{ij}
observation in j position from group i

SSW
 sum of squares within
 (y_{i}_{j}  mean_{i})^{2} + (y_{ij}  mean_{i})^{2} + all js

df within
n dot  number of categories

MSW
 mean squares within
 = SSW/dfW

SSB
 sum of squares between
 n_{i}(y_{i}  y_{grand})^{2} + ...
 Will go up if one mean is higher (reject)

y double bar
 overall mean
 (n_{i} * group mean) + (n_{i} * group mean) / n dot


MSB
 Mean Squares between
 = SSB/dfB

SS(total)
 SS(total) = SSW + SSB
 df = total observations 1

distance of y_{ij} from grand mean
(y_{ij}  y_{i mean}) + (y_{i mean}  grand mean)

Values in ANOVA table
 df (w & b) total
 SS (w & b) total
 MS (w & b) no total
 F_{s}

F_{s}
 test statistic for ANOVA
 MSB/MSW
 High = reject
 distribution like X^{2}

relationship btw Ftest and ttest
 F_{2} = t_{s}^{2}_{}^{ }
 using pooled SE
 F_{alpha} = t_{alpha/2}^{2}_{ }
 df = n_{1} + n_{2} 2
 nondirectional
 will have same decision

eg
F(4,20)_{.05}
 F(dfB, dfW)_{alpha}
 can get I and ndot from

Conditions of validity for ANOVA
 Population sds are equal
 within each group the samples are from normal populations are large n
 independent and random samples

how to check if sd's are similar
 residuals:
 add/subract average  to center on zero
 want parallel lines

checking for normality?
plot qq in straight line

response variable
explanatory variable (controlled)
 Y
 X, predictor, covariate


measures strength of relationship X and Y

Hypothosis for correlation btw X and Y
 H_{0}: rho = 0
 H_{A}: rho ≠ 0

t_{s} for X, Y correlation
 ttest
 = r* √(n2)/(1r^{2})
 df = n  2

e_{i}
 residuals
 = (y_{i}  b_{o}  mx_{i})
 want them to be smallest

