Question

2) You may do this problem by hand or in R. It is recornmended you use R to gain more familiarity with it. The following table summarizes some statistics for r 4 groups. Yi 1 10 10.69 0.830 2 12 12.17 0.894 3 8 11.95 0.510 4 9 11.82 0.741 ni Use it to complete the ANOVA table. df SS MS F p-value Treatment -- ".. “ -- Error Total

Answer 1

r=4 is the number of groups

The total number of observations is

$n=\sum_{i=1}^rn_i$

the grand mean is

$\bar{\bar{Y}}=\frac{\sum_{i=1}^rn_i\bar{Y}_i}{n}$
The degrees of freedom for Treatment

$df_{\text{treat}}=r-1$

The sum of square Treatment is

$\text{SSTR}=\sum_{i=1}^rn_i(\bar{Y}_i-\bar{\bar{Y}})^2$

Mean square of treatment is

$\text{MSTR}=\frac{\text{SSTR}}{df_{\text{treat}}}$

The degrees of freedom for error

$df_{\text{error}}=n-r$

The sum of square Error is

$\text{SSE}=\sum_{i=1}^r\sum_{j=1}^{n_i}(Y_{ij}-\bar{Y}_i)^2=\sum_{i=1}^r(n_i-1)s_i^2$

Mean square of Error is

$\text{MSE}=\frac{\text{SSE}}{df_{\text{error}}}$

Total Degrees of freedom

$df_{total}=n-1$

Sum of square total is

$\text{SST}=\text{SSTR}+\text{SSE}$

F statistics is

$F=\frac{\text{MSTR}}{\text{MSE}}$

p-value is

$\text{p-value}=P(F>\text{Fstat})$

R code for all these (all statements starting with # are comments)

#set the number of groups
r<-4;
#set the sample size for each group
n<-c(10,12,8,9)
#set the sample mean
ybar<-c(10.69,12.17,11.95,11.82)
#set the sample variances
s2<-c(0.830,0.894,0.510,0.741)
#calculate the total number of observations
nT<-sum(n)
#calculate the grand mean
ybarbar<-sum(n*ybar)/nT

#Treatment
#degrees of freedom
dftr<-r-1
#sum of square treatment
sstr<-sum(n*(ybar-ybarbar)^2)
#mean square treatment
mstr<-sstr/dftr

#Error
#degrees of freedom
dfe<-nT-r
#sum of square Error
sse<-sum((n-1)*s2)
#mean square error
mse<-sse/dfe

#df total
dftot<-nT-1
#sum of square total
sst<-sse+sstr

#F statistics
f<-mstr/mse

#p-value, P(F>f)
pvalue<-pf(f,dftr,dfe,lower.tail=FALSE)

#print the ANOVA values
sprintf("Treatment: df=%d, SS=%.4f, MS=%.4f, F=%.4f, p-value=%.4f",dftr,round(sstr,4),round(mstr,4),round(f,4),round(pvalue,4))
sprintf("Error: df=%d, SS=%.4f, MS=%.4f",dfe,round(sse,4),round(mse,4))
sprintf("Total: df=%d, SS=%.4f",dftot,round(sst,4))

##get this output

Make this ANOVA table

Source	df	SS	MS	F	p-values
Treatment	3	13.3426	4.4775	5.8471	0.0024
Error	35	26.802	0.7658
Total	38	40.2346