##14

ybars <- c(14.62, 34.12, 33.61, 29.1, 27, 26.97, 26.8) 
sds <- c(5.039, 11.942, 6.582, 4.593, 3.818, 9.010, 5.969)
ns <- c(9, 5, 6, 9, 2, 6, 9) 

I <- length(ns)

SSE <- sum((ns-1)*sds^2)
# a)
 print(sp <- sqrt(SSE/sum(ns-1)) )

# b)  this test should use pooled estimate of s
  
print( t <- (14.62 - 34.12)/sp/sqrt(1/9 + 1/5) )
print(pvalue <- 1 - pt(abs(t), df = sum(ns-1)) )


##15   

 data <- read.csv("Spock.csv", header = TRUE)    
 Y <- data[,1]
 
 var(Y)
 SST <- var(Y)*(length(Y)-1)
c(SST, SSE)
 n <- length(Y)

 
anova.table <- function(SSE, SST, n, I){
  SSB <- SST - SSE
  MSB <- SSB/(I-1)
  MSE <- SSE/(n-I)
  F <- MSB/MSE
  pval <- 1 - pf(F, df1 = I-1, df2 = n-I)
  print("Source   DF      SS      MS     F    P") 
  print(paste("Between", I-1,  round(SSB,1), "  ", round(MSB,1), "  ", round(F,2), "  ",  round(pval,3))) 
  print(paste("Within ", n-I,  round(SSE,1), "  ", round(MSE,1)))
  print(paste("Total  ", n-1,  round(SST,1)))
}

anova.table(SSE, SST, n, I)
  
 SOURCE          DF     SS       MS       F      P 
  BETWEEN         6    1927.1    321.2  6.72    0.000
  WITHIN(Error)  39    1864.5    47.8
  TOTAL          45    3791.5


##17 

 anova.table(35088, 70907, 32, 8)

 Source   DF      SS      MS     F      P
 Between  7     35819    5117    3.5    0.01
 Within   24    35088    1462
 Total    31    70907


##21 

 attach(data)
   lm1 <- lm(PERCENT ~ JUDGE)
 detach(data)

 res.sq <- lm1$res^2
  lm2 <- lm(res.sq ~ data$JUDGE)
 anova(lm2)

 # p-value for Levene's test  = 0.078: not a significant difference in variances
 # 


##23 
data <- read.csv("trex.csv", header = TRUE)       
  attach(data)

boxplot(OXYGEN ~ BONEGRP)
lm3 <- lm(OXYGEN ~ BONEGRP)
anova(lm3)
qqnorm(lm3$resid)   # the QQ plot looks ok
plot(lm3$fit, lm3$resid) 

# p-value for the test is about 1e-06  =>  evidence that the temperatures *are* different throughout the body 
# however, the boxplots show potential difficulty with equal variances assumption

kruskal.test(OXYGEN ~ BONEGRP)     # Kruskal-Wallis test arrives at a similar conclusion

  detach(data)