#Set working directory
setwd("~/Dropbox/ISI SCB/Data sets")
# read in csv file
ovtm <-read.csv("hyp1example4.csv", TRUE)
#assign the list of overtime values to the variable x
x <- ovtm$overtime

#assumed population variance
sigmasq <- 25
#sample size
n = length(x)
n
#sample standard deviation 
s <- sd(x)
s

#hypotheses
#H0: sigmasq = 25
#H1: sigmasq not= 25

#chi-square test statistic
ts <- (n-1)*(s^2)/sigmasq
ts

#lower critical value - 2.5th percentile (0.025th quantile)
chi025 <- qchisq(.025,n-1)
chi025
 
 #upper critical value - 97.5th percentile (0.975th quantile)
chi975 <- qchisq(.975,n-1)
chi975

#plot of chi-square density function with 
#15 degrees of freedom showing test statistic
#and the critical values
curve(dchisq(x, df = n-1), from = 0, to = 40,
                main = 'Chi Square Distribution:15 df',
                ylab = 'Density', xlab = '')
abline(v=ts, col = "blue")
abline(v=qchisq(.975,df=n-1), col = "red")
abline(v=qchisq(.025,df=n-1), col = "red")


#determination of the p-value
#plot of chi-square density function with 
#15 degrees of freedom
curve(dchisq(x, df = n-1), from = 0, to = 40,
      main = 'Chi Square Distribution:15 df',
      ylab = 'Density', xlab = '')
#location of the test statistic
abline(v=ts,col="blue")


# first determination of the value of the density function
# for the quantile which is the value of the test statistic
ts_density <-dchisq(ts,n-1)
ts_density

abline(h=ts_density, col="red")

#trial and error to determine other qu atile qhich has the same
#density function value as the statistic
dchisq(ts,n-1)
dchisq(9.5,n-1)
dchisq(9.6,n-1)
dchisq(9.7,n-1)
dchisq(9.8,n-1)
dchisq(9.9,n-1)

dchisq(ts,n-1)
dchisq(9.8,n-1)
dchisq(9.75,n-1)
dchisq(9.76,n-1)
dchisq(9.77,n-1)
dchisq(9.78,n-1)
dchisq(9.79,n-1)

dchisq(ts,n-1)
dchisq(9.79,n-1)
dchisq(9.795,n-1)
dchisq(9.796,n-1)

dchisq(ts,n-1)
dchisq(9.7952,n-1)
dchisq(9.7953,n-1)

dchisq(ts,n-1)
dchisq(9.7952,n-1)
dchisq(9.79521,n-1)
dchisq(9.79522,n-1)
dchisq(9.79523,n-1)
dchisq(9.79524,n-1)


#after trial and error, arrive at a value of the
#quantile of the chi-square distribution with 15  
#degrees of freedom which gives approximately the 
#same value of the density function as the test statistic
ts_other = 9.79523
ts_other_density <- dchisq(ts_other,n-1)
ts_other_density

#location of this quantile
abline(v=ts_other,col="green")

#horizontal line showing that the density function associated
#with the test statistic and ts_other are equal
abline(h=ts_density, col="red")


#determination of the p-value
#P(chisq > ts)
pv_up <- pchisq(ts, n-1, lower.tail = FALSE)
pv_up
#P(chisq < ts_other)
pv_lo <- pchisq(ts_other,n-1, lower.tail = TRUE)
pv_lo
#p-value of the test
pvalue = pv_up + pv_lo
pvalue


#check for outliers
boxplot(x)
#check for normality of data set
qqnorm(x)
shapiro.test(x)