#Example 2
# Testing hypothesis about the population mean when the
# population standard deviation is unknown

#Set working directory
setwd("~/Dropbox/ISI SCB/Data sets")
# read in csv file
bc <-read.csv("hyp1example2.csv", TRUE)
#assign the list of bacteria counts to the variable x
x <- bc$bacteria

#assumed population mean
mu <- 190
#sample size
n <- length(x)
n
#sample mean 
xbar <- mean(x)
xbar
#sample standard deviation 
s =sd(x)
s

#hypotheses
#H0: mu <= 190
#H1: mu > 190

#test statistic
ts <- (xbar - mu)/(s/sqrt(n))
ts

#upper critical value - 95th percentile (0.95th quantile)
upper_t05_9 <- qt(.95,n-1)
upper_t05_9

#plot of density function of t-distribution
curve(dt(x,n-1), from = -4, to = 4,
      main = 't-Distribution',
      ylab = 'Density', xlab = '')

#location of the test statistic and critical value
abline(v=ts,col="blue")
#location of the lower critical value
abline(v=upper_t05_9,col="red")

#p-value
#plot of density function of t-distribution
curve(dt(x,n-1), from = -4, to = 4,
      main = 't-Distribution',
      ylab = 'Density', xlab = '')
#location of the test statistic 
abline(v=ts,col="blue")

#p-value
#P(t > ts)
pvalue <-pt(ts,n-1,lower.tail = FALSE)
pvalue

#using t.test in Stats R package
t.test(x,
       alternative = "greater",
       mu = 190,
       conf.level = 0.95)





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