##### Lesson 91: One-Sample Hypothesis Tests in R #####

# Setting the working directory 
setwd("path to your folder")

## Data Entry ##

# FORD MPG - reported # 
x = c(21.7, 29, 28.1, 31.5, 24, 21.5, 28.7, 29)

# Compared to Baseline
mu = 26.2
rangex = c(22,32)
sigma = diff(range_ford)/6
n = length(x)
p = 1/n


## Hypothesis Test on the Mean ##

# T-Test #
  xbar = mean(x)
  s = sd(x)
  to = (xbar-mu)/(s/sqrt(n))
  
  alpha = 0.05
  pvalue_ttest = pt(to,df=(n-1))
  tcritical = qt(alpha,df=(n-1))

  # using t-Test function in R #
  t.test(x,alternative = "less", mu = mu, conf.level = 0.95)

  # Visual Perspective # 
  library(plotrix)
  null_t = rt(100000,df=(n-1))
  plot(density(null_t),xlim=c(-4,4),font=2,font.lab=2,xlab="",main="Null Distribution Assuming Ho is True")
  abline(h=0)
  ablineclip(v = tcritical, y1 = 0, y2 = dt(tcritical,df=(n-1)), lty=1, lwd=2,col = "red")
  ablineclip(h = 0, x1 = -4, x2 = tcritical, lty=1, lwd=2,col = "red")
  points(to,0.01,pch=15,col="blue",cex=2)

# Sign-Test #
  splus = length(which(x>mu))
  
  round(dbinom(0:n,n,prob=0.5),4)
  round(cumsum(dbinom(0:n,n,prob=0.5)),4)
  
  pvalue_signtest = sum(dbinom(0:splus,n,prob=0.5))

  # Using Binom Test function in R 
  binom.test(splus,n,p=0.5,alternative = "less")

  # Visual Perspective # 
  y1 = c("-","+","+","+","-","-","+","+")
  
  stripchart(x,ylim=c(0.99,1.5),font=2,font.lab=2)
  text(x,rep(1.05,length(x)),y1)
  ablineclip(v = mu, y1 = 0, y2 = 1.1, lty=1, lwd=2,col = "red")
  
  plot(0:n,dbinom(0:n,n,prob=0.5),type="o",font=2,font.lab=2,xlab="k",ylab="P(S+ = k)")
  cord.x <- c(splus,seq(0,splus),splus) 
  cord.y <- c(0, dbinom(0:splus,n,prob=0.5), 0) 
  polygon(cord.x,cord.y,col="grey")
  text(3,0.1,"P(S+ <= 5) = 0.855")


# Bootstrap Test #
  N = 10000
  xmean_null = matrix(NA,nrow=N,ncol=1)

  for (i in 1:N)
  {
    xboot = sample(x,n,replace=T)
    xmean_null[i,1] = mean(xboot)
  }
  
  hist(xmean_null,col="pink",xlab="Mean of the Distribution",font=2,font.lab=2,main="Null Distribution Assuming Ho is True")
  points(mu,10,pch=15,cex=2,col="blue")
  abline(v=c(quantile(xmean_null,0.95)),col="black",lwd=2,lty=2)
  pvalue_bootstrap = length(which(xmean_null>mu))/N


## Hypothesis Test on the Variance ##

# Chi-square Test 
  chi0 = ((n-1)*s^2)/sigma^2
  pvalue_chi0 = pchisq(chi0,df=(n-1))
  
  N = 100000
  
  nulldist = rchisq(N,df=(n-1))
  plot(density(nulldist),xlim=c(0,40),xlab="",font=2,font.lab=2,main="Null Distribution assuming Ho is True")
  abline(h=0)
  chilimit_right = qchisq(0.975,df=(n-1))
  chilimit_left = qchisq(0.025,df=(n-1))
  
  ablineclip(v = chilimit_right, y1 = 0, y2 = dchisq(chilimit_right,df=(n-1)), lty=1, lwd=2,col = "red")
  ablineclip(h = 0, x1 = chilimit_right, x2 = 24, lty=1, lwd=2,col = "red")
  ablineclip(v = chilimit_left, y1 = 0, y2 = dchisq(chilimit_left,df=(n-1)), lty=1, lwd=2,col = "red")
  ablineclip(h = 0, x1 = 0, x2 = chilimit_left, lty=1, lwd=2,col = "red")
  
  points(chi0,0.002,pch=15,col="blue",cex=2)
  
# Bootstrap Test for Standard Deviation #
  N = 10000
  xsd_null = matrix(NA,nrow=N,ncol=1)
  
  for (i in 1:N)
  {
    xboot = sample(x,n,replace=T)
    xsd_null[i,1] = sd(xboot)
  }
  
  hist(xsd_null,col="pink",xlab="Standard Deviation of the Distribution",font=2,font.lab=2,main="Null Distribution Assuming Ho is True")
  points(sigma,10,pch=15,cex=2,col="blue")
  abline(v=c(quantile(xsd_null,0.025),quantile(xsd_null,0.975)),col="black",lwd=2,lty=2)
  pvalue_bootstrap = length(which(xsd_null>sigma))/N
  
  
## Hypothesis Test on the Proportion ##
  ncars = length(which(x<22))
  plot(0:n,dbinom(0:n,n,prob=p),type="o",xlab="Number of cars with MPG less than the range",ylab="P(X=x)",font=2,font.lab=2)
  x_at_alpha = qbinom(0.95,n,prop) # approx quantile for 10% rejection 
  #abline(v=x_at_alpha)
  
  cord.x <- c(x_at_alpha,seq(x_at_alpha,n),x_at_alpha) 
  cord.y <- c(0, dbinom(x_at_alpha:n,n,prob=p), 0) 
  polygon(cord.x,cord.y,col="pink")
  points(ncars,0.002,pch=15,col="blue",cex=2)
  pval = sum(dbinom(ncars:n,n,prob=p))
  print(pval)
  
# Bootstrap Test #
  N = 10000
  xprop_null = matrix(NA,nrow=N,ncol=1)
  
  for (i in 1:N)
  {
    xboot = sample(x,n,replace=T)
    xprop_null[i,1] = length(which(xboot<22))/n
  }
  
  hist(xprop_null,col="pink",xlab="Proportion of the Distribution",font=2,font.lab=2,main="Null Distribution Assuming Ho is True")
  points(prop,10,pch=15,cex=2,col="blue")
  abline(v=c(quantile(xprop_null,0.95)),col="black",lwd=2,lty=2)
  pval=length(which(xprop_null>prop))/N