# Enter data
x <- c(1.2, 2.4, 1.3, 1.3, 0.9, 1.0, 1.8, 0.8, 4.6, 1.4)



## T-test 
n <- length(x) # sample size
# Compute 'tobs' - the observed test statistic
tobs <- (mean(x) - 0) / (sd(x) / sqrt(n))
# Compute the p-value as a tail-probability
# in the relevant t-distribution:
2 * (1 - pt(abs(tobs), df = n-1))


## Den nemme måde:
t.test(x, mu = 0)
t.test(x)


t.test(x, mu = 2) ## her testes H_0: mu = 2. 

### Konfidensinterval

t.test(x, conf.level = 0.95)






### Højde af studerende 
x <- c(168,161,167,179,184,166,198,187,191,179)
t.test(x, conf.level = 0.95) ## tester H_0 : mu = 0

t.test(x, mu = 180, conf.level = 0.95) ## tester H_0 : mu = 180




#### Simulation



N = 10 ## antal obs.
K = 500 ## antal gentagelser

mu <- 4 
sigma <- 3

cis <- numeric(K)
for (i in 1:K)
  cis[i] <- t.test(rnorm(N, mu, sigma), mu = 4)$p.val 
## H0 er sand, altså er p-værdien tilfældigt mellem 0 og 1. 

hist(cis)
mean(cis < 0.05) # På 5% sig. niveau afvises ca. 5%


## Samme, men med t-fordeling og kritisk værdi
tv <- numeric(K)

for (i in 1:K)
  tv[i] <- t.test(rnorm(N, mu, sigma), mu = 4)$statistic ## t-værdi

hist(tv)

qt(0.975, df = 9) # kritisk værdi for testet
mean( abs(tv) > qt(0.975, df = 9))





# Studenterhøjder
x <- c(168,161,167,179,184,166,198,187,191,179)
xr <- rnorm(100, mean(x), sd(x))
hist(xr, xlab = "Height", main = "", freq = FALSE, ylim = c(0, 0.032))
lines(seq(130, 230, 1), dnorm(seq(130, 230, 1), mean(x), sd(x)))


### QQ-plot
qqnorm(x)
qqline(x)


## 
xr <- rnorm(1000, mean(x), sd(x))
qqnorm(xr)
qqline(xr)




## Reading in the data
radon <- c(2.4, 4.2, 1.8, 2.5, 5.4, 2.2, 4.0, 1.1, 1.5, 5.4, 6.3,
           1.9, 1.7, 1.1, 6.6, 3.1, 2.3, 1.4, 2.9, 2.9)
## Histogram and q-q plot of data
par(mfrow = c(1,2))
hist(radon)
qqnorm(radon)
qqline(radon)

xr <- rnorm(20)
qqnorm(xr)
qqline(xr)