N = 100
n = 10

uzorak = c(2,5,1,4,4,3,2,5,2,3)

t_ocena = N*mean(uzorak)
t_ocena

D_t_ocena = N*(N-n)*var(uzorak)/n
D_t_ocena

xn = mean(uzorak)
xn

D_xn_ocena = (N-n)*var(uzorak)/(N*n)
D_xn_ocena
D_t_ocena/N^2

n
# <30, studentov kvantil
alpha = 1 - 0.9
z_student = qt(1-alpha/2, n-1)
I_poverenja_za_t_90 = c(t_ocena-z_student*sqrt((N-n)*N*var(uzorak)/n), t_ocena+z_student*sqrt((N-n)*N*var(uzorak)/n))
I_poverenja_za_t_90

I_poverenja_za_xn_90 = c(xn-z_student*sqrt((N-n)*var(uzorak)/(N*n)), xn+z_student*sqrt((N-n)*var(uzorak)/(N*n)))
I_poverenja_za_xn_90
I_poverenja_za_t_90/N