# INSA - Typst Template
Typst Template for short documents and letters for the french engineering school INSA.

## Example
By default, the template initializes with the `insa-short` show rule, with parameters that you must fill in by yourself.

Here is an example of filled template:
````typst
#import "@preview/silky-letter-insa:0.1.0": *
#show: doc => insa-short(
  author: "<NAME>, <NAME>",
  doc)

#v(15pt)
#align(center, text(size: 22pt, weight: "bold", smallcaps("Probabilités - Annale 2022 (V1)")))
#v(5pt)

#set heading(numbering: "1.")
#show heading.where(level: 2): it => [
  #counter(heading).display()
  #text(weight: "medium", style: "italic", size: 13pt, it.body)

]

= Intervalle de confiance
== Calculer sur l’échantillon une estimation de la moyenne.
$ overline(x_n) = 1/n sum_(i=1)^n x_i = 1885 $

== Calculer sur l’échantillon une estimation de la variance.
$
"Variance biaisée :" s^2 &= 1/n sum_(i=1)^n (x_i - overline(x_n))^2 = 218^2\
"Variance corrigée :" s'^2 &=  n/(n-1) s^2 = 231^2
$

Le bon estimateur est le second.

== Écrire le code R permettant d’évaluer les deux bornes de l’intervalle de confiance du temps d’exécution avec une confiance de 92%.
Nous sommes dans le cas d'une recherche de moyenne avec variance inconnue, l'intervalle sera donc
$ [overline(X) + t_(n-1)(alpha/2) S'/sqrt(n), quad overline(X) + t_(n-1)(1 - alpha/2) S'/sqrt(n)] $
En R, avec l'échantillon nommé `data`, ça donne
```R
data = c(1653, 2059, 2281, 1813, 2180, 1721, 1857, 1677, 1728)
moyenne = mean(data)
s_prime = sqrt(var(data)) # car la variance de R est déjà corrigée
n = 9
alpha = 0.08

IC_min = moyenne + qt(alpha / 2, df = n - 1) * s_prime / sqrt(n)
IC_max = moyenne + qt(1 - alpha / 2, df = n - 1) * s_prime / sqrt(n)
```

Ici on a $[1730, 2040]$.
````

## Notes
This template is being developed by <NAME> from the INSA de Rennes in [this repository](https://github.com/SkytAsul/INSA-Typst-Template).

For now it includes assets from the INSA de Rennes graphic charter, but users from other INSAs can open an issue on the repository with the correct assets for their INSA.

If you have any other feature request, open an issue on the repository as well.

## License
This package is licensed with the [MIT license](https://github.com/SkytAsul/INSA-Typst-Template/blob/main/LICENSE).