Estimations Tests

INFERENCE ABOUT A POPULATION

*Nous allons simuler une distribution de Revenus dans une population de taille \(N=1000000\).

*On va supposer que le revenu moyen \(\mu=2500\) et l’ecart type \(\sigma=300\).

*On va utiliser pour cela la Loi Log_Normale pour generer nos donnes.

• Calcule en \(https://www.wolframalpha.com/\)solve[exp(m+s^2/2)==2500&& (exp(s^{2)-1)exp(2m+s}2)==(300^2),{m,s}

set.seed(123) #Pour qu'on ait les meme data
Income=rlnorm(1000000,7.8169,0.119571)
N=length(Income)
mu=mean(Income) # mu
sigma=sd(Income) # sigma
hist(Income)

replace = Tirage sans remise

set.seed(123)
Echantillion=sample(Income,50,replace = FALSE)
n=length(Echantillion)
xbar=mean(Echantillion)
s=sd(Echantillion)
n
xbar
s

• On va tirer plusiers fois la moyenne \(\bar{x}\) de l’echantillion e regarder sa distribution. *Cela nous permet d’entrevoir la distribution d’echantiollnnage de \[\bar{X}\].

xbars=c()
for(i in 1:1000)
{
xbars=c(xbars,mean(sample(Income,50,replace = FALSE)))
}
hist(xbars)

• Lhistograme ressemble a une cloche en raison du Theoreme de la Limite Centrale.

sd(xbars) ## Cette valeur est Ecart Type de x bar, est le sigma /racine de n.
sigma/sqrt(50)

EXERCICE

• On preleve un echantillion de taille \(n=20\) dans une population grande et pas “trop non normale”.

• on trouve \(\bar{x}=4.6\) et un Ecart-Type \(sd=2.1\).

• Donez une intervalle de confiance pour la moyenne \(\mu\).

*on a Loi de Stdent $=frac(xbar - ) (S/sqrt n)

qt(0.025,19,lower.tail = FALSE)
4.6-2.09*2.1/sqrt(20) ## S'appelle LCL_Lower Confidence Limit
4.6+2.09*2.1/sqrt(20)  ## S'appelle UCL_Upper Confidence Limit

EXAMPLE 12.2 Tax Collected from Audited Returns (Import Data Xm 12-02)

*In 2007 (the latest year reported), 134,543,000 tax returns were filed in the United States (Source: Statistical Abstract of the United States, 2009, Table 463). The Internal Revenue Service (IRS) examined 1.03% or 1,385,000 of them to determine if they were correctly done. To determine how well the auditors are performing, a random sample of these returns was drawn and the additional tax was reported, which is listed next. Estimate with 95% confidence the mean additional income tax collected from the 1,385,000 files audited. (Adapted from U.S. Internal Revenue Service, IRS Data Book, annual, Publication 55B.)

hist(Xm12_02$Taxes)
n = length(Xm12_02$Taxes)
xbar = mean(Xm12_02$Taxes)
s = sd(Xm12_02$Taxes)

xbar-qt(0.025,n-1, lower.tail = FALSE)*s/sqrt(n) ## S'appelle LCL_Lower Confidence Limit
xbar+qt(0.025,n-1, lower.tail = FALSE)*s/sqrt(n) ## S'appelle UCL_Upper Confidence Limit