TU Wien:Statistik und Wahrscheinlichkeitstheorie UE (Levajkovic)/Übungen 2023W/HW10.2

Elections

Anna is one of the candidates in the upcoming elections for the student representative at the TU Wien. Her team wants to determine whether or not more than $3/4$ of all students would vote for her. In a random poll sampling of $n=137$ students, the responses $x_{1},x_{2},\dots ,x_{n}$ were collected (each is 1 or 0, if they would vore for her or not). Among them, there were observed

131 ”yes” responses, i.e. $\sum _{i=1}^{n}x_{i}=131$. Perform a hypothesis test with the level of significance $\alpha =0.01$ and state your conclusion based on the information given.

Dieses Beispiel ist als solved markiert. Ist dies falsch oder ungenau? Aktualisiere den Lösungsstatus (Details: Vorlage:Beispiel)

Lösungsvorschlag von Lessi

--Lessi 2024-02-07T13:04:11Z

Let $X_{1},\dots ,X_{n}$ be i.i.d RV with $X_{i}\sim ber(p)$ and $n=137$. The significance level is $\alpha =0.01$ and the hypothesis $H_{0}:p\geq 3/4=p_{0}$

p0 <- 3 / 4
n <- 137
alpha <- 0.01
h <- 131 / n
se <- sqrt(h * (1 - h) / n)
z <- (h - p0) / se

h
se
z

We will perform a left-sided test $p=\mathbb {P} _{H_{0}}(Z\leq z)=\Phi (z)$

p <- pnorm(z)
p

$p>\alpha$, so we do not reject the null hypothesis

Check again with prop.test

prop.test(131, 137, p = 3 / 4, alternative = "less")