TU Wien:Statistik und Wahrscheinlichkeitstheorie UE (Levajkovic)/Übungen 2023W/HW11.2

Rolling die, part 2

Test the null hypothesis that the die is fair with a ${\mathcal {X}}^{2}$-test on the 5%-significance level, without using R-command chisq.test().

(a) What are the observed (absolute) frequencies?

(b) What are the expected frequencies under the null hypothesis?

(c) What is the value $x^{2}$ of the ${\mathcal {X}}^{2}$-statistic?

(d) How is the ${\mathcal {X}}^{2}$-statistic $X^{2}$ distributed under the null hypothesis (in the context of the associated model)

(e) What is the rejection area $R$?

(f) Do you reject the null hypothesis?

(g) Compute the $p$-value.

(h) Interpret your result.

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Lösung von Tutorin

(a)

x <- as.vector(b); x

blue   naranja rot     verte
35     58      25      32

(b)

e <- rep(n/4,4); e

37.5    37.5    37.5    37.5

(c)

x_sq <- as.vector(b); x

16.34667

(d)

$$X^{2}\sim {\mathcal {X}}^{2}(3)$$

(e)

R <- c(qchisq(0.95, df=3), Inf); R

7.814728        Inf

(f)

Yes, since $x^{2}\in R$

(g)

P <- 1-pchisq(x_sq, df=3); P

0.0009627063

(h)

The data are difficult to reconcile with the claim. If the null hypothesis is correct, then in less than one case in 1000 we observe a discrepancy that is at least as extreme as in the data.