TU Wien:Statistik und Wahrscheinlichkeitstheorie UE (Levajkovic)/Übungen 2023W/HW10.1

One-sample test for proportions

You are at a fair and a lottery booth showman claims that every second lottery ticket is a win. You observe the hustle and bustle around the lottery booth for a while and count that from 58 tickets sold 17 won. Does this observation let you doubt the claim?

(a) Calculate the ${\textstyle p}$-value in the context of the (two-sided) one-sample test for proportions.

(b) What do you answer the showman?

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Lösungsvorschlag von Friday

--Friday Sa 30 Jan 2021 18:15:06 CET

1a)

$${\begin{aligned}H_{0}:p&=p_{0}=0.5\\h&={\frac {17}{58}}\approx 0.2931\\se_{h}&={\sqrt {\frac {h(1-h)}{n}}}\approx 0.0597\\|h-p_{0}|&\approx 3.461\cdot se_{h}\\p&=\mathbb {P} _{H_{0}}(|Z|\geq |z|)\approx 5.37\cdot 10^{-4}\end{aligned}}$$

1b)

The observed win-rate is extremly unlikley for the claim and has a chance lower than one in thousand.