Hypothesis test for the parameter of a geometric distributionEdexcel A-Level Further Maths: Mind map

Exam questions on Hypothesis test for the parameter of a geometric distribution

1. A scratch-card company claims that the probability that a card wins a prize is $0.2$, independently for each card. A customer buys cards one at a time and has her first win on the $12$th card. She suspects that the probability of winning is lower than the company claims and carries out a test at the $5\%$ significance level. Let $X$ be the number of cards bought up to and including the first win.
   Find the critical region for the test.2 marks
2. A manufacturer says that the probability that a screen it makes is defective is $0.02$, independently for each screen. An inspector tests screens one at a time and finds the first defective screen at the $2$nd screen tested. She suspects that the probability of a defective screen is higher than $0.02$ and carries out a test at the $5\%$ significance level. Let $X$ be the number of screens tested up to and including the first defective one.
   Write down the conclusion of the test, in context.2 marks
3. A website claims that each visitor clicks on an advert with probability $0.1$, independently of other visitors. An analyst suspects that the probability is lower. She counts the number of visitors $X$ up to and including the first visitor who clicks, and will test the claim at the $5\%$ significance level.
   State suitable hypotheses for the test, where $p$ is the probability that a visitor clicks on the advert, and state the distribution of $X$ under $\mathrm{H}_0$.3 marks