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

1. 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.

   (a)

   Which of the following gives the correct hypotheses, where $p$ is the probability that a card wins a prize?

   [1 mark]

   • A$\mathrm{H}_0:p=0.2,\ \mathrm{H}_1:p<0.2$
   • B$\mathrm{H}_0:p=0.2,\ \mathrm{H}_1:p>0.2$
   • C$\mathrm{H}_0:p=0.2,\ \mathrm{H}_1:p\neq0.2$
   • D$\mathrm{H}_0:p<0.2,\ \mathrm{H}_1:p=0.2$

   (b)

   Under $\mathrm{H}_0$, find the probability of the first win being on the $12$th card or later, $\mathrm{P}(X\geq12)$.

   [1 mark]

   • A$0.0687$
   • B$0.9141$
   • C$0.1074$
   • D$0.0859$

   (c)

   Find the critical region for the test.

   [2 marks]

   Total for question 1: 4 marks

2. 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.

   (a)

   Under $\mathrm{H}_0:p=0.02$, find the probability of the first defective screen being found at the $2$nd screen or earlier, $\mathrm{P}(X\leq2)$.

   [1 mark]

   • A$0.9604$
   • B$0.0196$
   • C$0.0396$
   • D$0.0400$

   (b)

   Which of the following is the critical region for the test at the $5\%$ significance level?

   [1 mark]

   • A$X\leq1$
   • B$X\leq2$
   • C$X\leq3$
   • D$X\geq3$

   (c)

   Write down the conclusion of the test, in context.

   [2 marks]

   Total for question 2: 4 marks

3. 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.

   (a)

   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]

   (b)

   Find the critical region for the test, and state the actual significance level.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   In a video game, the designer says that a player completes a level at each attempt with probability $0.3$, independently of other attempts. A teacher thinks that the probability for her pupils is different. She records the number of attempts $X$ one pupil makes up to and including the first attempt that completes the level, and uses a two-tailed test at the $10\%$ significance level with $5\%$ in each tail.

   (a)

   (i) State suitable hypotheses and the distribution of $X$ under $\mathrm{H}_0$.
   (ii) Find the critical region for the test and the actual significance level.

   [6 marks]

   (b)

   The pupil needs $11$ attempts to complete the level.
   (i) Carry out the test.
   (ii) Explain why the model with a constant probability of success might not be suitable for a pupil learning the level, and what this means for your conclusion.

   [6 marks]

   Total for question 4: 12 marks

End of questions