Language of hypothesis testingAQA A-Level Maths: Subtopic test

1. 1
   A seed company claims that 60% of its seeds germinate. A gardener suspects that the true percentage is lower. She plants 20 seeds and records the number $X$ that germinate. She will test her suspicion at the 5% significance level, assuming that $X\sim B(20,p)$, where $p$ is the probability that a seed germinates. For reference, $P(X\le7)=0.0210$ and $P(X\le8)=0.0565$ when $p=0.6$.

   (a)

   Which pair of hypotheses should she use?

   [1 mark]

   • A$H_0:p=0.6,\ H_1:p\ne0.6$
   • B$H_0:p=0.6,\ H_1:p<0.6$
   • C$H_0:p<0.6,\ H_1:p=0.6$
   • D$H_0:p=0.6,\ H_1:p>0.6$

   (b)

   What is the largest critical region for this test?

   [1 mark]

   • A$X\le8$
   • B$X\le6$
   • C$X\ge13$
   • D$X\le7$

   (c)

   State the critical value of the test and write down the acceptance region.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A spinner has four equal-looking sections, one of which is red. The manufacturer says that the probability of the spinner landing on red is $0.25$. A teacher thinks the spinner may be biased, in either direction. She spins it 20 times and records the number $Y$ of reds, using $Y\sim B(20,p)$ and a 10% significance level. When $p=0.25$: $P(Y\le1)=0.0243$, $P(Y\le2)=0.0913$, $P(Y\ge8)=0.1018$ and $P(Y\ge9)=0.0409$.

   (a)

   How much probability is placed in each tail when finding the critical region?

   [1 mark]

   • A$0.05$
   • B$0.10$
   • C$0.20$
   • D$0.025$

   (b)

   What is the critical region for the test?

   [1 mark]

   • A$Y\le2$ or $Y\ge8$
   • B$Y\le1$ or $Y\ge8$
   • C$Y\le1$ or $Y\ge9$
   • D$Y\le0$ or $Y\ge10$

   (c)

   The teacher says: "A 10% significance level means there is a 10% chance that the spinner is biased." Explain why this is incorrect.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A café owner believes that 30% of customers order a cold drink. She runs an advert that is intended to increase this proportion. Afterwards she picks 15 customers at random and finds that 9 of them order a cold drink. Let $X$ be the number of cold drinks ordered by 15 customers, and test at the 5% significance level whether the proportion has increased.

   (a)

   Write down the null and alternative hypotheses, and state the distribution of $X$ if the null hypothesis is true.

   [3 marks]

   (b)

   Find the $p$-value for the result of 9 cold drinks, and state the conclusion of the test in context.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A website claims that 35% of its visitors click on a banner advert. The marketing manager thinks that the proportion has changed. She will record the number $X$ of clicks from a random sample of 30 visitors, using $X\sim B(30,p)$, and test at the 5% significance level. When $p=0.35$: $P(X\le4)=0.0075$, $P(X\le5)=0.0233$, $P(X\le6)=0.0586$, $P(X\ge15)=0.0652$, $P(X\ge16)=0.0301$, $P(X\ge17)=0.0124$.

   (a)

   Write down suitable hypotheses and find the critical region for the test.

   [6 marks]

   (b)

   (i) Find the actual significance level of the test.
   (ii) Write down the acceptance region.
   (iii) In the sample, 16 visitors click on the advert. State and justify the conclusion, and explain why the sample proportion being much higher than $0.35$ does not lead to rejection.

   [6 marks]

   Total for question 4: 12 marks

End of questions