Binomial distribution and hypothesis testing (AS)AQA A-Level Maths: Topic test

1. 1
   A shop sells scratch cards. Each card independently wins a prize with probability $0.2$. Marcus buys $8$ cards. Let $X$ be the number of cards that win a prize.

   (a)

   Which distribution can be used to model $X$?

   [1 mark]

   • A$B(8,0.8)$
   • B$B(8,0.2)$
   • C$B(0.2,8)$
   • D$B(2,0.8)$

   (b)

   What is $P(X=2)$?

   [1 mark]

   • A$0.294$
   • B$0.0105$
   • C$0.797$
   • D$0.2$

   (c)

   Find the probability that at least one of the cards wins a prize.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A manufacturer claims that 10% of its phone chargers are faulty. An inspector believes that the proportion $p$ of faulty chargers is greater than this. She takes a random sample of $20$ chargers, counts the number $X$ that are faulty, and tests at the 5% significance level. When $p=0.1$, $P(X\le3)=0.8670$ and $P(X\le4)=0.9568$.

   (a)

   Which pair of hypotheses should she use?

   [1 mark]

   • A$H_0:p=0.1$, $H_1:p<0.1$
   • B$H_0:p=0.1$, $H_1:p\ne0.1$
   • C$H_0:p=0.1$, $H_1:p>0.1$
   • D$H_0:p>0.1$, $H_1:p=0.1$

   (b)

   What is the critical region for the test?

   [1 mark]

   • A$X\ge3$
   • B$X\ge4$
   • C$X\le4$
   • D$X\ge5$

   (c)

   State the actual significance level of the test, and explain what it represents.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A university reports that 30% of its graduates move abroad within a year. A careers adviser thinks that the proportion is higher this year. She chooses $20$ graduates at random and finds that $10$ of them have moved abroad within a year. She tests at the 5% significance level.

   (a)

   State suitable hypotheses and find the probability of obtaining a result at least as extreme as $10$, assuming that the university's figure is correct.

   [3 marks]

   (b)

   Complete the test and state your conclusion in context. Explain whether the result proves that the proportion has risen.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A coffee chain says that 20% of its customers use its loyalty app. A branch manager believes that the proportion at her branch is lower. She takes a random sample of $30$ customers and records the number $X$ who use the app. She will test at the 5% significance level.

   (a)

   State the hypotheses, and find the critical region for the test and the actual significance level.

   [6 marks]

   (b)

   In the sample, $3$ customers use the app. (i) Carry out the test and state your conclusion in context. (ii) The manager says, 'This proves that exactly 20% of my customers use the app.' Comment. (iii) State one assumption needed for the binomial model and one way the sampling could break it.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A call centre finds that each call it makes is answered with probability $0.35$. An operator makes $12$ calls. Let $Y$ be the number of calls that are answered.

   (a)

   Which assumption is needed to model $Y$ as $B(12,0.35)$?

   [1 mark]

   • AExactly $12$ calls are answered
   • BThe probability of an answer changes after each call
   • CEach call has three possible outcomes
   • DThe calls are independent and each has the same probability of being answered

   (b)

   What is $P(Y=4)$?

   [1 mark]

   • A$0.151$
   • B$0.237$
   • C$0.35$
   • D$0.0424$

   (c)

   Find the probability that at most $2$ calls are answered.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A die manufacturer wants to test whether the probability $p$ of a particular die showing a six differs from $\frac{1}{6}$. The die is rolled $40$ times and the number of sixes, $X$, is recorded. The test is two-tailed at the 10% significance level, with 5% in each tail. When $p=\frac16$: $P(X\le2)=0.0274$, $P(X\le3)=0.0811$, $P(X\ge11)=0.0584$ and $P(X\ge12)=0.0261$.

   (a)

   Which pair of hypotheses should be used?

   [1 mark]

   • A$H_0:p=\frac16$, $H_1:p\ne\frac16$
   • B$H_0:p=\frac16$, $H_1:p>\frac16$
   • C$H_0:p\ne\frac16$, $H_1:p=\frac16$
   • D$H_0:p=\frac16$, $H_1:p<\frac16$

   (b)

   What is the critical region?

   [1 mark]

   • A$X\le3$ or $X\ge11$
   • B$X\le2$ or $X\ge11$
   • C$X\le2$ or $X\ge12$
   • D$X\le3$ or $X\ge12$

   (c)

   Find the actual significance level of the test.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A manufacturer says that 90% of its batteries last at least $500$ hours. A consumer group suspects that the proportion is lower. It tests a random sample of $20$ batteries and finds that $14$ last at least $500$ hours. The group tests at the 1% significance level.

   (a)

   State suitable hypotheses and find the probability of obtaining a result at least as extreme as $14$ if the manufacturer's claim is correct.

   [3 marks]

   (b)

   Complete the test and state your conclusion in context. Explain what would have happened if the group had used the 5% significance level instead.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A bank says that 12% of its online transactions are flagged for review. After a software upgrade, the fraud team thinks that the proportion flagged may have changed. The team takes a random sample of $50$ transactions and lets $X$ be the number flagged. It tests at the 5% significance level, using a two-tailed test with 2.5% in each tail.

   (a)

   State the hypotheses and find the critical region for the test. Find the actual significance level and explain what it means.

   [6 marks]

   (b)

   In the sample, $11$ transactions are flagged. (i) Carry out the two-tailed test and state your conclusion in context. (ii) The team had instead decided before sampling that it only suspected an increase, using $H_1:p>0.12$ at the 5% level. Show that the conclusion would then differ, and explain why the alternative hypothesis must be chosen before the sample is taken.

   [6 marks]

   Total for question 8: 12 marks

End of questions