Further Statistics 1: Quality of testsEdexcel A-Level Further Maths: Topic test

1. 1
   A basketball coach believes that a player has improved from a free-throw success probability of $0.4$. In a trial, the player takes $15$ throws and $X$ is the number scored. The coach tests $\mathrm{H}_0: p=0.4$ against $\mathrm{H}_1: p>0.4$ and rejects $\mathrm{H}_0$ if $X\geq10$.

   (a)

   What is the size of the test?

   [1 mark]

   • A$0.0950$
   • B$0.0338$
   • C$0.9662$
   • D$0.2131$

   (b)

   What is a Type I error in this test?

   [1 mark]

   • AConcluding that $p=0.4$ when the player has really improved
   • BScoring $10$ or more throws when the player has really improved
   • CConcluding that the player has improved when $p$ is really $0.4$
   • DScoring $9$ or fewer throws when $p$ is really $0.4$

   (c)

   Find the probability of a Type II error when the player's true success probability is $0.6$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The number of complaints received by a council helpline in one week is modelled by $\mathrm{Po}(\lambda)$. A manager believes that $\lambda$ has increased from $6$. She tests $\mathrm{H}_0:\lambda=6$ against $\mathrm{H}_1:\lambda>6$ using one week's count $X$, and rejects $\mathrm{H}_0$ if $X\geq11$.

   (a)

   What is the probability of a Type I error?

   [1 mark]

   • A$0.0426$
   • B$0.0839$
   • C$0.9574$
   • D$0.0201$

   (b)

   What is the power of the test when the true value of $\lambda$ is $8$?

   [1 mark]

   • A$0.8159$
   • B$0.0426$
   • C$0.5830$
   • D$0.1841$

   (c)

   Find the probability of a Type II error when the true value of $\lambda$ is $9$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The length, in mm, of a bolt made by a machine is normally distributed with mean $\mu$ and standard deviation $0.5$. The machine is set to make bolts with $\mu=40$. A sample of $25$ bolts is taken and the sample mean is $\bar X$. A two-tailed test of $\mathrm{H}_0:\mu=40$ against $\mathrm{H}_1:\mu\neq40$ is carried out at the $5\%$ significance level.

   (a)

   Find the critical region for $\bar X$.

   [3 marks]

   (b)

   The mean of the lengths of the bolts is in fact $40.3$. Find the probability of a Type II error.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A call-centre worker claims that $30\%$ of her calls end in a sale. Her manager suspects that the proportion has risen after training. She monitors $20$ calls, and $X$ is the number that end in a sale, so $X\sim\mathrm{B}(20,p)$. The manager tests $\mathrm{H}_0:p=0.3$ against $\mathrm{H}_1:p>0.3$ and rejects $\mathrm{H}_0$ if $X\geq10$.

   (a)

   Find the size of the test and the power of the test when $p=0.5$. Interpret the power in context.

   [6 marks]

   (b)

   The manager considers changing the critical region to $X\geq11$. Find the new size and the new power when $p=0.5$, and comment on the effect of the change.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A game developer states that a rare item drops with probability $0.05$ each time a boss is defeated. A player suspects that the drop probability $p$ is higher. Let $X$ be the number of boss defeats up to and including the first drop, so $X$ has a geometric distribution. The player tests $\mathrm{H}_0:p=0.05$ against $\mathrm{H}_1:p>0.05$ and rejects $\mathrm{H}_0$ if $X\leq2$.

   (a)

   What is the size of the test?

   [1 mark]

   • A$0.0475$
   • B$0.9025$
   • C$0.0975$
   • D$0.05$

   (b)

   What is the probability of a Type II error when the true value of $p$ is $0.2$?

   [1 mark]

   • A$0.36$
   • B$0.64$
   • C$0.128$
   • D$0.8$

   (c)

   Find the power of the test when $p=0.3$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The lifetime, $X$ hours, of a type of battery is normally distributed with mean $\mu$ and standard deviation $12$. A technician tests $\mathrm{H}_0:\mu=100$ against $\mathrm{H}_1:\mu<100$ using the lifetime of a single battery, and rejects $\mathrm{H}_0$ if $X<80$.

   (a)

   What is the probability of a Type I error?

   [1 mark]

   • A$0.9522$
   • B$0.0228$
   • C$0.1587$
   • D$0.0478$

   (b)

   The critical value is changed from $80$ to $75$. Which statement is correct?

   [1 mark]

   • AThe probability of a Type I error decreases and the probability of a Type II error increases
   • BThe probability of a Type I error increases and the probability of a Type II error decreases
   • CBoth error probabilities decrease
   • DNeither error probability changes

   (c)

   Find the power of the original test when $\mu=70$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A spinner is thought to land on red with probability $p=0.25$. It is spun $20$ times and $X$ is the number of times it lands on red. A two-tailed test of $\mathrm{H}_0:p=0.25$ against $\mathrm{H}_1:p\neq0.25$ has critical region $X\leq1$ or $X\geq10$.

   (a)

   Find the size of the test.

   [3 marks]

   (b)

   The spinner actually lands on red with probability $0.4$. Find the probability of a Type II error.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A supplier of optical cable claims that the number of faults in a $1$ km length follows a Poisson distribution with mean $3.5$. An engineer suspects the mean $\lambda$ is higher. She tests $\mathrm{H}_0:\lambda=3.5$ against $\mathrm{H}_1:\lambda>3.5$ using the number of faults $X$ in one randomly chosen $1$ km length. She wants the critical region of the form $X\geq c$ with the largest possible size that does not exceed $5\%$.

   (a)

   Find the critical region and its size. Find the probability of a Type II error when $\lambda=6$.

   [6 marks]

   (b)

   Find the power of the test when $\lambda=5$ and when $\lambda=8$. Comment on what this shows about the effectiveness of the test.

   [6 marks]

   Total for question 8: 12 marks

End of questions