Further Statistics 1: Type I and Type II errorsAQA A-Level Further Maths: Topic test

1. 1
   A coach company states that the probability that one of its coaches arrives late is $0.1$. An inspector checks a random sample of 20 coaches and records the number, $X$, that arrive late. She tests $\mathrm{H}_0:p=0.1$ against $\mathrm{H}_1:p>0.1$ and rejects $\mathrm{H}_0$ if $X\ge5$.

   (a)

   Which statement describes a Type I error in this context?

   [1 mark]

   • AThe inspector concludes that the probability of a coach being late is $0.1$ when it is in fact greater than $0.1$.
   • BThe inspector concludes that the probability of a coach being late is less than $0.1$ when it is in fact $0.1$.
   • CThe inspector concludes that the probability of a coach being late is greater than $0.1$ when it is in fact $0.1$.
   • DThe inspector counts $5$ or more late coaches when fewer than $5$ were late.

   (b)

   Find the probability that the inspector makes a Type I error.

   [1 mark]

   • A$0.0432$
   • B$0.1330$
   • C$0.9568$
   • D$0.0113$

   (c)

   In fact the probability that a coach is late is $0.25$. Find the probability that the inspector makes a Type II error.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The number of meteors visible per hour on a clear August night is modelled by $Y\sim\mathrm{Po}(\lambda)$. An astronomer claims that $\lambda=4$. A sceptic observes for one hour and tests $\mathrm{H}_0:\lambda=4$ against $\mathrm{H}_1:\lambda>4$. She rejects $\mathrm{H}_0$ if $Y\ge9$.

   (a)

   Which statement describes a Type II error in this context?

   [1 mark]

   • AThe sceptic sees $9$ or more meteors, so rejects $\mathrm{H}_0$, when the true rate is $4$ per hour.
   • BThe sceptic sees $9$ or more meteors, so rejects $\mathrm{H}_0$, when the true rate is greater than $4$ per hour.
   • CThe sceptic sees $8$ or fewer meteors, so does not reject $\mathrm{H}_0$, when the true rate is $4$ per hour.
   • DThe sceptic sees $8$ or fewer meteors, so does not reject $\mathrm{H}_0$, when the true rate is greater than $4$ per hour.

   (b)

   In fact $\lambda=6$. What is the probability that the sceptic makes a Type II error?

   [1 mark]

   • A$0.1528$
   • B$0.8472$
   • C$0.9786$
   • D$0.0214$

   (c)

   Calculate the probability that the sceptic makes a Type I error.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The daily output of a solar panel, in kWh, is normally distributed with standard deviation $1.2$ kWh. The supplier claims that the mean daily output is $8.0$ kWh. A customer takes a random sample of 16 days and tests $\mathrm{H}_0:\mu=8.0$ against $\mathrm{H}_1:\mu<8.0$. She rejects $\mathrm{H}_0$ if the sample mean $\bar X<7.5$.

   (a)

   Find the probability that the customer makes a Type I error.

   [3 marks]

   (b)

   The true mean daily output is $7.4$ kWh. Find the probability of a Type II error and the power of the test.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A school council states that $40\%$ of students support a longer lunch break. A journalist asks a random sample of 12 students and records the number, $X$, who support it. She carries out a two-tailed test of $\mathrm{H}_0:p=0.4$ against $\mathrm{H}_1:p\ne0.4$ at the $5\%$ significance level, using the largest critical region in each tail that has probability at most $2.5\%$.

   (a)

   Find the critical region for the test and state the actual probability of a Type I error.

   [6 marks]

   (b)

   In fact $60\%$ of the students support a longer lunch break. Find the power of the test. Comment on the power and suggest how it could be increased.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A pharmacist states that the probability that a patient taking a certain drug suffers a side effect is $0.05$. In a trial, 40 randomly chosen patients take the drug and the number, $X$, who suffer a side effect is recorded. A regulator tests $\mathrm{H}_0:p=0.05$ against $\mathrm{H}_1:p>0.05$ and rejects $\mathrm{H}_0$ if $X\ge5$.

   (a)

   Which statement about the probability of a Type I error in this test is correct?

   [1 mark]

   • AIt is $P(X\le4)$ calculated using $p=0.05$, which is $0.9520$.
   • BIt is $P(X\ge5)$ calculated using $p=0.05$, which is $0.0480$.
   • CIt is exactly $0.05$, because that is the value of $p$ in $\mathrm{H}_0$.
   • DIt cannot be found without knowing the true value of $p$.

   (b)

   The regulator changes the critical region to $X\ge4$. How does this affect the probabilities of the two types of error, when the true value of $p$ is greater than $0.05$?

   [1 mark]

   • AThe probabilities of both types of error increase.
   • BThe probabilities of both types of error decrease.
   • CThe probability of a Type I error decreases and the probability of a Type II error increases.
   • DThe probability of a Type I error increases and the probability of a Type II error decreases.

   (c)

   Calculate the probability of a Type I error when the critical region is $X\ge4$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The number of bees entering a hive in one minute is modelled by $W\sim\mathrm{Po}(\lambda)$. A beekeeper states that $\lambda=7$ in summer. A researcher believes the rate is lower after pesticide was used nearby. She counts the bees entering in one minute and tests $\mathrm{H}_0:\lambda=7$ against $\mathrm{H}_1:\lambda<7$, rejecting $\mathrm{H}_0$ if $W\le3$.

   (a)

   Find the probability that the researcher makes a Type I error.

   [1 mark]

   • A$0.0818$
   • B$0.0296$
   • C$0.1730$
   • D$0.9182$

   (b)

   In fact $\lambda=4$. Find the probability that the researcher makes a Type II error.

   [1 mark]

   • A$0.4335$
   • B$0.9182$
   • C$0.5665$
   • D$0.0818$

   (c)

   The power of the test when $\lambda=4$ is $0.4335$. Explain what this value means in context.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The time taken to complete a laboratory assay is normally distributed with standard deviation $2.5$ minutes. A technician claims that the mean time is $40$ minutes. A manager takes a random sample of 25 assays and tests $\mathrm{H}_0:\mu=40$ against $\mathrm{H}_1:\mu\ne40$. She rejects $\mathrm{H}_0$ if the sample mean $\bar X<39$ or $\bar X>41$.

   (a)

   Find the probability that the manager makes a Type I error.

   [3 marks]

   (b)

   The true mean time is $41.5$ minutes. Find the probability of a Type II error and the power of the test.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   Two separate tests are described. Test 1: The number of power cuts per month in a town is modelled by $V\sim\mathrm{Po}(\lambda)$. The council states that $\lambda=6$. After an upgrade, an engineer observes one month and tests $\mathrm{H}_0:\lambda=6$ against $\mathrm{H}_1:\lambda<6$ at the $5\%$ significance level, using the largest critical region with probability at most $5\%$. Test 2: The distance an electric scooter travels on a full charge, in km, is normally distributed with standard deviation $3$ km. The manufacturer claims that the mean distance is $40$ km. A consumer group tests $\mathrm{H}_0:\mu=40$ against $\mathrm{H}_1:\mu<40$ using the mean $\bar X$ of a random sample of 36 scooters. It rejects $\mathrm{H}_0$ if $\bar X<c$, where $c$ is chosen so that the probability of a Type I error is $0.01$.

   (a)

   For Test 1, find the critical region and the actual probability of a Type I error. Then find the probability of a Type II error when in fact $\lambda=3$.

   [6 marks]

   (b)

   For Test 2, find the value of $c$. Hence find the power of the test when the true mean distance is $38.5$ km.

   [6 marks]

   Total for question 8: 12 marks

End of questions