4.18 Hypothesis testing: means, proportions, correlation and errorsIB Maths: Applications and Interpretation HL: Subtopic test

1. 1
   A machine is set to fill bags of sugar with a mean mass of $500$ g. The mass of a bag is normally distributed with a known standard deviation of $8$ g. An inspector suspects that the machine is under-filling the bags. She takes a random sample of $36$ bags, which has a mean mass of $497.4$ g, and tests the claim at the $5\%$ significance level.

   (a)

   Which of the following are the correct hypotheses for the inspector's test?

   [1 mark]

   • A$\mathrm{H}_0:\mu=500,\ \mathrm{H}_1:\mu\ne500$
   • B$\mathrm{H}_0:\mu=500,\ \mathrm{H}_1:\mu<500$
   • C$\mathrm{H}_0:\mu<500,\ \mathrm{H}_1:\mu=500$
   • D$\mathrm{H}_0:\bar{x}=500,\ \mathrm{H}_1:\bar{x}<500$

   (b)

   Use your GDC to find the $p$-value of the test.

   [1 mark]

   • A$0.0256$
   • B$0.0512$
   • C$0.974$
   • D$0.373$

   (c)

   State the conclusion of the test, giving a reason for your answer.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A trainer records the $100$ m sprint times of $8$ athletes before and after a training programme. The differences in time, in seconds, calculated as before minus after, are $0.12,\ 0.05,\ -0.03,\ 0.20,\ 0.08,\ 0.15,\ 0.02,\ 0.10$. The differences are normally distributed. The trainer tests, at the $5\%$ significance level, whether the programme reduces the mean sprint time. Let $\mu_d$ be the population mean difference. Use your GDC where helpful.

   (a)

   Which of the following are the correct hypotheses for the trainer's test?

   [1 mark]

   • A$\mathrm{H}_0:\mu_d=0,\ \mathrm{H}_1:\mu_d\ne0$
   • B$\mathrm{H}_0:\mu_d>0,\ \mathrm{H}_1:\mu_d=0$
   • C$\mathrm{H}_0:\mu_d=0,\ \mathrm{H}_1:\mu_d<0$
   • D$\mathrm{H}_0:\mu_d=0,\ \mathrm{H}_1:\mu_d>0$

   (b)

   Use your GDC to find the $p$-value of the test.

   [1 mark]

   • A$0.0126$
   • B$0.994$
   • C$0.00630$
   • D$0.000436$

   (c)

   State the conclusion of the test in context, giving a reason for your answer.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A manufacturer claims that $30\%$ of its customers choose the premium model of a phone. A retailer believes that the proportion is higher. She asks $25$ randomly chosen customers and finds that $12$ of them choose the premium model. She tests the manufacturer's claim at the $5\%$ significance level.

   (a)

   Let $p$ be the proportion of customers who choose the premium model. Write down the null and alternative hypotheses, and state the distribution of $X$, the number of customers in the sample who choose the premium model, if the null hypothesis is true.

   [3 marks]

   (b)

   (i) Use your GDC to find the $p$-value for the test.
   (ii) State the conclusion of the test in context, giving a reason.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A call centre manager states that $70\%$ of calls are resolved at the first attempt. Calls arrive independently at a uniform average rate of $6$ per hour.

   (a)

   A random sample of $20$ calls is used to test $\mathrm{H}_0:p=0.7$ against $\mathrm{H}_1:p<0.7$ at the $5\%$ significance level, where $p$ is the proportion of calls resolved at the first attempt.
   (i) Find the critical region for this test.
   (ii) Write down the probability of a Type I error.
   (iii) The true value of $p$ is $0.5$. Find the probability of a Type II error.

   [6 marks]

   (b)

   The manager believes that the rate of calls has increased during the busiest hour. She records the number of calls $X$ in one busy hour and tests the claim at the $5\%$ significance level.
   (i) Write down suitable null and alternative hypotheses, where $\lambda$ is the mean number of calls per hour.
   (ii) Find the critical region for the test.
   (iii) The true mean is $\lambda=9$ calls per hour. Find the probability of a Type II error.

   [6 marks]

   Total for question 4: 12 marks

End of questions