Hypothesis tests for a binomial proportionEdexcel A-Level Maths: Subtopic test

1. 1
   A drug is known to cure 30% of patients with a certain condition. A new drug is tested on 20 randomly chosen patients, of whom 10 are cured. Let $p$ be the probability that a patient given the new drug is cured and let $X$ be the number cured in a sample of 20. The company tests at the 5% significance level whether the new drug is more effective.

   (a)

   State the distribution of $X$ if the new drug is no more effective than the old one.

   [1 mark]

   • A$B(20,0.3)$
   • B$B(20,0.5)$
   • C$B(10,0.3)$
   • D$B(0.3,20)$

   (b)

   Find the $p$-value for this test.

   [1 mark]

   • A$0.1133$
   • B$0.0480$
   • C$0.9520$
   • D$0.0171$

   (c)

   State the conclusion of the test, in context.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A company claims that 85% of its parcels are delivered on time. A customer believes that the proportion is lower. In a random sample of 15 parcels, 10 are delivered on time. Let $p$ be the proportion of all the company's parcels that are delivered on time and let $X$ be the number delivered on time in a sample of 15. The customer tests at the 5% significance level.

   (a)

   Which hypotheses should the customer use?

   [1 mark]

   • A$H_0:p=0.85,\ H_1:p>0.85$
   • B$H_0:p=\frac{10}{15},\ H_1:p<\frac{10}{15}$
   • C$H_0:p=0.85,\ H_1:p<0.85$
   • D$H_0:p<0.85,\ H_1:p=0.85$

   (b)

   Find the $p$-value for this test.

   [1 mark]

   • A$0.0168$
   • B$0.9383$
   • C$0.1773$
   • D$0.0617$

   (c)

   State the conclusion of the test, in context.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A network provider claims that 40% of its customers are on a premium plan. A researcher believes that the proportion has changed. In a random sample of 30 customers, 17 are on a premium plan. Let $X$ be the number of customers on a premium plan in a sample of 30.

   (a)

   State the hypotheses for a test at the 5% significance level, and say what $p$ represents.

   [3 marks]

   (b)

   Carry out the test at the 5% significance level and state your conclusion in context.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A manufacturer says that 5% of its light bulbs are faulty. Let $p$ be the proportion of all its bulbs that are faulty.

   (a)

   A customer believes that the proportion of faulty bulbs is higher than 5%, and tests a random sample of 40 bulbs at the 1% significance level.
   (i) Find the critical region for the test.
   (ii) State the actual significance level of the test.
   (iii) The sample contains 6 faulty bulbs. State the conclusion of the test, in context.

   [6 marks]

   (b)

   After a change to the production process, the manufacturer believes that the proportion of faulty bulbs has fallen. A random sample of 60 bulbs contains 1 faulty bulb. Test the manufacturer's belief at the 5% significance level, stating your hypotheses and conclusion.

   [6 marks]

   Total for question 4: 12 marks

End of questions