Language of hypothesis testingEdexcel A-Level Maths: Subtopic test

1. 1
   A coin is suspected of being biased towards heads. Let $p$ be the probability that the coin lands heads. The coin is tossed 20 times and $X$ is the number of heads. A hypothesis test is carried out at the 5% significance level.

   (a)

   Which pair of hypotheses should be used?

   [1 mark]

   • A$H_0:p>0.5,\ H_1:p=0.5$
   • B$H_0:p=0.5,\ H_1:p\ne0.5$
   • C$H_0:X=10,\ H_1:X>10$
   • D$H_0:p=0.5,\ H_1:p>0.5$

   (b)

   Which of the following is the test statistic?

   [1 mark]

   • AThe probability $p$ that the coin lands heads
   • BThe significance level, 5%
   • CThe number of heads, $X$, in the 20 tosses
   • DThe critical value of $X$

   (c)

   The critical region for this test is $X\ge15$. Find the actual probability of rejecting $H_0$ when it is true.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A die is thought to be biased with respect to sixes. Let $p$ be the probability that the die lands on six. The die is rolled 30 times and $X$ is the number of sixes. The test of $H_0:p=\frac16$ against $H_1:p\ne\frac16$ is carried out at the 10% significance level.

   (a)

   Why is this a two-tailed test?

   [1 mark]

   • AThe die has six faces
   • B$H_1$ states that $p$ differs from $\frac16$, in either direction
   • CThe significance level is 10%
   • DThe sample size is 30

   (b)

   If $H_0$ is true, what is the expected value of $X$?

   [1 mark]

   • A$5$
   • B$\frac{25}{6}$
   • C$\frac16$
   • D$30$

   (c)

   Find the critical region for the lower tail of this test.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A manufacturer claims that 20% of its chocolate eggs contain a toy. A child believes that the proportion is greater than 20%. In a random sample of 25 eggs, 9 contain a toy. Let $p$ be the proportion of all the manufacturer's eggs that contain a toy, and let $X$ be the number of eggs in a sample of 25 that contain a toy.

   (a)

   State the null and alternative hypotheses, and state the distribution of $X$ if the null hypothesis is true.

   [3 marks]

   (b)

   Carry out the test at the 5% significance level, using the $p$-value, and state your conclusion in context.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A spinner has four sectors, one of which is red. It is suspected that the spinner is biased with respect to red. Let $p$ be the probability that the spinner lands on red. It is spun 40 times and $X$ is the number of times it lands on red. The test of $H_0:p=0.25$ against $H_1:p\ne0.25$ is carried out at the 5% significance level.

   (a)

   Find the critical region for the test, and find the actual significance level of the test.

   [6 marks]

   (b)

   The spinner lands on red 3 times in the 40 spins, and the critical region found in (a) is $X\le4$ or $X\ge17$.
   (i) State, with a reason, whether $H_0$ is rejected.
   (ii) A student says, 'This proves that $p$ is not 0.25.' Comment on this statement.
   (iii) The test is repeated at the 1% significance level. Explain what happens to the critical region and to the probability of wrongly rejecting $H_0$.

   [6 marks]

   Total for question 4: 12 marks

End of questions