Hypothesis test for a binomial proportionAQA A-Level Maths: Subtopic test

1. 1
   A bus company knows that, historically, 25% of its buses arrive late. After the timetable is changed, a manager records a random sample of 12 journeys and finds that exactly 1 bus is late. She wants to test, at the 5% significance level, whether the proportion of late buses has decreased. Let $X$ be the number of late buses in a sample of 12.

   (a)

   Which expression gives the $p$-value for this test?

   [1 mark]

   • A$P(X=1)$ where $X\sim B(12,0.25)$
   • B$P(X\ge1)$ where $X\sim B(12,0.25)$
   • C$P(X\le1)$ where $X\sim B(12,0.25)$
   • D$P(X\le1)$ where $X\sim B\left(12,\frac1{12}\right)$

   (b)

   The $p$-value is $0.1584$. Which conclusion is correct?

   [1 mark]

   • ADo not reject $H_0$; there is insufficient evidence that the proportion of late buses has decreased
   • BReject $H_0$ because $0.1584>0.05$
   • CDo not reject $H_0$; this proves that 25% of buses are still late
   • DReject $H_0$ because $0.1584<0.5$

   (c)

   Describe what it would mean, in context, to reject $H_0$ incorrectly, and state the significance level of this test (the greatest probability of this happening).

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Ali suspects that a coin is biased. He tosses it 10 times and gets 9 heads. He tests, at the 5% significance level, whether the probability $p$ of heads is different from $0.5$. Let $X$ be the number of heads in 10 tosses.

   (a)

   Assuming the coin is fair, what is the value of $P(X\ge9)$?

   [1 mark]

   • A$0.0547$
   • B$0.0010$
   • C$0.0098$
   • D$0.0107$

   (b)

   Which value should $P(X\ge9)$ be compared with?

   [1 mark]

   • A$0.05$
   • B$0.025$
   • C$0.01$
   • D$0.95$

   (c)

   State the conclusion of Ali's test in context.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A factory knows that 8% of the items from its usual supplier are defective. It tries a new supplier and the quality manager suspects that the proportion of defective items will be higher. A random sample of 30 items from the new supplier contains 6 defective items. Let $X$ be the number of defective items in a sample of 30, and let $p$ be the probability that an item from the new supplier is defective.

   (a)

   Write down suitable hypotheses and find the $p$-value for the sample result.

   [3 marks]

   (b)

   (i) State the conclusion of the test at the 5% significance level, in context.
   (ii) Would the conclusion be the same at the 1% significance level? Justify your answer.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A standard treatment cures 70% of patients. A doctor tries a new treatment on 20 patients chosen at random, and 17 of them are cured. She wants to know, at the 5% significance level, whether the new treatment has a higher cure rate. Let $X$ be the number of patients cured out of 20 and let $p$ be the probability that a patient given the new treatment is cured.

   (a)

   Carry out the test, stating your hypotheses and your conclusion in context.

   [6 marks]

   (b)

   (i) Find the critical region for this test.
   (ii) State the actual probability of incorrectly rejecting $H_0$.
   (iii) The sample proportion of cures, $0.85$, is much higher than $0.7$. Suggest one change to the trial that would make it more likely to detect a genuine improvement, and explain why it helps.

   [6 marks]

   Total for question 4: 12 marks

End of questions