Hypothesis test for a binomial proportionAQA A-Level Maths: Mind map

Exam questions on Hypothesis test for a binomial proportion

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.
   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
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.
   State the conclusion of Ali's test in context.2 marks
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.
   Write down suitable hypotheses and find the $p$-value for the sample result.3 marks