State hypotheses with additional parts

2 A shop obtains apples from a certain farm. It has been found that 5\% of apples from this farm are Grade A. Following a change in growing conditions at the farm, the shop management plan to carry out a hypothesis test to find out whether the proportion of Grade A apples has increased. They select 25 apples at random. If the number of Grade A apples is more than 3 they will conclude that the proportion has increased.

1. State suitable null and alternative hypotheses for the test.
2. Find the probability of a Type I error.
   In fact 2 of the 25 apples were Grade A .
3. Which of the errors, Type I or Type II, is possible? Justify your answer.

2 A headteacher models the number of children who bring a 'healthy' packed lunch to school on any day by the distribution \(\mathrm { B } ( 150 , p )\). In the past, she has found that \(p = \frac { 1 } { 3 }\). Following the opening of a fast food outlet near the school, she wishes to test, at the \(1 \%\) significance level, whether the value of \(p\) has decreased.

1. State the null and alternative hypotheses for this test.
   On a randomly chosen day she notes the number, \(N\), of children who bring a 'healthy' packed lunch to school. She finds that \(N = 36\) and then, assuming that the null hypothesis is true, she calculates that \(\mathrm { P } ( N \leqslant 36 ) = 0.0084\).
2. State, with a reason, the conclusion that the headteacher should draw from the test.
3. According to the model, what is the largest number of children who might bring a packed lunch to school?

9 Last year, market research showed that \(8 \%\) of adults living in a certain town used a particular local coffee shop. Following an advertising campaign, it was expected that this proportion would increase. In order to test whether this had happened, a random sample of 150 adults in the town was chosen. The random variable \(X\) denotes the number of these 150 adults who said that they used the local coffee shop.

1. 1. Assuming that the proportion of adults using the local coffee shop is unchanged from the previous year, state a suitable binomial distribution with which to model the variable \(X\).
   2. The probabilities given by this model are the terms of the binomial expansion of an expression of the form \(( a + b ) ^ { n }\). Write down this expression, using appropriate values of \(a , b\) and \(n\). It was found that 18 of these 150 adults said that they use the local coffee shop.
2. Test, at the 5\% significance level, whether the proportion of adults in the town who use the local coffee shop has increased. It was later discovered by a statistician that the random sample of 150 adults had been chosen from shoppers in the town on a Friday and a Saturday.
3. Explain why this suggests that the assumptions made when using a binomial model for \(X\) may not be valid in this context.

8 In 2018 research showed that 81\% of young adults in England had never donated blood.
Following an advertising campaign in 2021, it is believed that the percentage of young adults in England who had never donated blood in 2021 is less than \(81 \%\). Ling decides to carry out a hypothesis test at the 5\% level.
Ling collects data from a random sample of 400 young adults in England.

1. State the null and alternative hypotheses for the test, defining the parameter used.
2. Write down the probability that the null hypothesis is rejected when it should in fact be accepted.
3. Assuming the null hypothesis is correct, calculate the expected number of young adults in the sample who had never donated blood.
4. Calculate the probability that there were no more than 308 young adults who had never donated blood in the sample.
5. Determine the critical region for the test. In fact, the sample contained 314 young adults who had never donated blood.
6. Carry out the test, giving the conclusion in the context of the question.

12 Data collected in the twentieth century showed that the probability of a randomly selected person having blue eyes was 0.08 . A medical researcher believes that the probability in 2024 is less than this so they decide to carry out a hypothesis test at the \(5 \%\) significance level.

1. Write down suitable hypotheses for the test, defining the parameter used.
2. Assuming that the probability that a person selected at random has blue eyes is still 0.08 , calculate the probability that 3 or fewer people in a random sample of 92 people have blue eyes. The researcher collects a random sample of 92 people and finds that 3 of them have blue eyes.
3. Use your answer to part (b) to carry out the test, giving your conclusion in context.