Statistical Hypothesis Testing

1. Planets is the name given to a new box of chocolates. Each box contains 18 chocolates and each chocolate is one of four types: Venus, Saturn, Jupiter and Neptune. The same number of each type is manufactured but a random process is used to fill the boxes.
   (a) Find the probability that a box contains no more than 7 Saturns.

Through online forums it is clear that Saturn is the most popular. In order to increase sales, the manufacturer claims to have increased the proportion of Saturns. You decide to test the manufacturer’s claim by purchasing a box of Planets.
(b) The box contains 8 Saturns. Does this support the manufacturer’s claim at the significance level? Write down your hypotheses clearly, state the p -value and show your working in full.
2. The figures released by a social media platform said that of Year 9 students in England had a registered username on the platform.

A school wishes to investigate whether or not the proportion of its Year 9 students signed up to the platform is . A random sample of 20 Year 9 students was found to contain 12 who had a username.

Carry out a suitable hypothesis test at the significance level, stating your hypothesis and conclusions carefully. Find the critical region for the test.
3. The school tombola at the Christmas Fayre displays a sign saying ’ 1 in every 5 tickets is a winner’. Meghana, an A level Maths student, decides to test this claim.
(a) State suitable null and alternative hypotheses for the test.

Meghana buys 12 tickets for the tombola and finds none of them are winning tickets.
(b) Carry out the hypothesis test at the significance level, clearly stating the conclusion you draw.
How many tickets would Meghana have to buy for the critical region to have a nonempty lower tail?
4. Two Year 12 students, Jade and Yunus, are the cycling safety officers for their school. One day they examined the bicycle tyres of all the bikes in the school bike shed and were disappointed to find that had at least one tyre with an unsafe tread. As a result, they decided to undertake a campaign, presenting assemblies to all students on the importance of road-safety and arranging for a related email to be sent to all parents. At the start of the following term, 18 bikes were stopped at random on the way in to school and their tyres were inspected. Just one of the bikes had an unsafe tyre.

Jade and Yunus ask you to carry out a suitable hypothesis test to examine whether their campaign appears to have been successful.

(a) State your hypotheses clearly, justifying the form of the alternative hypothesis.
(b) Carry out the test at the significance level, stating your conclusions clearly.
State, with a reason, the critical value for the test.
(d) Give a level of significance such that you would come to the opposite conclusion for your test. Explain your reasoning.