State test assumptions or distributions

3 The proportion of adults in a large village who support a proposal to build a bypass is denoted by \(p\). A random sample of size 20 is selected from the adults in the village, and the members of the sample are asked whether or not they support the proposal.

1. Name the probability distribution that would be used in a hypothesis test for the value of \(p\).
2. State the properties of a random sample that explain why the distribution in part (i) is likely to be a good model. \(4 X\) is a continuous random variable.

1. A machine fills packets with sweets and \(\frac { 1 } { 7 }\) of the packets also contain a prize.

The packets of sweets are placed in boxes before being delivered to shops. There are 40 packets of sweets in each box. The random variable \(T\) represents the number of packets of sweets that contain a prize in each box.

1. State a condition needed for \(T\) to be modelled by \(\mathrm { B } \left( 40 , \frac { 1 } { 7 } \right)\) A box is selected at random.
2. Using \(T \sim \mathrm {~B} \left( 40 , \frac { 1 } { 7 } \right)\) find
   1. the probability that the box has exactly 6 packets containing a prize,
   2. the probability that the box has fewer than 3 packets containing a prize. Kamil's sweet shop buys 5 boxes of these sweets.
3. Find the probability that exactly 2 of these 5 boxes have fewer than 3 packets containing a prize. Kamil claims that the proportion of packets containing a prize is less than \(\frac { 1 } { 7 }\) A random sample of 110 packets is taken and 9 packets contain a prize.
4. Use a suitable test to assess Kamil's claim. You should
   • state your hypotheses clearly
   • use a \(5 \%\) level of significance

1. Rowan believes that \(35 \%\) of type \(A\) vacuum tubes shatter when exposed to alternating high and low temperatures.

Rowan takes a random sample of 15 of these type \(A\) vacuum tubes and uses a two-tailed test, at the \(5 \%\) level of significance, to test his belief.

1. Give two assumptions, in context, that Rowan needs to make for a binomial distribution to be a suitable model for the number of these type \(A\) vacuum tubes that shatter when exposed to alternating high and low temperatures.
2. Using a binomial distribution, find the critical region for the test. You should state the probability of rejection in each tail, which should be as close as possible to 0.025
3. Find the actual level of significance of the test based on your critical region from part (b) Rowan records that in the latest batch of 15 type \(A\) vacuum tubes exposed to alternating high and low temperatures, 4 of them shattered.
4. With reference to part (b), comment on Rowan's belief. Give a reason for your answer. Rowan changes to type \(B\) vacuum tubes. He takes a random sample of 40 type \(B\) vacuum tubes and finds that 8 of them shatter when exposed to alternating high and low temperatures.
5. Test, at the \(5 \%\) level of significance, whether or not there is evidence that the proportion of type \(B\) vacuum tubes that shatter when exposed to alternating high and low temperatures is lower than \(35 \%\) You should state your hypotheses clearly.

A company has 20 boats to hire out. Payment is always taken in advance and all 20 boats are hired out each day. A manager at the company notices that 10% of groups do not turn up to take the boats, despite having already paid to hire them. The manager wishes to investigate whether the numbers of boats that do not get taken each day can be modelled by the binomial distribution B\((20, 0 \cdot 1)\). The numbers of boats that were not taken for 110 randomly selected days are given below.

Number of boats not taken	0	1	2	3	4	5 or more
Frequency	10	35	29	25	8	3

1. State suitable hypotheses to carry out a goodness of fit test. [1]
2. Here is part of the table for a \(\chi^2\) goodness of fit test on the data.

   Number of boats not taken	0	1	2	3	4	5 or more
   Observed	10	35	29	25	8	3
   Expected	\(f\)	29·72	\(g\)	20·91	9·88	4·75

   1. Calculate the values of \(f\) and \(g\).
   2. By completing the test, give the conclusion the manager should reach. [10]

The cost of hiring a boat is £15. Since demand is high and the proportion of groups that do not turn up is also relatively high, the manager decides to take payment for 22 boats each day. She would give £20 (a full refund and some compensation) to any group that has paid and turned up, but cannot take a boat out due to the overselling. Assume that the proportion of groups not turning up stays the same.

1. 1. Suggest a binomial model that the manager could use for the number of groups arriving expecting to hire a boat.
   2. Hence calculate the expected daily net income for the company following the manager's decision. [8]
2. Is the manager justified in her decision? Give a reason for your answer. [1]