Modelling and Hypothesis Testing

4 The diagram represents a very simple maze with two vertices, A and B. At each vertex a rat either exits the maze or runs to the other vertex, each with probability 0.5 . The rat starts at vertex A .
\includegraphics[max width=\textwidth, alt={}, center]{dab87ac5-eda4-433f-b07a-0a609aca2f65-4_79_930_534_571}

1. Describe how to use 1-digit random numbers to simulate this situation.
2. Use the random digits provided in your answer book to run 10 simulations, each starting at vertex A. Hence estimate the probability of the rat exiting at each vertex, and calculate the mean number of times it runs between vertices before exiting. The second diagram represents a maze with three vertices, A, B and C. At each vertex there are three possibilities, and the rat chooses one, each with probability \(1 / 3\). The rat starts at vertex A.
   \includegraphics[max width=\textwidth, alt={}, center]{dab87ac5-eda4-433f-b07a-0a609aca2f65-4_566_889_1082_589}
3. Describe how to use 1-digit random numbers to simulate this situation.
4. Use the random digits provided in your answer book to run 10 simulations, each starting at vertex A. Hence estimate the probability of the rat exiting at each vertex.

6 An apple tree has 6 apples left on it. Each day each remaining apple has a probability of \(\frac { 1 } { 3 }\) of falling off the tree during the day.

1. Give a rule for using one-digit random numbers to simulate whether or not a particular apple falls off the tree during a given day.
2. Use the random digits given in your answer book to simulate how many apples fall off the tree during day 1 . Give the total number of apples that fall during day 1 .
3. Continue your simulation from the end of day 1 , which you simulated in part (ii), for successive days until there are no apples left on the tree. Use the same list of random digits, continuing from where you left off in part (ii). During which day does the last apple fall from the tree? Now suppose that at the start of each day the gardener picks one apple from the tree and eats it.
4. Repeat your simulation with the gardener picking the lowest numbered apple remaining on the tree at the start of each day. Give the day during which the last apple falls or is picked. Use the same string of random digits, a copy of which is provided for your use in this part of the question.
5. How could your results be made more reliable?

6 Answer part (iv) of this question on the insert provided. There are two types of customer who use the shop at a service station. \(70 \%\) buy fuel, the other \(30 \%\) do not. There is only one till in operation.

1. Give an efficient rule for using one-digit random numbers to simulate the type of customer arriving at the service station. Table 6.1 shows the distribution of time taken at the till by customers who are buying fuel.

   Time taken (mins)	1	1.5	2	2.5
   Probability	\(\frac { 3 } { 10 }\)	\(\frac { 2 } { 5 }\)	\(\frac { 1 } { 5 }\)	\(\frac { 1 } { 10 }\)

   \section*{Table 6.1}
2. Specify an efficient rule for using one-digit random numbers to simulate the time taken at the till by customers purchasing fuel. Table 6.2 shows the distribution of time taken at the till by customers who are not buying fuel.

   Time taken (mins)	1	1.5	2	2.5	3
   Probability	\(\frac { 1 } { 7 }\)	\(\frac { 2 } { 7 }\)	\(\frac { 2 } { 7 }\)	\(\frac { 1 } { 7 }\)	\(\frac { 1 } { 7 }\)

   \section*{Table 6.2}
3. Specify an efficient rule for using two-digit random numbers to simulate the time taken at the till by customers not buying fuel. What is the advantage in using two-digit random numbers instead of one-digit random numbers in this part of the question? The table in the insert shows a partially completed simulation study of 10 customers arriving at the till.
4. Complete the table using the random numbers which are provided.
5. Calculate the mean total time spent queuing and paying.

2. Statistical models can be used to describe real world problems. Explain the process involved in the formulation of a statistical model.
(4)

1 Martin is considering paying for a vaccination against a disease. If he catches the disease he would not be able to work and would lose \(\pounds 900\) in income because he would have to stay at home recovering. The vaccination costs \(\pounds 20\). The vaccination would reduce his risk of catching the disease during the year from 0.02 to 0.001 .

1. Draw a decision tree for Martin.
2. Evaluate the EMV of Martin's loss at each node of your tree, and give the action that Martin should take to minimise the EMV of his loss. Martin can answer a medical questionnaire which will give an estimate of his susceptibility to the disease. If he is found to be susceptible, then his chance of catching the disease is 0.05 . Vaccination will reduce that to 0.0025 . If he is found not to be susceptible, then his chance of catching the disease is 0.01 and vaccination will reduce it to 0.0005 . Historically, \(25 \%\) of people are found to be susceptible.
3. What is the EMV of this questionnaire? Martin decides not to answer the questionnaire. He also decides that there is more than just his EMV to be considered in deciding whether or not to have the vaccination. The vaccination itself is likely to have side effects, but catching the disease would be very unpleasant. Martin estimates that he would find the effects of the disease 1000 times more unpleasant than the effects of the vaccination.
4. Analyse which course of action would minimise the unpleasantness for Martin.

2 Adrian is considering selling his house and renting a flat.
Adrian still owes \(\pounds 150000\) on his house. He has a mortgage for this, for which he has to pay \(\pounds 4800\) annual interest. If he sells he will pay off the \(\pounds 150000\) and invest the remainder of the proceeds at an interest rate of \(2.5 \%\) per annum. He will use the interest to help to pay his rent. His estate agent estimates that there is a \(30 \%\) chance that the house will sell for \(\pounds 225000\), a \(50 \%\) chance that it will sell for \(\pounds 250000\), and a \(20 \%\) chance that it will sell for \(\pounds 275000\). A flat will cost him \(\pounds 7500\) per annum to rent.

1. Draw a decision tree to help Adrian to decide whether to keep his house, or to sell it and rent a flat. Compare the EMVs of Adrian's annual outgoings, and ignore the costs of selling.
2. Would the analysis point to a different course of action if Adrian were to use a square root utility function, instead of EMVs? Adrian's circumstances change so that he has to decide now whether to sell or not in one year's time. Economic conditions might then be less favourable for the housing market, the same, or more favourable, these occurring with probabilities \(0.3,0.3\) and 0.4 respectively. The possible selling prices and their probabilities are shown in the table.

   Economic conditions and probabilities		Selling prices ( £) and probabilities
   less favourable	0.3	200000	0.2	225000	0.3	250000	0.5
   unchanged	0.3	225000	0.3	250000	0.5	275000	0.2
   more favourable	0.4	250000	0.3	300000	0.5	350000	0.2

3. Draw a decision tree to help Adrian to decide what to do. Compare the EMVs of Adrian's annual outgoings. Assume that he will still owe \(\pounds 150000\) in one year's time, and that the cost of renting and interest rates do not change.

2 Zoe is preparing for a Decision Maths test on two topics, Decision Analysis (D) and Simplex (S). She has to decide whether to devote her final revision session to D or to S . There will be two questions in the test, one on D and one on S . One will be worth 60 marks and the other will be worth 40 marks. Historically there is a 50\% chance of each possibility. Zoe is better at \(D\) than at \(S\). If her final revision session is on \(D\) then she would expect to score \(80 \%\) of the \(D\) marks and \(50 \%\) of the \(S\) marks. If her final session is on \(S\) then she would expect to score \(70 \%\) of the S marks and \(60 \%\) of the D marks.

1. Compute Zoe's expected mark under each of the four possible circumstances, i.e. Zoe revising \(D\) and the D question being worth 60 marks, etc.
2. Draw a decision tree for Zoe. Michael claims some expertise in forecasting which question will be worth 60 marks. When he forecasts that it will be the D question which is worth 60 , then there is a \(70 \%\) chance that the D question will be worth 60 . Similarly, when he forecasts that it will be the S question which is worth 60 , then there is a \(70 \%\) chance that the S question will be worth 60 . He is equally likely to forecast that the D or the S question will be worth 60.
3. Draw a decision tree to find the worth to Zoe of Michael's advice.

4 A fair spinner has five sides numbered \(1,2,3,4,5\). The score on one spin is denoted by \(X\).

1. Show that \(\operatorname { Var } ( X ) = 2\).
   Fiona has another spinner, also with five sides numbered \(1,2,3,4,5\). She suspects that it is biased so that the expected score is less than 3 . In order to test her suspicion, she plans to spin her spinner 40 times. If the mean score is less than 2.6 she will conclude that her spinner is biased in this way.
2. Find the probability of a Type I error.
3. State what is meant by a Type II error in this context.

4 A survey is to be carried out to draw conclusions about the proportion \(p\) of residents of a town who support the building of a new supermarket. It is proposed to carry out the survey by interviewing a large number of people in the high street of the town, which attracts a large number of tourists.

1. Give two different reasons why this proposed method is inappropriate.
2. Suggest a good method of carrying out the survey.
3. State two statistical properties of your survey method that would enable reliable conclusions about \(p\) to be drawn.

2. Suggest, with reasons, suitable distributions for modelling each of the following:

1. the number of times the letter J occurs on each page of a magazine,
2. the length of string left over after cutting as many 3 metre long pieces as possible from partly used balls of string,
3. the number of heads obtained when spinning a coin 15 times.

4 A ski-lift gondola can carry 4 people. The weight restriction sign in the gondola says "4 people - 325 kg ". The table models the distribution of weights of people using the gondola.

1. Give an efficient rule for using 2-digit random numbers to simulate the weight of a person entering the gondola.
2. Give a reason for using 2-digit rather than 1-digit random numbers in these circumstances.
3. Using the random numbers given in your answer book, simulate the weights of four people entering the gondola, and hence give its simulated load.
4. Using the random numbers given in your answer book, repeat your simulation 9 further times. Hence estimate the probability of the load of a fully-laden gondola exceeding 325 kg .
5. What in reality might affect the pattern of loading of a gondola which is not modelled by your simulation?

2 The manager of a video hire shop wishes to estimate the proportion of videos damaged by customers. He takes a random sample of 120 videos and finds that 33 of them are damaged. Find a \(95 \%\) confidence interval for the true proportion of videos that are being damaged when hired from this shop.

5 A fair six-sided die has faces numbered \(1,2,3,4,5,6\). The score on one throw is denoted by \(X\).

1. Write down the value of \(\mathrm { E } ( X )\) and show that \(\operatorname { Var } ( X ) = \frac { 35 } { 12 }\). Fayez has a six-sided die with faces numbered \(1,2,3,4,5,6\). He suspects that it is biased so that when it is thrown it is more likely to show a low number than a high number. In order to test his suspicion, he plans to throw the die 50 times. If the mean score is less than 3 he will conclude that the die is biased.
2. Find the probability of a Type I error.
3. With reference to this context, describe circumstances in which Fayez would make a Type II error.

1. The small village of Tornep has a preservation society which is campaigning for a new by-pass to be built. The society needs to measure
   1. the strength of opinion amongst the residents of Tornep for the scheme and
   2. the flow of traffic through the village on weekdays.
   The society wants to know whether to use a census or a sample survey for each of these measures.
   (a) In each case suggest which they should use and specify a suitable sampling frame. For the measurement of traffic flow through Tornep,
   (b) suggest a suitable statistic and a possible statistical model for this statistic.

1. Explain what is meant by
   1. a population,
   2. a sampling unit.
   Suggest suitable sampling frames for surveys of
2. families who have holidays in Greece,
3. mothers with children under two years old.

1. The height, \(h\) metres, of a tree, \(t\) years after being planted, is modelled by the equation

$$h ^ { 2 } = a t + b \quad 0 \leqslant t < 25$$ where \(a\) and \(b\) are constants.
Given that

• the height of the tree was 2.60 m , exactly 2 years after being planted
• the height of the tree was 5.10 m , exactly 10 years after being planted
  1. find a complete equation for the model, giving the values of \(a\) and \(b\) to 3 significant figures.

Given that the height of the tree was 7 m , exactly 20 years after being planted

evaluate the model, giving reasons for your answer.

2 Bill is at a horse race meeting. He has \(\pounds 2\) left with two races to go. He only ever bets \(\pounds 1\) at a time. For each race he chooses a horse and then decides whether or not to bet on it. In both races Bill's horse is offered at "evens". This means that, if Bill bets \(\pounds 1\) and the horse wins, then Bill will receive back his \(\pounds 1\) plus \(\pounds 1\) winnings. If Bill's horse does not win then Bill will lose his \(\pounds 1\).

1. Draw a decision tree to model this situation. Show Bill's payoffs on your tree, i.e. how much money Bill finishes with under each possible outcome. Assume that in each race the probability of Bill's horse winning is the same, and that it has value \(p\).
2. Find Bill's EMV when
   (A) \(p = 0.6\),
   (B) \(p = 0.4\). Give his best course of action in each case.
3. Suppose that Bill uses the utility function utility \(= ( \text { money } ) ^ { x }\), to decide whether or not to bet \(\pounds 1\) on one race. Show that, with \(p = 0.4\), Bill will not bet if \(x = 0.5\), but will bet if \(x = 1.5\).