Mean-variance comparison for Poisson validation

6 It is known that 1 in 5000 people in Atalia have a certain condition. A random sample of 12500 people from Atalia is chosen for a medical trial. The number having the condition is denoted by \(X\).

1. Use an appropriate approximating distribution to find \(\mathrm { P } ( X \leqslant 3 )\).
2. Find the values of \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\), and explain how your answers suggest that the approximating distribution used in (a) is likely to be appropriate.

5 Chai packs china mugs into cardboard boxes. Chai's manager suspects that breakages occur at random times and that the number of breakages may follow a Poisson distribution. He takes a small sample of observations and finds that the number of breakages in a one-hour period has a mean of 2.4 and a standard deviation of 1.5.

1. Explain how this information tends to support the manager's suspicion.
   The manager now takes a larger sample and claims that the numbers of breakages in a one-hour period follow a Poisson distribution. The numbers of breakages in a random sample of 180 one-hour periods are summarised in the following table.

   Number of breakages	0	1	2	3	4	5	6	7 or more
   Frequency	21	33	46	31	23	16	10	0

   The mean number of breakages calculated from this sample is 2.5.
2. Use the data from this larger sample to carry out a goodness of fit test, at the \(10 \%\) significance level, to test the claim.

3 An electrical retailer gives customers extended guarantees on washing machines. Under this guarantee all repairs in the first 3 years are free. The retailer records the numbers of free repairs made to 80 machines.

1. Show that the sample mean is 0.4375 .
2. The sample standard deviation \(s\) is 0.6907 . Explain why this supports a suggestion that a Poisson distribution may be a suitable model for the distribution of the number of free repairs required by a randomly chosen washing machine. The random variable \(X\) denotes the number of free repairs required by a randomly chosen washing machine. For the remainder of this question you should assume that \(X\) may be modelled by a Poisson distribution with mean 0.4375.
3. Find \(\mathrm { P } ( X = 1 )\). Comment on your answer in relation to the data in the table.
4. The manager decides to monitor 8 washing machines sold on one day. Find the probability that there are at least 12 free repairs in total on these 8 machines. You may assume that the 8 machines form an independent random sample.
5. A launderette with 8 washing machines has needed 12 free repairs. Why does your answer to part (iv) suggest that the Poisson model with mean 0.4375 is unlikely to be a suitable model for free repairs on the machines in the launderette? Give a reason why the model may not be appropriate for the launderette. The retailer also sells tumble driers with the same guarantee. The number of free repairs on a tumble drier in three years can be modelled by a Poisson distribution with mean 0.15 . A customer buys a tumble drier and a washing machine.
6. Assuming that free repairs are required independently, find the probability that
   (A) the two appliances need a total of 3 free repairs between them,
   (B) each appliance needs exactly one free repair.

5 The manager of an emergency response hotline believes that calls are made to the hotline independently and at constant average rate throughout the day. From a small random sample of the population, the manager finds that the mean number of calls made in a 1-hour period is 14.4. Let \(R\) denote the number of calls made in a randomly chosen 1-hour period.

1. Using evidence from the small sample, state a suitable distribution with which to model \(R\). You should give the value(s) of any parameter(s).
2. In this part of the question, use the distribution and value(s) of the parameter(s) from your answer to part (a).
   1. Find \(\mathrm { P } ( R > 20 )\).
   2. Given that \(\mathrm { P } ( \mathrm { R } = \mathrm { r } ) > \mathrm { P } ( \mathrm { R } = \mathrm { r } + 1 )\), show algebraically that \(r > 13.4\).
   3. Hence write down the mode of the distribution. The manager also finds, from records over many years, that the modal value of \(R\) is 10 .
3. Use this result to comment on the validity of the distribution used in part (b).
4. Assume now that the type of distribution used in part (b) is valid. Find the range(s) of values of the parameter(s) of this distribution that would correspond to the modal value of \(R\) being 10.

6 A statistician investigates the number, \(F\), of signal failures per week on a railway network.

1. The statistician assumes that signal failures occur randomly. Explain what this statement means.
2. State two further assumptions needed for \(F\) to be well modelled by a Poisson distribution. In a random sample of 50 weeks, the statistician finds that the mean number of failures per week is 1.61, with standard deviation 1.28.
3. Explain whether this suggests that \(F\) is likely to be well modelled by a Poisson distribution. Assume first that \(F \sim \operatorname { Po } ( 1.61 )\).
4. Write down an exact expression for \(\mathrm { P } ( F = 0 )\).
5. Complete the table in the Printed Answer Booklet to show the probabilities of different values of \(F\), correct to three significant figures.

   Value of \(F\)	0	1	\(\geqslant 2\)
   Probability	0.200

   After further investigation, the statistician decides to use a different model for the distribution of \(F\). In this model it is now assumed that \(\mathrm { P } ( F = 0 )\) is still 0.200 , but that if one failure occurs, there is an increased probability that further failures occur.
6. Explain the effect of this assumption on the value of \(\mathrm { P } ( F = 1 )\).

1. A student is investigating the numbers of cherries in a Rays fruit cake. A random sample of Rays fruit cakes is taken and the results are shown in the table below.

1. Calculate the mean and the variance of these data.
2. Explain why the results in part (a) suggest that a Poisson distribution may be a suitable model for the number of cherries in a Rays fruit cake. The number of cherries in a Rays fruit cake follows a Poisson distribution with mean 1.5 A Rays fruit cake is to be selected at random. Find the probability that it contains
3. 1. exactly 2 cherries,
   2. at least 1 cherry. Rays fruit cakes are sold in packets of 5
4. Show that the probability that there are more than 10 cherries, in total, in a randomly selected packet of Rays fruit cakes, is 0.1378 correct to 4 decimal places. Twelve packets of Rays fruit cakes are selected at random.
5. Find the probability that exactly 3 packets contain more than 10 cherries. \href{http://PhysicsAndMathsTutor.com}{PhysicsAndMathsTutor.com}

4. A company director monitored the number of errors on each page of typing done by her new secretary and obtained the following results:

1. Show that the mean number of errors per page in this sample of pages is 2 .
2. Find the variance of the number of errors per page in this sample.
3. Explain how your answers to parts (a) and (b) might support the director's belief that the number of errors per page could be modelled by a Poisson distribution.
   (1) Some time later the director notices that a 4-page report which the secretary has just typed contains only 3 errors. The director wishes to test whether or not this represents evidence that the number of errors per page made by the secretary is now less than 2 .
4. Assuming a Poisson distribution and stating your hypothesis clearly, carry out this test. Use a \(5 \%\) level of significance.
   (6)

5 Peter, a geologist, is studying pebbles on a beach. He uses a frame, called a quadrat, to enclose an area of the beach. He then counts the number of quartz pebbles, \(X\), within the quadrat. He repeats this procedure 40 times, obtaining the following summarised data. $$\sum x = 128 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 126.4$$ Peter believes that the distribution of \(X\) can be modelled by a Poisson distribution with \(\lambda = 3.2\).

1. Use the summarised data to support Peter's belief.
2. Using Peter's model, calculate the probability that:
   1. a single placing of the quadrat contains more than 5 quartz pebbles;
   2. a single placing of the quadrat contains at least 3 quartz pebbles but fewer than 8 quartz pebbles;
   3. when the quadrat is placed twice, at least one placing contains no quartz pebbles.
3. Peter also models the number of flint pebbles enclosed by the quadrat by a Poisson distribution with mean 5 . He assumes that the number of flint pebbles enclosed by the quadrat is independent of the number of quartz pebbles enclosed by the quadrat. Using Peter's models, calculate the probability that a single placing of the quadrat contains a total of either 9 or 10 pebbles which are quartz or flint.
   [0pt] [3 marks]

1. In a packet of 40 biscuits, the number of currants in each biscuit is as follows

1. Find the mean and variance of the random variable \(X\) representing the number of currants per biscuit.
2. State an appropriate model for the distribution of \(X\), giving two reasons for your answer. Another machine produces biscuits with a mean of 1.9 currants per biscuit.
3. Determine which machine is more likely to produce a biscuit with at least two currants.

4. A Geiger counter is observed in the presence of a radioactive source. In 100 one-minute intervals, the number of counts recorded are as follows:

1. Find the mean and variance of this data, and show that it supports the idea that the random variable \(X\) is following a Poisson distribution.
2. Use a Poisson distribution with the mean found in part (a) to calculate, to 3 decimal places, the probability that more than 6 counts will be recorded in any particular minute.
3. Find the number of one-minute intervals, in the sample of 100 , in which more than 6 counts would be expected. \section*{STATISTICS 2 (A) TEST PAPER 10 Page 2}

1 The random variable \(X\) represents the number of cars arriving at a car wash per 10-minute period. From observations over a number of days, an estimate was made of the probability distribution of \(X\). Table 1 shows this estimated probability distribution. \begin{table}[h] \end{table}

1. In this question you must show detailed reasoning. Use Table 1 to calculate estimates of each of the following.
   • \(\mathrm { E } ( X )\)
   • \(\operatorname { Var } ( X )\)
   • Explain how your answers to part (a) indicate that a Poisson distribution may be a suitable model for \(X\).
   You should now assume that \(X\) can be modelled by a Poisson distribution with mean equal to the value which you calculated in part (a).
2. Find each of the following.
   • \(\mathrm { P } ( X = 2 )\)
   • \(\mathrm { P } ( X > 3 )\)
   • Given that the probability that there is at least 1 car arriving in a period of \(k\) minutes is at least 0.99 , find the least possible value of \(k\).

8 A company records the number of complaints, \(X\), that it receives over 60 months. The summarised results are $$\sum x = 102 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 103.25$$ 8

1. Using this data, explain why it may be appropriate to model the number of complaints received by the company per month by a Poisson distribution with mean 1.7
   8
2. The company also receives enquiries as well as complaints. The number of enquiries received is independent of the number of complaints received. The company models the number of complaints per month with a Poisson distribution with mean 1.7 and the number of enquiries per month with a Poisson distribution with mean 5.2 The company starts selling a new product.
   The company records a total of 3 complaints and enquiries in one randomly chosen month. Investigate if the mean total number of complaints and enquiries received per month has changed following the introduction of the new product, using the \(10 \%\) level of significance.
   8
3. It is later found that the mean total number of complaints and enquiries received per month is 6.1 Find the power of the test carried out in part (b), giving your answer to four decimal places.
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