Poisson hypothesis test

5 The number of absences per week by workers at a factory has the distribution \(\operatorname { Po } ( 2.1 )\).

1. Find the standard deviation of the number of absences per week.
2. Find the probability that the number of absences in a 2-week period is at least 2 .
3. Find the probability that the number of absences in a 3-week period is more than 4 and less than 8 .
   Following a change in working conditions, the management wished to test whether the mean number of absences has decreased. They found that, in a randomly chosen 3-week period, there were exactly 2 absences.
4. Carry out the test at the \(10 \%\) significance level.
5. State, with a reason, which of the errors, Type I or Type II, might have been made in carrying out the test in part (d).

6 Pieces of metal discovered by people using metal detectors are found randomly in fields in a certain area at an average rate of 0.8 pieces per hectare. People using metal detectors in this area have a theory that ploughing the fields increases the average number of pieces of metal found per hectare. After ploughing, they tested this theory and found that a randomly chosen field of area 3 hectares yielded 5 pieces of metal.

1. Carry out the test at the \(10 \%\) level of significance.
2. What would your conclusion have been if you had tested at the \(5 \%\) level of significance? Jack decides that he will reject the null hypothesis that the average number is 0.8 pieces per hectare if he finds 4 or more pieces of metal in another ploughed field of area 3 hectares.
3. If the true mean after ploughing is 1.4 pieces per hectare, calculate the probability that Jack makes a Type II error.

6 The number of accidents on a certain road has a Poisson distribution with mean 3.1 per 12-week period.

1. Find the probability that there will be exactly 4 accidents during an 18-week period. Following the building of a new junction on this road, an officer wishes to determine whether the number of accidents per week has decreased. He chooses 15 weeks at random and notes the number of accidents. If there are fewer than 3 accidents altogether he will conclude that the number of accidents per week has decreased. He assumes that a Poisson distribution still applies.
2. Find the probability of a Type I error.
3. Given that the mean number of accidents per week is now 0.1 , find the probability of a Type II error.
4. Given that there were 2 accidents during the 15 weeks, explain why it is impossible for the officer to make a Type II error.

4. Accidents occur randomly at a crossroads at a rate of 0.5 per month. A researcher records the number of accidents, \(X\), which occur at the crossroads in a year.

1. Find \(\mathrm { P } ( 5 \leqslant X < 7 )\) A new system is introduced at the crossroads. In the first 18 months, 4 accidents occur at the crossroads.
2. Test, at the \(5 \%\) level of significance, whether or not there is reason to believe that the new system has led to a reduction in the mean number of accidents per month. State your hypotheses clearly.

1. In the manufacture of cloth in a factory, defects occur randomly in the production process at a rate of 2 per \(5 \mathrm {~m} ^ { 2 }\)

The quality control manager randomly selects 12 pieces of cloth each of area \(15 \mathrm {~m} ^ { 2 }\).

1. Find the probability that exactly half of these 12 pieces of cloth will contain at most 7 defects. The factory introduces a new procedure to manufacture the cloth. After the introduction of this new procedure, the manager takes a random sample of \(25 \mathrm {~m} ^ { 2 }\) of cloth from the next batch produced to test if there has been any change in the rate of defects.
2. 1. Write down suitable hypotheses for this test.
   2. Describe a suitable test statistic that the manager should use.
   3. Explain what is meant by the critical region for this test.
3. Using a 5\% level of significance, find the critical region for this test. You should choose the largest critical region for which the probability in each tail is less than 2.5\%
4. Find the actual significance level for this test.

5. Vehicles pass a particular point on a road at a rate of 51 vehicles per hour.

1. Give two reasons to support the use of the Poisson distribution as a suitable model for the number of vehicles passing this point. Find the probability that in any randomly selected 10 minute interval
2. exactly 6 cars pass this point,
3. at least 9 cars pass this point. After the introduction of a roundabout some distance away from this point it is suggested that the number of vehicles passing it has decreased. During a randomly selected 10 minute interval 4 vehicles pass the point.
4. Test, at the \(5 \%\) level of significance, whether or not there is evidence to support the suggestion that the number of vehicles has decreased. State your hypotheses clearly.
   (6)

6. A doctor expects to see, on average, 1 patient per week with a particular disease.

1. Suggest a suitable model for the distribution of the number of times per week that the doctor sees a patient with the disease. Give a reason for your answer.
2. Using your model, find the probability that the doctor sees more than 3 patients with the disease in a 4 week period. The doctor decides to send information to his patients to try to reduce the number of patients he sees with the disease. In the first 6 weeks after the information is sent out, the doctor sees 2 patients with the disease.
3. Test, at the \(5 \%\) level of significance, whether or not there is reason to believe that sending the information has reduced the number of times the doctor sees patients with the disease. State your hypotheses clearly. Medical research into the nature of the disease discovers that it can be passed from one patient to another.
4. Explain whether or not this research supports your choice of model. Give a reason for your answer.

1. (a) Explain briefly why it is often useful to take a sample from a population.
   (b) Suggest a suitable sampling frame for a local council to use to survey attitudes towards a proposed new shopping centre.
2. A certain type of lettuce seed has a \(12 \%\) failure rate for germination. In a new sample of 25 genetically modified seeds, only 1 did not germinate.
   Clearly stating your hypotheses, test, at the \(1 \%\) significance level, whether the GM seeds are better.
3. A random variable \(X\) has a Poisson distribution with a mean, \(\lambda\), which is assumed to equal 5 .
   (a) Find \(\mathrm { P } ( X = 0 )\).
   (b) In 100 measurements, the value 0 occurs three times. Find the highest significance level at which you should reject the original hypothesis in favour of \(\lambda < 5\).
4. The waiting time, in minutes, at a dentist is modelled by the continuous random variable \(T\) with probability density function

$$\begin{array} { l l } \mathrm { f } ( t ) = k ( 10 - t ) & 0 \leq t \leq 10
\mathrm { f } ( t ) = 0 & \text { otherwise. } \end{array}$$ (a) Sketch the graph of \(\mathrm { f } ( t )\) and find the value of \(k\).
(b) Find the mean value of \(T\).
(c) Find the 95th percentile of \(T\).
(d) State whether you consider this function to be a sensible model for \(T\) and suggest how it could be modified to provide a better model.

1. The number of errors made by a secretary is modelled by a Poisson distribution with a mean of 2.4 per 100 words.

A 100-word piece of work completed by the secretary is selected at random.

1. Find the probability that
   1. there are exactly 3 errors,
   2. there are fewer than 2 errors. After a long holiday, a randomly selected piece of work containing 250 words completed by the secretary is examined to see if the rate of errors has changed.
2. Stating your hypotheses clearly, and using a \(5 \%\) level of significance, find the critical region for a suitable test.
3. Find P (Type I error) for the test in part (b)

1. Bacteria are randomly distributed in a river at a rate of 5 per litre of water. A new factory opens and a scientist claims it is polluting the river with bacteria. He takes a sample of 0.5 litres of water from the river near the factory and finds that it contains 7 bacteria. Stating your hypotheses clearly test, at the \(5 \%\) level of significance, whether there is evidence that the level of pollution has increased.

\section*{Q uestion 1 continued}

5 The number of customers arriving at a store between 8.50 am and 9 am on Saturday mornings is a random variable which can be modelled by the distribution \(\operatorname { Po } ( 11.0 )\). Following a series of price cuts, on one particular Saturday morning 19 customers arrive between 8.50 am and 9 am . The store's management claims, first, that the mean number of customers has increased, and second, that this is due to the price cuts.

1. Test the first part of the claim, at the \(5 \%\) significance level.
2. Comment on the second part of the claim.

8 There are two hospitals in a city. Over a period of time, the first hospital recorded an average of 20 births a day.
Over the same period of time, the second hospital recorded an average of 5 births a day. Stuart claims that birth rates in the hospitals have changed over time.
On a randomly chosen day, he records a total of 16 births from the two hospitals.
8

1. Investigate Stuart's claim, using a suitable test at the \(5 \%\) level of significance.
   8
2. For a test of the type carried out in part (a), find the probability of making a Type I error, giving your answer to two significant figures.
   \includegraphics[max width=\textwidth, alt={}, center]{0d592978-08eb-40a8-ab3b-88339956b89d-16_2490_1735_219_139}

6 The number of computers sold per day by a shop can be modelled by the random variable \(Y\) where \(Y \sim \operatorname { Po } ( 42 )\) 6

1. State the variance of \(Y\) 6
2. One month ago, the shop started selling a new model of computer.
   On a randomly chosen day in the last month, the shop sold 53 computers.
   Carry out a hypothesis test, at the \(5 \%\) level of significance, to investigate whether the mean number of computers sold per day has increased in the last month.
   [0pt] [6 marks]
   6
3. Describe, in the context of the hypothesis test in part (b), what is meant by a Type II error.

7 Over a period of time, it has been shown that the mean number of customers entering a small store is 6 per hour. The store runs a promotion, selling many products at lower prices. 7

1. Luke randomly selects an hour during the promotion and counts 11 customers entering the store. He claims that the promotion has changed the mean number of customers per hour entering the store. Investigate Luke's claim, using the \(5 \%\) level of significance.
   7
2. Luke randomly selects another hour and carries out the same investigation as in part (a). Find the probability of a Type I error, giving your answer to four decimal places.
   Fully justify your answer.
   7
3. When observing the store, Luke notices that some customers enter the store together as a group. Explain why the model used in parts (a) and (b) might not be valid.
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5 Emily claims that the average number of runners per minute passing a shop during a long distance run is 8 Emily conducts a hypothesis test to investigate her claim.
5

1. State the hypotheses for Emily's test. 5
2. Emily counts the number of runners, \(X\), passing the shop in a randomly chosen minute. The critical region for Emily's test is \(X \leq 2\) or \(X \geq 14\)
   During a randomly chosen minute, Emily counts 3 runners passing the shop.
   Determine the outcome of Emily's hypothesis test.
   5
3. The actual average number of runners per minute passing the shop is 7 Find the power of Emily's hypothesis test, giving your answer to three significant figures.

7 A newspaper reports that the average price of unleaded petrol in the UK is 110.2 p per litre. The price, in pence, of a litre of unleaded petrol at a random sample of 15 petrol stations in Yorkshire is shown below together with some output from software used to analyse the data. \begin{table}[h] \end{table}

1. Select a suitable hypothesis test to investigate whether there is any evidence that the average price of unleaded petrol in Yorkshire is different from 110.2 p. Justify your choice of test.
2. Conduct the hypothesis test at the \(5 \%\) level of significance.