Hypothesis test of a Poisson distribution

4 The number of floods in a certain river plain is known to have a Poisson distribution. It is known that up until 10 years ago the mean number of floods per year was 0.32 . During the last 10 years there were 6 floods. Test at the \(1 \%\) significance level whether there is evidence of an increase in the mean number of floods per year.

1 The number of new enquiries per day at an office has a Poisson distribution. In the past the mean has been 3 . Following a change of staff, the manager wishes to test, at the \(5 \%\) significance level, whether the mean has increased.

1. State the null and alternative hypotheses for this test. The manager notes the number, \(N\), of new enquiries during a certain 6 -day period. She finds that \(N = 25\) and then, assuming that the null hypothesis is true, she calculates that \(\mathrm { P } ( N \geqslant 25 ) = 0.0683\).
2. What conclusion should she draw?

7

1. The random variable \(X\) has a Poisson distribution with parameter \(\lambda\).
   1. Prove, from first principles, that \(\mathrm { E } ( X ) = \lambda\).
   2. Given that \(\mathrm { E } \left( X ^ { 2 } - X \right) = \lambda ^ { 2 }\), deduce that \(\operatorname { Var } ( X ) = \lambda\).
2. The number of faults in a 100-metre ball of nylon string may be modelled by a Poisson distribution with parameter \(\lambda\).
   1. An analysis of one ball of string, selected at random, showed 15 faults. Using an exact test, investigate the claim that \(\lambda > 10\). Use the \(5 \%\) level of significance.
   2. A subsequent analysis of a random sample of 20 balls of string showed a total of 241 faults.
      (A) Using an approximate test, re-investigate the claim that \(\lambda > 10\). Use the \(5 \%\) level of significance.
      (B) Determine the critical value of the total number of faults for the test in part (b)(ii)(A).
      (C) Given that, in fact, \(\lambda = 12\), estimate the probability of a Type II error for a test of the claim that \(\lambda > 10\) based upon a random sample of 20 balls of string and using the \(5 \%\) level of significance.
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3 The random variable \(G\) has the distribution \(\operatorname { Po } ( \lambda )\). A test is carried out of the null hypothesis \(\mathrm { H } _ { 0 } : \lambda = 4.5\) against the alternative hypothesis \(\mathrm { H } _ { 1 } : \lambda \neq 4.5\), based on a single observation of \(G\). The critical region for the test is \(G \leqslant 1\) and \(G \geqslant 9\).

1. Find the significance level of the test.
2. Given that \(\lambda = 5.5\), calculate the probability that the test results in a Type II error.

1. (a) Explain what you understand by
   1. a hypothesis test,
   2. a critical region.
   During term time, incoming calls to a school are thought to occur at a rate of 0.45 per minute. To test this, the number of calls during a random 20 minute interval, is recorded.
   (b) Find the critical region for a two-tailed test of the hypothesis that the number of incoming calls occurs at a rate of 0.45 per 1 minute interval. The probability in each tail should be as close to \(2.5 \%\) as possible.
   (c) Write down the actual significance level of the above test. In the school holidays, 1 call occurs in a 10 minute interval.
   (d) Test, at the \(5 \%\) level of significance, whether or not there is evidence that the rate of incoming calls is less during the school holidays than in term time.

1. A single observation \(x\) is to be taken from a Poisson distribution with parameter \(\lambda\) This observation is to be used to test, at a \(5 \%\) level of significance,

$$\mathrm { H } _ { 0 } : \lambda = k \quad \mathrm { H } _ { 1 } : \lambda \neq k$$ where \(k\) is a positive integer.
Given that the critical region for this test is \(( X = 0 ) \cup ( X \geqslant 9 )\)

1. find the value of \(k\), justifying your answer.
2. Find the actual significance level of this test.

The number of times per day a computer fails and has to be restarted is recorded for 200 days. The results are summarised in the table. Test whether or not a Poisson model is suitable to represent the number of restarts per day. Use a 5\% level of significance and state your hypothesis clearly. (Total 12 marks)

4. The number of emergency plumbing calls received per day by a local council was recorded over a period of 80 days. The results are summarised in the table below.

1. Show that the mean number of emergency plumbing calls received per day is 3.5 A council officer suggests that a Poisson distribution can be used to model the number of emergency plumbing calls received per day. He uses the mean from the sample above and calculates the expected frequencies shown in the table below.

   \(\boldsymbol { x }\)	0	1	2	3	4	5	6	7	8 or more
   Expected frequency	2.42	8.46	14.80	\(r\)	15.10	10.57	6.17	3.08	\(s\)

2. Calculate the value of \(r\) and the value of \(s\), giving your answers correct to 2 decimal places.
3. Test, at the \(5 \%\) level of significance, whether or not the Poisson distribution is a suitable model for the number of emergency plumbing calls received per day. State your hypotheses clearly.

The number of accidents, \(x\), that occur each day on a motorway are recorded over a period of 40 days. The results are shown in the following table.

Number of accidents	0	1	2	3	4	5	6	\(\geqslant 7\)
Observed frequency	3	5	8	10	5	7	2	0

\begin{enumerate}[label=(\roman*)] \item Show that the mean number of accidents each day is 2.95 and calculate the variance for this sample. Explain why these values suggest that a Poisson distribution might fit the data. [3] \item A Poisson distribution with mean 2.95, as found from the data, is used to calculate the expected frequencies, correct to 2 decimal places. The results are shown in the following table.

Number of accidents	0	1	2	3	4	5	6	\(\geqslant 7\)
Observed frequency	3	5	8	10	5	7	2	0
Expected frequency	2.09	6.18	9.11	8.96	6.61	3.90	1.92	1.23

Show how the expected frequency of 6.61 for \(x = 4\) is obtained. [2] \item Test at the 5% significance level the goodness of fit of this Poisson distribution to the data. [7] \end{enumerate]

4 At a doctors' surgery, the number of missed appointments per day has a Poisson distribution. In the past the mean number of missed appointments per day has been 0.9 . Following some publicity, the manager carries out a hypothesis test to determine whether this mean has decreased. If there are fewer than 3 missed appointments in a randomly chosen 5-day period, she will conclude that the mean has decreased.

1. Find the probability of a Type I error.
2. State what is meant by a Type I error in this context.
3. Find the probability of a Type II error if the mean number of missed appointments per day is 0.2 .

5. The number of tornadoes per year to hit a particular town follows a Poisson distribution with mean \(\lambda\). A weatherman claims that due to climate changes the mean number of tornadoes per year has decreased. He records the number of tornadoes \(x\) to hit the town last year. To test the hypotheses \(\mathrm { H } _ { 0 } : \lambda = 7\) and \(\mathrm { H } _ { 1 } : \lambda < 7\), a critical region of \(x \leq 3\) is used.

1. Find, in terms \(\lambda\) the power function of this test.
2. Find the size of this test.
3. Find the probability of a Type II error when \(\lambda = 4\).

4 The number of severe floods per year in a certain country over the last 100 years has followed a Poisson distribution with mean 1.8. Scientists suspect that global warming has now increased the mean. A hypothesis test, at the \(5 \%\) significance level, is to be carried out to test this suspicion. The number of severe floods, \(X\), that occur next year will be used for the test.

1. Show that the rejection region for the test is \(X > 4\).
2. Find the probability of making a Type II error if the mean number of severe floods is now actually 2.3.

The number of accidents per month at a certain road junction has a Poisson distribution with mean 4.8. A new road sign is introduced warning drivers of the danger ahead, and in a subsequent month 2 accidents occurred.

1. A hypothesis test at the 10% level is used to determine whether there were fewer accidents after the new road sign was introduced. Find the critical region for this test and carry out the test. [5]
2. Find the probability of a Type I error. [2]

4 At a certain company, computer faults occur randomly and at a constant mean rate. In the past this mean rate has been 2.1 per week. Following an update, the management wish to determine whether the mean rate has changed. During 20 randomly chosen weeks it is found that 54 computer faults occur. Use a suitable approximation to test at the \(5 \%\) significance level whether the mean rate has changed.

7

1. Test the scientist's claim, using the 10\% level of significance.
   7
2. For the context of the test carried out in part (a), state the meaning of a Type I error. [1 mark]

1. The number of telephone calls per hour received by a business is a random variable with distribution \(\operatorname { Po } ( \lambda )\).

Charlotte records the number of calls, \(C\), received in 4 hours. A test of the null hypothesis \(\mathrm { H } _ { 0 } : \lambda = 1.5\) is carried out. \(\mathrm { H } _ { 0 }\) is rejected if \(C > 10\)

1. Write down the alternative hypothesis.
2. Find the significance level of the test. Given that \(\mathrm { P } ( C > 10 ) < 0.1\)
3. find the largest possible value of \(\lambda\) that can be found by using the tables.

2. The cloth produced by a certain manufacturer has defects that occur randomly at a constant rate of \(\lambda\) per square metre. If \(\lambda\) is thought to be greater than 1.5 then action has to be taken. Using \(\mathrm { H } _ { 0 } : \lambda = 1.5\) and \(\mathrm { H } _ { 1 } : \lambda > 1.5\) a quality control officer takes a \(4 \mathrm {~m} ^ { 2 }\) sample of cloth and rejects \(\mathrm { H } _ { 0 }\) if there are 11 or more defects. If there are 8 or fewer defects she accepts \(\mathrm { H } _ { 0 }\). If there are 9 or 10 defects a second sample of \(4 \mathrm {~m} ^ { 2 }\) is taken and \(\mathrm { H } _ { 0 }\) is rejected if there are 11 or more defects in this second sample, otherwise it is accepted.

1. Find the size of this test.
2. Find the power of this test when \(\lambda = 2\)

Car parking in a market town's high street was, until 31 May 2014, limited to one hour free of charge between 8 am and 6 pm. Records show that, during a period of 30 days prior to this date, a total of 315 penalty tickets were issued. Car parking in the high street later became limited to thirty minutes free of charge between 8 am and 6 pm. A subsequent investigation revealed that, during a period of 60 days from 1 October 2014, a total of 747 penalty tickets were issued. The daily numbers of penalty tickets issued may be modelled by independent Poisson distributions with means \(\lambda_A\) until 31 May 2014 and \(\lambda_B\) from 1 October 2014. Investigate, at the 1\% level of significance, a claim by traders on the high street that \(\lambda_B > \lambda_A\). [7 marks]

8 In excavating an archaeological site, Roman coins are found scattered throughout the site.

1. State two assumptions needed to model the number of coins found per square metre of the site by a Poisson distribution. Assume now that the number of coins found per square metre of the site can be modelled by a Poisson distribution with mean \(\lambda\).
2. Given that \(\lambda = 0.75\), calculate the probability that exactly 3 coins are found in a region of the site of area \(7.20 \mathrm {~m} ^ { 2 }\). A test is carried out, at the \(5 \%\) significance level, of the null hypothesis \(\lambda = 0.75\), against the alternative hypothesis \(\lambda > 0.75\), in Region LVI which has area \(4 \mathrm {~m} ^ { 2 }\).
3. Determine the smallest number of coins that, if found in Region LVI, would lead to rejection of the null hypothesis, stating also the values of any relevant probabilities.
4. Given that, in fact, \(\lambda = 1.2\) in Region LVI, find the probability that the test results in a Type II error.