Assess model suitability before testing

A family was asked to record the number of letters delivered to their house on each of 200 randomly chosen weekdays. The results are summarised in the following table. It is suggested that the number of letters delivered each weekday has a Poisson distribution. By finding the mean and variance for this sample, comment on the appropriateness of this suggestion. The following table includes some of the expected values, correct to 3 decimal places, using a Poisson distribution with mean equal to the sample mean for the above data.

1. Show that \(p = 46.304\), correct to 3 decimal places, and find \(q\).
2. Carry out a goodness of fit test at the \(10 \%\) significance level.

6 A machine is used to toss a coin repeatedly. Rosa believes that the outcome of each toss made by the machine is not independent of the previous toss. Rosa gets the machine to toss a coin 6 times and record the number of heads, \(X\), obtained. After recording the number of heads obtained, Rosa resets the machine and gets it to toss the coin 6 more times. Rosa again records the number of heads obtained and she repeats this procedure until she has recorded 88 independent values of \(X\).

1. The sample mean and sample variance of \(X\) are 3.35 and 3.392 respectively. Explain what these results suggest about the validity of a binomial model \(\mathrm { B } ( 6 , p )\) for the data. Rosa uses a computer spreadsheet to work out the probabilities for a more sophisticated model in which the outcome of each toss is dependent on the outcome of the previous toss. Her model suggests that the probabilities \(\mathrm { P } ( X = x )\), for \(x = 0,1,2,3,4,5,6\), are approximately in the ratio \(5 : 6 : 7 : 8 : 7 : 6 : 5\). She carries out a \(\chi ^ { 2 }\) test to investigate whether this model is a good fit for the data. The following table shows the full results of the experiments, together with some of the calculations needed for the test.

   \(x\)	0	1	2	3	4	5	6	Total
   Observed frequency	7	10	16	15	15	11	14	88
   Expected frequency
   Contribution to \(\chi ^ { 2 }\) statistic	0.9	0.3333	0.2857	0.0625	0.0714

2. In the Printed Answer Booklet, complete the table.
3. Carry out the test, using a 10\% significance level.
4. Rosa says that the results definitely show that one of the two proposed models is correct. Comment on this statement.

5 Some bird-watchers study the song of chaffinches in a particular wood. They investigate whether the number, \(N\), of separate bursts of song in a 5 minute period can be modelled by a Poisson distribution. They assume that a burst of song can be considered as a single event, and that bursts of song occur randomly. \section*{(a) State two further assumptions needed for \(N\) to be well modelled by a Poisson distribution.} The bird-watchers record the value of \(N\) in each of 60 periods of 5 minutes. The mean and variance of the results are 3.55 and 5.6475 respectively.
(b) Explain what this suggests about the validity of a Poisson distribution as a model in this context. The complete results are shown in the table. The bird-watchers carry out a \(\chi ^ { 2 }\) goodness of fit test at the \(5 \%\) significance level.
(c) State suitable hypotheses for the test.
(d) Determine the contribution to the test statistic for \(n = 3\).
(e) The total value of the test statistic, obtained by combining the cells for \(n \leqslant 1\) and also for \(n \geqslant 6\), is 9.202 , correct to 4 significant figures. Complete the goodness of fit test.
(f) It is known that chaffinches are more likely to sing in the presence of other chaffinches. Explain whether this fact affects the validity of a Poisson model for \(N\).

6 Cosmic rays passing through the upper atmosphere cause muons, and other types of particle, to be formed. Muons can be detected when they reach the surface of the earth. It is known that the mean number of muons reaching a particular detector is 1.7 per second. The numbers of muons reaching this detector in 200 randomly selected periods of 1 second are shown in Fig. 6.1. \begin{table}[h] \end{table}

1. Use the values of the sample mean and sample variance to discuss the suitability of a Poisson distribution as a model. The screenshot in Fig. 6.2 shows part of a spreadsheet to assess the goodness of fit of the distribution Po(1.7). \begin{table}[h]

   	A	B	C	D	E
   1	Number of muons	Observed frequency	Poisson probability	Expected frequency	Chi-squared contribution
   2	0	34	0.1827	36.5367	0.1761
   3	1	65
   4	2	55	0.2640	52.7955	0.0920
   5	3	24	0.1496	29.9175	1.1704
   6	4	14			0.1299
   7	\(\geqslant 5\)	8	0.0296	5.9230	0.7284

   \captionsetup{labelformat=empty} \caption{Fig. 6.2}
   \end{table}
2. Calculate the missing values in each of the following cells.
   Carry out the test at the 5\% significance level.

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. [2]

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.

1. Use the data from this larger sample to carry out a goodness of fit test, at the 10% significance level, to test the claim. [8]