5.06b Fit prescribed distribution: chi-squared test

4 A scientist is investigating the numbers of a particular type of butterfly in a certain region. He claims that the numbers of these butterflies found per square metre can be modelled by a Poisson distribution with mean 2.5. He takes a random sample of 120 areas, each of one square metre, and counts the number of these butterflies in each of these areas. The following table shows the observed frequencies together with some of the expected frequencies using the scientist's Poisson distribution.

1. Find the values of \(p\) and \(q\), correct to 2 decimal places.
2. Carry out a goodness of fit test, at the \(10 \%\) significance level, to test the scientist's claim.

3 A random sample of 50 values of the continuous random variable \(X\) was taken. These values are summarised in the following table. It is required to test the goodness of fit of the distribution with probability density function \(f\) given by $$f ( x ) = \begin{cases} \frac { 1 } { 24 } \left( \frac { 4 } { x ^ { 2 } } + x ^ { 2 } \right) & 1 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$ The expected frequencies, correct to 4 decimal places, are given in the following table.

1. Show that \(a = 4.6007\) and find the value of \(b\).
2. Carry out a goodness of fit test, at the \(10 \%\) significance level, to test whether f is a satisfactory model for the data.

3 Apples are sold in bags of 5. Based on her previous experience, Freya claims that the probability of any apple weighing more than 100 grams is 0.35 , independently of other apples in the bag. The apples in a random sample of 150 bags are checked and the number, \(x\), in each bag weighing more than 100 grams is recorded. The results are shown in the following table. Carry out a goodness of fit test at the \(5 \%\) significance level and hence comment on Freya's claim.

3 A supermarket sells pears in packs of 8 . Some of the pears in a pack may not be ripe, and the supermarket manager claims that the number of unripe pears in a pack can be modelled by the distribution \(\mathrm { B } ( 8,0.15 )\). A random sample of 150 packs was selected and the number of unripe pears in each pack was recorded. The following table shows the observed frequencies together with some of the expected frequencies using the manager's binomial distribution.

1. Find the values of \(p\) and \(q\).
2. Carry out a goodness of fit test, at the \(5 \%\) significance level, to test whether the manager's claim is justified.

2 An organisation runs courses to train students to become engineers. These students are taught in groups of 8 . The director of the organisation claims that on average \(60 \%\) of the students in a group achieve a pass. A random sample of 150 groups of 8 students is chosen. The following table shows the observed frequencies together with some of the expected frequencies using the appropriate binomial distribution.

1. Find the values of \(p , q\) and \(r\) giving your answers correct to 3 decimal places.
2. Carry out a goodness of fit test, at the \(10 \%\) significance level, to test whether there is evidence to reject the director's claim.

3 A statistician believes that the number of telephone calls received by an advice centre in a 10 -minute interval can be modelled by the Poisson distribution \(\mathrm { Po } ( 1.9 )\). The number of calls received in a randomly chosen 10-minute interval was recorded on each of 100 days. The results are summarised in the table, together with some of the expected frequencies corresponding to the distribution \(\operatorname { Po } ( 1.9 )\).

1. Complete the table.
2. Carry out a goodness of fit test, at the \(10 \%\) significance level, to determine whether the statistician's belief is reasonable. \includegraphics[max width=\textwidth, alt={}, center]{e2a45d19-7d48-4aa5-93f9-6ef90f99d7c4-07_2726_35_97_20}

3 Rosie sows 5 seeds in each of 150 plant pots. The number of seeds that germinate is recorded for each pot. The results are summarised in the following table. Rosie suggests that the number of seeds that germinate follows the binomial distribution \(\mathrm { B } ( 5 , p )\).

1. Use Rosie's results to show that \(p = 0.42\).
2. Carry out a goodness of fit test, at the \(10 \%\) significance level, to test whether the distribution \(\mathrm { B } ( 5,0.42 )\) is a good fit for the data. \includegraphics[max width=\textwidth, alt={}, center]{b9cbf607-4f40-41bb-8374-6b2c39f945ac-06_2720_38_109_2010} \includegraphics[max width=\textwidth, alt={}, center]{b9cbf607-4f40-41bb-8374-6b2c39f945ac-07_2726_35_97_20}

3 A statistician believes that the number of telephone calls received by an advice centre in a 10 -minute interval can be modelled by the Poisson distribution \(\mathrm { Po } ( 1.9 )\). The number of calls received in a randomly chosen 10-minute interval was recorded on each of 100 days. The results are summarised in the table, together with some of the expected frequencies corresponding to the distribution \(\operatorname { Po } ( 1.9 )\).

1. Complete the table.
2. Carry out a goodness of fit test, at the \(10 \%\) significance level, to determine whether the statistician's belief is reasonable. \includegraphics[max width=\textwidth, alt={}, center]{8b2a13d7-62f4-45a7-84c5-7d5bc870b8ce-07_2726_35_97_20}

5 A music store sells both upright and grand pianos. Grand pianos are sold at random times and at a constant average weekly rate \(\lambda\). The probability that in one week no grand pianos are sold is 0.45 .

1. Show that \(\lambda = 0.80\), correct to 2 decimal places. Upright pianos are sold, independently, at random times and at a constant average weekly rate \(\mu\). During a period of 100 weeks the store sold 180 upright pianos.
2. Calculate the probability that the total number of pianos sold in a randomly chosen week will exceed 3.
3. Calculate the probability that over a period of 3 weeks the store sells a total of 6 pianos during the first week and a total of 4 pianos during the next fortnight.

4 An experiment by Lord Rutherford at Cambridge in 1909 involved measuring the numbers of \(\alpha\)-particles emitted during radioactive decay. The following table shows emissions during 2608 intervals of 7.5 seconds. It is given that the mean number of particles emitted per interval, calculated from the data, is 3.87 , correct to 3 significant figures.

1. Find the contribution to the \(\chi ^ { 2 }\) value of the frequency of 273 corresponding to \(x = 6\) in a goodness of fit test for a Poisson distribution.
2. Given that no cells need to be combined, state why the number of degrees of freedom is 10 .
3. Given also that the calculated value of \(\chi ^ { 2 }\) is 13.0 , correct to 3 significant figures, carry out the test at the 10\% significance level.

4

1. An amateur weather forecaster has been keeping records of air pressure, measured in atmospheres. She takes the measurement at the same time every day using a barometer situated in her garden. A random sample of 100 of her observations is summarised in the table below. The corresponding expected frequencies for a Normal distribution, with its two parameters estimated by sample statistics, are also shown in the table.

   Pressure ( \(a\) atmospheres)	Observed frequency	Frequency as given by Normal model
   \(a \leqslant 0.98\)	4	1.45
   \(0.98 < a \leqslant 0.99\)	6	5.23
   \(0.99 < a \leqslant 1.00\)	9	13.98
   \(1.00 < a \leqslant 1.01\)	15	23.91
   \(1.01 < a \leqslant 1.02\)	37	26.15
   \(1.02 < a \leqslant 1.03\)	21	18.29
   \(1.03 < a\)	8	10.99

   Carry out a test at the \(5 \%\) level of significance of the goodness of fit of the Normal model. State your conclusion carefully and comment on your findings.
2. The forecaster buys a new digital barometer that can be linked to her computer for easier recording of observations. She decides that she wishes to compare the readings of the new barometer with those of the old one. For a random sample of 10 days, the readings (in atmospheres) of the two barometers are shown below.

   Day	A	B	C	D	E	F	G	H	I	J
   Old	0.992	1.005	1.001	1.011	1.026	0.980	1.020	1.025	1.042	1.009
   New	0.985	1.003	1.002	1.014	1.022	0.988	1.030	1.016	1.047	1.025

   Use an appropriate Wilcoxon test to examine at the \(10 \%\) level of significance whether there is any reason to suppose that, on the whole, readings on the old and new barometers do not agree.

1 Design engineers are simulating the load on a particular part of a complex structure. They intend that the simulated load, measured in a convenient unit, should be given by the random variable \(X\) having probability density function $$f ( x ) = 12 x ^ { 3 } - 24 x ^ { 2 } + 12 x , \quad 0 \leqslant x \leqslant 1 .$$

1. Find the mean and the mode of \(X\).
2. Find the cumulative distribution function \(\mathrm { F } ( x )\) of \(X\). $$\text { Verify that } \mathrm { F } \left( \frac { 1 } { 4 } \right) = \frac { 67 } { 256 } , \mathrm {~F} \left( \frac { 1 } { 2 } \right) = \frac { 11 } { 16 } \text { and } \mathrm { F } \left( \frac { 3 } { 4 } \right) = \frac { 243 } { 256 } .$$ The engineers suspect that the process for generating simulated loads might not be working as intended. To investigate this, they generate a random sample of 512 loads. These are recorded in a frequency distribution as follows.

   Load \(x\)	\(0 \leqslant x \leqslant \frac { 1 } { 4 }\)	\(\frac { 1 } { 4 } < x \leqslant \frac { 1 } { 2 }\)	\(\frac { 1 } { 2 } < x \leqslant \frac { 3 } { 4 }\)	\(\frac { 3 } { 4 } < x \leqslant 1\)
   Frequency	126	209	131	46

3. Use a suitable statistical procedure to assess the goodness of fit of \(X\) to these data. Discuss your conclusions briefly.

4 An experiment is carried out to compare five industrial paints, A, B, C, D, E, that are intended to be used to protect exterior surfaces in polluted urban environments. Five different types of surface (I, II, III, IV, V) are to be used in the experiment, and five specimens of each type of surface are available. Five different external locations ( \(1,2,3,4,5\) ) are used in the experiment. The paints are applied to the specimens of the surfaces which are then left in the locations for a period of six months. At the end of this period, a "score" is given to indicate how effective the paint has been in protecting the surface.

1. Name a suitable experimental design for this trial and give an example of an experimental layout. Initial analysis of the data indicates that any differences between the types of surface are negligible, as also are any differences between the locations. It is therefore decided to analyse the data by one-way analysis of variance.
2. State the usual model, including the accompanying distributional assumptions, for the one-way analysis of variance. Interpret the terms in the model.
3. The data for analysis are as follows. Higher scores indicate better performance.

   Paint A	Paint B	Paint C	Paint D	Paint E
   64	66	59	65	64
   58	68	56	78	52
   73	76	69	69	56
   60	70	60	72	61
   67	71	63	71	58

   [The sum of these data items is 1626 and the sum of their squares is 106838 .]
   Construct the usual one-way analysis of variance table. Carry out the appropriate test, using a 5\% significance level. Report briefly on your conclusions.
   [0pt] [12]

4 An agricultural company conducts a trial of five fertilisers (A, B, C, D, E) in an experimental field at its research station. The fertilisers are applied to plots of the field according to a completely randomised design. The yields of the crop from the plots, measured in a standard unit, are analysed by the one-way analysis of variance, from which it appears that there are no real differences among the effects of the fertilisers. A statistician notes that the residual mean square in the analysis of variance is considerably larger than had been anticipated from knowledge of the general behaviour of the crop, and therefore suspects that there is some inadequacy in the design of the trial.

1. Explain briefly why the statistician should be suspicious of the design.
2. Explain briefly why an inflated residual leads to difficulty in interpreting the results of the analysis of variance, in particular that the null hypothesis is more likely to be accepted erroneously. Further investigation indicates that the soil at the west side of the experimental field is naturally more fertile than that at the east side, with a consistent 'fertility gradient' from west to east.
3. What experimental design can accommodate this feature? Provide a simple diagram of the experimental field indicating a suitable layout. The company decides to conduct a new trial in its glasshouse, where experimental conditions can be controlled so that a completely randomised design is appropriate. The yields are as follows.

   Fertiliser A	Fertiliser B	Fertiliser C	Fertiliser D	Fertiliser E
   23.6	26.0	18.8	29.0	17.7
   18.2	35.3	16.7	37.2	16.5
   32.4	30.5	23.0	32.6	12.8
   20.8	31.4	28.3	31.4	20.4

   [The sum of these data items is 502.6 and the sum of their squares is 13610.22 .]
4. Construct the usual one-way analysis of variance table. Carry out the appropriate test, using a \(5 \%\) significance level. Report briefly on your conclusions.
5. State the assumptions about the distribution of the experimental error that underlie your analysis in part (iv).

2 In a study of the inheritance of skin colouration in corn snakes, a researcher found 865 snakes with black and orange bodies, 320 snakes with black bodies, 335 snakes with orange bodies and 112 snakes with bodies of other colours. Theory predicts that snakes of these colours should occur in the ratios \(9 : 3 : 3 : 1\). Test, at the \(5 \%\) significance level, whether these experimental results are compatible with theory.

6 In each of 38 randomly selected weeks of the English Premier Football League there were 10 matches. Table 1 summarises the number of home wins in 10 matches, \(X\), and the corresponding number of weeks. \begin{table}[h] \end{table} A researcher investigates whether \(X\) can be modelled by the distribution \(\mathrm { B } ( 10 , p )\). He calculates the expected frequencies using a value of \(p\) obtained from the sample mean.

1. Show that \(p = 0.45\). Table 2 shows the observed and expected number of weeks. \begin{table}[h]

   Number of home wins	0	1	2	3	4	5	6	7	8	9	10	Totals
   Observed number of weeks	0	1	2	8	8	9	7	1	2	0	0	38
   Expected number of weeks	0.096	0.788	2.899	6.326	9.058	8.893	6.064	2.835	0.870	0.158	0.013	38

   \captionsetup{labelformat=empty} \caption{Table 2
2. Show how the value of 2.835 for 7 home wins is obtained.}