Chi-squared goodness of fit: Given ratios

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 A large population of plants consists of five species \(A , B , C , D\) and \(E\) in the proportions \(p _ { A } , p _ { B } , p _ { C } , p _ { D }\) and \(p _ { E }\) respectively. A random sample of 120 plants consisted of \(23,14,24,27\) and 32 of \(A , B , C , D\) and \(E\) respectively. Carry out a test at the \(10 \%\) significance level of the null hypothesis that the proportions are \(p _ { \mathrm { A } } = p _ { \mathrm { B } } = 0.15 , p _ { \mathrm { C } } = p _ { \mathrm { D } } = 0.25\) and \(p _ { \mathrm { E } } = 0.2\).

2

1. A genetic model involving body colour and eye colour of fruit flies predicts that offspring will consist of four phenotypes in the ratio \(9 : 3 : 3 : 1\). A random sample of 200 such offspring is taken. Their phenotypes are found to be as follows.

   Phenotype	Brown body Red eye	Brown body Brown eye	Black body Red eye	Black body Brown eye
   Frequency	125	37	32	6
   Relative proportion from model	9	3	3	1

   Carry out a test, using a \(2.5 \%\) level of significance, of the goodness of fit of the genetic model to these data.
2. The median length of European fruit flies is 2.5 mm . South American fruit flies are believed to be larger than European fruit flies. A random sample of 12 South American fruit flies is taken. The flies are found to have the following lengths (in mm). \(1.7 \quad 1.4\) \(3.1 \quad 3.5\) 3.8
   4.2
   2.2
   2.9
   4.4
   2.6 \(3.9 \quad 3.2\) Carry out a Wilcoxon signed rank test, using a \(5 \%\) level of significance, to test this belief.

9 The proportions of blood types \(\mathrm { A } , \mathrm { B } , \mathrm { AB }\) and O in the Australian population are \(38 \% , 10 \% , 3 \%\) and \(49 \%\) respectively. In order to test whether the population in Sydney conforms to these figures, a random sample of 200 residents is selected. The table shows the observed frequencies of these types in the sample. Carry out a suitable test at the 5\% significance level. Find the smallest sample size that could be used for the test.

7 Benford's Law states that, in many tables containing large numbers of numerical values, the probability distribution of the leading non-zero digit \(D\) is given by $$\mathrm { P } ( D = d ) = \log _ { 10 } \left( \frac { d + 1 } { d } \right) , \quad d = 1,2 , \ldots , 9 .$$ The following table shows a summary of a random sample of 100 non-zero leading digits taken from a table of cumulative probabilities for the Poisson distribution. Carry out a suitable goodness of fit test at the 10\% significance level.

7 In a standard model from genetic theory, the ratios of types \(a , b , c\) and \(d\) of a characteristic from a genetic cross are predicted to be 9:3:3:1. Andrei collects 120 specimens from such a cross, and the numbers corresponding to each type of the characteristic are given in the table. Andrei tests, at the 1\% significance level, whether the observed frequencies are consistent with the standard model.

1. State appropriate hypotheses for the test.
2. Carry out the test.
3. State with a reason which one of the frequencies is least consistent with the standard model.
4. Suggest a different, improved model by changing exactly two of the ratio values.

9 The head teacher of a school believes that, on average, pupil absences on the days Monday, Tuesday, Wednesday, Thursday and Friday are in the ratio \(3 : 2 : 2 : 2 : 3\). The head teacher takes a random sample of 120 pupil absences. The results are as follows.

1. Test at the \(5 \%\) significance level whether these results are consistent with the head teacher's belief. A significance test at the \(5 \%\) level is also carried out on a second, independent, random sample of \(n\) pupil absences. All the numbers of absences are integers. The ratio of the numbers of absences for each day in this sample is identical to the ratio of the numbers of absences for each day in the original sample of size 120.
2. Determine the smallest value of \(n\) for which the conclusion of this significance test is that the data are not consistent with the head teacher's belief.

4. A company selects a random sample of five of its warehouses. The table below summarises the number of employees, in thousands, at each warehouse and the number of reported first aid incidents at each warehouse during 2017 The personnel manager claims that the mean number of reported first aid incidents per 1000 employees is the same at each of the company's warehouses.

1. Stating your hypotheses clearly, use a \(5 \%\) level of significance to test the manager's claim. Jean, the safety officer at warehouse \(C\), kept a record of each reported first aid incident at warehouse \(C\) in 2017. Jean wishes to select a systematic sample of 10 records from warehouse \(C\).
2. Explain, in detail, how Jean should obtain such a sample.

1. The manager of a company making ice cream believes that the proportions of people in the population who prefer vanilla, chocolate, strawberry and other are in the ratio \(10 : 5 : 2 : 3\)

The manager takes a random sample of 400 customers and records their age and favourite ice cream flavour. The results are shown in the table below.

1. Use the data in the table to test, at the \(5 \%\) level of significance, the manager's belief. You should state your hypotheses, test statistic, critical value and conclusion clearly. A researcher wants to investigate whether or not there is a relationship between the age of a customer and their favourite ice cream flavour. In order to test whether favourite ice cream flavour and age are related, the researcher plans to carry out a \(\chi ^ { 2 }\) test.
2. Use the table to calculate expected frequencies for the group
   1. teenagers whose favourite ice cream flavour is vanilla,
   2. adults whose favourite ice cream flavour is chocolate.
3. Write down the number of degrees of freedom for this \(\chi ^ { 2 }\) test.

2. Andy has some apple trees. Over many years she has graded each apple from her trees as \(A , B , C , D\) or \(E\) according to the quality of the apple, with \(A\) being the highest quality and \(E\) being the lowest quality. She knows that the proportion of apples in each grade produced by her trees is as follows. Raj advises Andy to add potassium to the soil around her apple trees. Andy believes that adding potassium will not affect the distribution of grades for the quality of the apples. To test her belief Andy adds potassium to the soil around her apple trees. The following year she counts the number of apples in each grade. The number of apples in each grade is shown in the table below. Test Andy's belief using a \(5 \%\) level of significance. Show your working clearly, stating your hypotheses, expected frequencies and degrees of freedom. 2 continued

1. A spinner used for a game is designed to give scores with the following probabilities

The spinner is spun 80 times and the results are as follows Test, at the \(10 \%\) level of significance, whether or not the spinner is giving scores as it is designed to do. Show your working and state your hypotheses clearly.

7 Josh is investigating whether sticking pins into a map at random, while blindfolded, provides a random sample of regions of the map. Josh divides the map into 49 squares of equal size and asks each of 98 friends to stick a pin into the map at random, while blindfolded. He then notes the number of pins in each square. To analyse the results he groups the squares as shown in the diagram. The results are summarised in the table.

1. Test at the 10\% significance level whether the use of pins in this way provides a random sample of regions of the map.
2. What can be deduced from considering the different contributions to the test statistic? \section*{OCR} \section*{Oxford Cambridge and RSA}

5 Hal designs a 4-edged spinner with edges labelled 1, 2, 3 and 4. He intends that the probability that the spinner will land on any edge should be proportional to the number on that edge. He spins the spinner 20 times and on each spin he records the number of the edge on which it lands. The results are shown in the table. Test at the \(10 \%\) significance level whether the results are consistent with the intended probabilities.

7 Sasha tends to forget his passwords. He investigates whether the number of attempts he needs to log on to a system with a password can be modelled by a geometric distribution. On 60 occasions he records the number of attempts he needs to log on, and the results are shown in the table.

1. Test at the \(1 \%\) significance level whether the results are consistent with the distribution Geo(0.4).
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2. Suggest which two probabilities should be changed, and in what way, to produce an improved model. (Numerical values are not required.) You should give a reason for your suggestion. [3]