Standard 2×2 contingency table

4 A study in 1981 investigated the effect of water fluoridation on children's dental health. In a town with fluoridation, 61 out of a random sample of 107 children showed signs of increased tooth decay after six months. In a town without fluoridation the corresponding number was 106 out of a random sample of 143 children. The population proportions of children with increased tooth decay are denoted by \(p _ { 1 }\) and \(p _ { 2 }\) for the towns with fluoridation and without fluoridation respectively. A test is carried out of the null hypothesis \(p _ { 1 } = p _ { 2 }\) against the alternative hypothesis \(p _ { 1 } < p _ { 2 }\). Find the smallest significance level at which the null hypothesis is rejected.

6 Certain types of food are now sold in metric units. A random sample of 1000 shoppers was asked whether they were in favour of the change to metric units or not. The results, classified according to age, were as shown in the table.

1. Use a \(\chi ^ { 2 }\) test to show that there is very strong evidence that shoppers' views about changing to metric units are not independent of their ages.
2. The data may also be regarded as consisting of two random samples of shoppers; one sample consists of 470 shoppers aged under 35 , of whom 187 were in favour of change, and the second sample consists of 530 shoppers aged 35 or over, of whom 161 were in favour of change. Determine whether a test for equality of population proportions supports the conclusion in part (i).

4

1. A random sample of 60 students studying mathematics was selected. Their grades in the Core 1 module are summarised in the table below, classified according to whether they worked less than 5 hours per week or at least 5 hours per week. Test, at the \(5 \%\) significance level, whether there is any association between grade and hours worked.

   		Hours worked
   \cline { 3 - 4 } \multicolumn{2}{|c|}{}	Less than 5	At least 5
   \multirow{2}{*}{Grade}	A or B	20	11
   \cline { 2 - 4 }	C or lower	13	16

2. At a canning factory, cans are filled with tomato purée. The machine which fills the cans is set so that the volume of tomato purée in a can, measured in millilitres, is Normally distributed with mean 420 and standard deviation 3.5. After the machine is recalibrated, a quality control officer wishes to check whether the mean is still 420 millilitres. A random sample of 10 cans of tomato purée is selected and the volumes, measured in millilitres, are as follows. $$\begin{array} { l l l l l l l l l l } 417.2 & 422.6 & 414.3 & 419.6 & 420.4 & 410.0 & 418.3 & 416.9 & 418.9 & 419.7 \end{array}$$ Carry out a test at the \(1 \%\) significance level to investigate whether the mean is still 420 millilitres. You should assume that the volumes are Normally distributed with unchanged standard deviation.

4

1. As part of an investigation into smoking, a random sample of 120 students was selected. The students were asked whether they were smokers, and also whether either of their parents were smokers. The results are summarised in the table below. Test, at the \(5 \%\) significance level, whether there is any association between the smoking habits of the students and their parents.

   	At least one parent smokes	Neither parent smokes
   Student smokes	21	27
   Student does not smoke	17	55

2. The manufacturer of a particular brand of cigarette claims that the nicotine content of these cigarettes is Normally distributed with mean 0.87 mg . A researcher suspects that the mean nicotine content of this brand is higher than the value claimed by the manufacturer. The nicotine content, \(x \mathrm { mg }\), is measured for a random sample of 100 cigarettes. The data are summarised as follows. $$\sum x = 88.20 \quad \sum x ^ { 2 } = 78.68$$ Carry out a test at the \(1 \%\) significance level to investigate the researcher's belief. \section*{END OF QUESTION PAPER}

8 After contracting a particular disease, patients from a hospital are advised to have their blood tested monthly for a year. In order to test whether patients comply with this advice the hospital management commissioned a survey of 100 patients. A hospital statistician selected the patients randomly from records and asked the patients whether or not they had complied with the advice. The results classified by gender are as follows.

1. Test at the \(5 \%\) significance level whether compliance with the advice is independent of gender.
2. A manager believed that a greater proportion of female patients than male patients comply with the advice. Carry out an appropriate test of proportions at the \(10 \%\) significance level.

3 The developers of a shopping mall sponsored a study of the shopping habits of its users. Each of a random sample of 100 users was asked whether their weekend shopping was mainly on Saturday or mainly on Sunday. The results, classified according to whether the user lived in the city or the country, are shown in the table.

1. Test, at the \(10 \%\) significance level, whether there is an association between the area in which shoppers live and the day on which they shop at the weekend.
2. State, with a reason, whether the conclusion of the test would be different at the \(3 \%\) significance level.

3

1. A personal trainer believes that drinking a glass of beetroot juice an hour before exercising enables endurance tests to be completed more quickly. To test his belief he takes a random sample of 12 of his trainees and, on two occasions, asks them to carry out 100 repetitions of a particular exercise as quickly as possible. Each trainee drinks a glass of water on one occasion and a glass of beetroot juice on the other occasion. The times in seconds taken by the trainees are given in the table.

   Trainee	Water	Beetroot juice
   A	75.1	72.9
   B	86.2	79.9
   C	77.3	71.6
   D	89.1	90.2
   E	67.9	68.2
   F	101.5	95.2
   G	82.5	76.5
   H	83.3	80.2
   I	102.5	99.1
   J	91.3	82.2
   K	92.5	90.1
   L	77.2	77.9

   The trainer wishes to test his belief using a paired \(t\) test at the \(1 \%\) level of significance. Assuming any necessary assumptions are valid, carry out a test of the hypotheses \(\mathrm { H } _ { 0 } : \mu _ { D } = 0 , \mathrm { H } _ { 1 } : \mu _ { D } < 0\), where \(\mu _ { D }\) is the population mean difference in times (time with beetroot juice minus time with water).
2. An ornithologist believes that the number of birds landing on the bird feeding station in her garden in a given interval of time during the morning should follow a Poisson distribution. In order to test her belief, she makes the following observations in 60 randomly chosen minutes one morning.

   Number of birds	0	1	2	3	4	5	6	\(\geqslant 7\)
   Frequency	2	5	10	17	14	7	4	1

   Given that the data in the table have a mean value of 3.3, use a goodness of fit test, with a significance level of \(5 \%\), to investigate whether the ornithologist is justified in her belief.

4 A school pupil keeps a note of whether her journeys to school and from school are delayed. The results for a random sample of journeys are shown in the table. Test at the 10\% significance level whether there is association between delays and the direction of the journey.

6 Yusha investigates the proportion of left-handed people living in two cities, \(A\) and \(B\). He obtains data from random samples from the two cities. His results are shown in the table, in which \(L\) denotes "left-handed".

1. Test at the 10\% significance level whether there is association between being left-handed and living in a particular city. A person is chosen at random from one of the cities \(A\) and \(B\).
   Let \(A\) denote "the person lives in city \(A\) ".
2. State the relationship between \(\mathrm { P } ( L )\) and \(P ( L \mid A )\) according to the model implied by the null hypothesis of your test.
3. Use the data in the table to suggest a value for \(P ( L \mid A )\) given by an improved model.

1. A doctor believes that the diet of her patients and their health are not independent.

She takes a random sample of 200 patients and records whether they are in good health or poor health and whether they have a good diet or a poor diet. The results are summarised in the table below. Stating your hypotheses clearly, test the doctor's belief using a \(5 \%\) level of significance. Show your working for your test statistic and state your critical value clearly.

4. People over the age of 65 are offered an annual flu injection. A health official took a random sample from a list of patients who were over 65 . She recorded their gender and whether or not the offer of an annual flu injection was accepted or rejected. The results are summarised below. Using a \(5 \%\) significance level, test whether or not there is an association between gender and acceptance or rejection of an annual flu injection. State your hypotheses clearly.

4. A new drug to treat the common cold was used with a randomly selected group of 100 volunteers. Each was given the drug and their health was monitored to see if they caught a cold. A randomly selected control group of 100 volunteers was treated with a dummy pill. The results are shown in the table below. Using a \(5 \%\) significance level, test whether or not the chance of catching a cold is affected by taking the new drug. State your hypotheses clearly.

1. A doctor takes a random sample of 100 patients and measures their intake of saturated fats in their food and the level of cholesterol in their blood. The results are summarised in the table below.

Using a \(5 \%\) level of significance, test whether or not there is an association between cholesterol level and intake of saturated fats. State your hypotheses and show your working clearly.

2 Syd, a snooker player, believes that the outcome of any frame of snooker in which he plays may be influenced by the time of day that the frame takes place. The results of 100 randomly selected frames of snooker, played by Syd, are recorded below. Use a \(\chi ^ { 2 }\) test, at the \(5 \%\) level of significance, to test Syd's belief.
(10 marks)

2

1. The continuous random variable \(X\) has a rectangular distribution defined by the probability density function $$f ( x ) = \begin{cases} 0.01 \pi & u \leqslant x \leqslant 11 u \\ 0 & \text { otherwise } \end{cases}$$ where \(u\) is a constant.
   1. Show that \(u = \frac { 10 } { \pi }\).
   2. Using the formulae for the mean and the variance of a rectangular distribution, find, in terms of \(\pi\), values for \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
   3. Calculate exact values for the mean and the variance of the circumferences of circles having diameters of length \(\left( X + \frac { 10 } { \pi } \right)\).
2. A machine produces circular discs which have an area of \(Y \mathrm {~cm} ^ { 2 }\). The distribution of \(Y\) has mean \(\mu\) and variance 25 . A random sample of 100 such discs is selected. The mean area of the discs in this sample is calculated to be \(40.5 \mathrm {~cm} ^ { 2 }\). Calculate a 95\% confidence interval for \(\mu\). Emily believed that the performances of 16-year-old students in their GCSEs are associated with the schools that they attend. To investigate her belief, Emily collected data on the GCSE results for 2010 from four schools in her area. The table shows Emily's collected data, denoted by \(O _ { i }\), together with the corresponding expected frequencies, \(E _ { i }\), necessary for a \(\chi ^ { 2 }\) test.

   \multirow{2}{*}{}	\(\boldsymbol { \geq } \mathbf { 5 }\) GCSEs		\(\mathbf { 1 } \boldsymbol { \leqslant }\) GCSEs < \(\mathbf { 5 }\)		No GCSEs
   	\(O _ { i }\)	\(E _ { i }\)	\(O _ { i }\)	\(E _ { i }\)	\(O _ { i }\)	\(E _ { i }\)
   Jolliffe College for the Arts	187	193.15	93	90.62	30	26.23
   Volpe Science Academy	175	184.43	97	86.52	24	25.05
   Radok Music School	183	183.81	78	86.23	34	24.96
   Bailey Language School	265	248.61	112	116.63	22	33.76

   Emily used these values to correctly conduct a \(\chi ^ { 2 }\) test at the \(1 \%\) level of significance.

9 A cyclist who lives on an island suspects that car drivers with locally registered number plates allow more space when passing her than those with non-locally registered number plates. She decides to carry out a hypothesis test and so over a period of time selects a random sample of 250 cars which pass her. For each car she estimates whether the car driver allows at least the recommended 1.5 metres when passing her. The table shows the data which she collected.

1. In this question you must show detailed reasoning. Carry out the test at the \(5 \%\) significance level to examine whether there is any association between where the car is registered and passing distance.
2. A friend of the cyclist suggests that there may be a problem with the data, since the cyclist may have introduced some bias in estimating whether cars were allowing the recommended distance. Explain how any bias might have arisen.

3 In this question you must show detailed reasoning. A student is investigating what people think about organic food. She wishes to see if there is any difference between the opinions of females and males. She takes a random sample of 100 people and asks each of them if they think that organic food is better for their health than non-organic food. She will use the data to conduct a hypothesis test. The table below shows the opinions of these 100 people.

1. Explain why the student should use a random sample.
2. Carry out a test at the \(5 \%\) significance level to examine whether there is any association between a person's sex and their opinion on organic food. Show your calculations.

3 In this question you must show detailed reasoning. In a survey into pet ownership, one of the questions was 'Do you own either a cat or a dog (or both)?'. A total of 121 people took part in the survey and you should assume that they form a random sample of people in a particular town. The results, classified by the age of the person being surveyed, are shown in Table 3. \begin{table}[h] \end{table} Carry out a test at the 10\% significance level to investigate whether, for people in this town, there is any association between age and ownership of a cat or dog.

1 Two groups of patients, suffering from the same medical condition, took part in a clinical trial of a new drug. One of the groups was given the drug whilst the other group was given a placebo, a drug that has no physical effect on their medical condition. The table shows the number of patients in each group and whether or not their condition improved. Conduct a \(\chi ^ { 2 }\) test, at the \(5 \%\) level of significance, to determine whether the condition of the patients at the conclusion of the trial is associated with the treatment that they were given.
(10 marks)

A survey in a college was commissioned to investigate whether or not there was any association between gender and passing a driving test. A group of 50 male and 50 female students were asked whether they passed or failed their driving test at the first attempt. All the students asked had taken the test. The results were as follows. Stating your hypotheses clearly test, at the 10\% level, whether or not there is any evidence of an association between gender and passing a driving test at the first attempt. [11]

It is claimed that a new drug is effective in the prevention of sickness in holiday-makers. A sample of \(100\) holiday-makers was surveyed, with the following results. Assuming that the \(100\) holiday-makers are a random sample, use a \(\chi^2\) test, at the \(5\%\) level of significance, to investigate the claim. [8 marks]

Two schools in the same town advertise at the same time for new heads of English and History departments. The number of applicants for each post are shown in the table below. Stating your hypotheses clearly, test at the 10\% level of significance whether or not there is evidence of the proportion of applicants for each job being different in the two schools. [11 marks]

Carl believes that the proportions of men and women who own black cars are different. He obtained a random sample of people who each owned exactly one car. The results are summarised in the table below.

	Black	Non-black
Men	69	71
Women	30	55

Test at the 5\% significance level whether Carl's belief is justified. [8]

Carl believes that the proportions of men and women who own black cars are different. He obtained a random sample of people who each owned exactly one car. The results are summarised in the table below. Test at the 5\% significance level whether Carl's belief is justified. [8]