Questions - OCR MEI S2 - A-Level Maths

State conditions for a Poisson distribution to be a suitable model for the number of faults occurring on a randomly selected day. Faults are detected at an average rate of 0.15 per day. You may assume that a Poisson distribution is a suitable model.
Find the probability that on a randomly chosen day there are
(A) no faults,
(B) at least 2 faults.
Find the probability that, in a randomly chosen period of 30 days, there are at most 3 faults. There is also a separate safety system to detect faults on the roller-coaster train itself. Faults are detected by this system at an average rate of 0.05 per day, independently of the faults detected on the track. You may assume that a Poisson distribution is also suitable for modelling the number of faults detected on the train.
State the distribution of the total number of faults detected by the two systems in a period of 10 days. Find the probability that a total of 5 faults is detected in a period of 10 days.
[0pt]
The roller-coaster is operational for 200 days each year. Use a suitable approximating distribution to find the probability that a total of at least 50 faults is detected in 200 days. [5]

OCR MEI S2 2006 January Q2

2 The drug EPO (erythropoetin) is taken by some athletes to improve their performance. This drug is in fact banned and blood samples taken from athletes are tested to measure their 'hematocrit level'. If the level is over 50 it is considered that the athlete is likely to have taken EPO and the result is described as 'positive'. The measured hematocrit level of each athlete varies over time, even if EPO has not been taken.

For each athlete in a large population of innocent athletes, the variation in measured hematocrit level is described by the Normal distribution with mean 42.0 and standard deviation 3.0.
(A) Show that the probability that such an athlete tests positive for EPO in a randomly chosen test is 0.0038 .
(B) Find the probability that such an athlete tests positive on at least 1 of the 7 occasions during the year when hematocrit level is measured. (These occasions are spread at random through the year and all test results are assumed to be independent.)
(C) It is standard policy to apply a penalty after testing positive. Comment briefly on this policy in the light of your answer to part (i)(B).
Suppose that 1000 tests are carried out on innocent athletes whose variation in measured hematocrit level is as described in part (i). It may be assumed that the probability of a positive result in each test is 0.0038 , independently of all other test results.
(A) State the exact distribution of the number of positive tests.
(B) Use a suitable approximating distribution to find the probability that at least 10 tests are positive.
Because of genetic factors, a particular innocent athlete has an abnormally high natural hematocrit level. This athlete's measured level is Normally distributed with mean 48.0 and standard deviation 2.0. The usual limit of 50 for a positive test is to be altered for this athlete to a higher value $h$. Find the value of $h$ for which this athlete would test positive on average just once in 200 occasions.

OCR MEI S2 2006 January Q3

3 A researcher is investigating the relationship between temperature and levels of the air pollutant nitrous oxide at a particular site. The researcher believes that there will be a positive correlation between the daily maximum temperature, $x$, and nitrous oxide level, $y$. Data are collected for 10 randomly selected days. The data, measured in suitable units, are given in the table and illustrated on the scatter diagram.

$x$	13.3	17.2	16.9	18.7	18.4	19.3	23.1	15.0	20.6	14.4
$y$	9	11	14	26	43	25	52	15	10	7

\includegraphics[max width=\textwidth, alt={}, center]{794b337f-6306-4d2e-bb5e-af8cedc9742e-4_823_1234_774_370}

Calculate the value of Spearman's rank correlation coefficient for these data.
Perform a hypothesis test at the $5 \%$ level to check the researcher's belief, stating your hypotheses clearly.
It is suggested that it would be preferable to carry out a test based on the product moment correlation coefficient. State the distributional assumption required for such a test to be valid. Explain how a scatter diagram can be used to check whether the distributional assumption is likely to be valid and comment on the validity in this case.
A statistician investigates data over a much longer period and finds that the assumptions for the use of the product moment correlation coefficient are in fact valid. Give the critical region for the test at the $1 \%$ level, based on a sample of 60 days.
In a different research project, into the correlation between daily temperature and ozone pollution levels, a positive correlation is found. It is argued that this shows that high temperatures cause increased ozone levels. Comment on this claim.

OCR MEI S2 2006 January Q4

4 The table summarises the usual method of travelling to school for 200 randomly selected pupils from primary and secondary schools in a city.

Write down null and alternative hypotheses for a test to examine whether there is any association between method of travel and type of school.
Calculate the expected frequency for primary school bus users. Calculate also the corresponding contribution to the test statistic for the usual $\chi ^ { 2 }$ test.
Given that the value of the test statistic for the usual $\chi ^ { 2 }$ test is 42.64 , carry out the test at the $5 \%$ level of significance, stating your conclusion clearly. The mean travel time for pupils who travel by bus is known to be 18.3 minutes. A survey is carried out to determine whether the mean travel time to school by car is different from 18.3 minutes. In the survey, 20 pupils who travel by car are selected at random. Their mean travel time is found to be 22.4 minutes.
Assuming that car travel times are Normally distributed with standard deviation 8.0 minutes, carry out a test at the $10 \%$ level, stating your hypotheses and conclusion clearly.
Comment on the suggestion that pupils should use a bus if they want to get to school quickly.

OCR MEI S2 2007 January Q1

1 In a science investigation into energy conservation in the home, a student is collecting data on the time taken for an electric kettle to boil as the volume of water in the kettle is varied. The student's data are shown in the table below, where $v$ litres is the volume of water in the kettle and $t$ seconds is the time taken for the kettle to boil (starting with the water at room temperature in each case). Also shown are summary statistics and a scatter diagram on which the regression line of $t$ on $v$ is drawn.

$v$	0.2	0.4	0.6	0.8	1.0
$t$	44	78	114	156	172

$$n = 5 , \Sigma v = 3.0 , \Sigma t = 564 , \Sigma v ^ { 2 } = 2.20 , \Sigma v t = 405.2 .$$ \includegraphics[max width=\textwidth, alt={}, center]{7ba30ff3-af90-4741-aab1-576efcbcb0b2-2_563_1376_742_386}

Calculate the equation of the regression line of $t$ on $v$, giving your answer in the form $t = a + b v$.
Use this equation to predict the time taken for the kettle to boil when the amount of water which it contains is
(A) 0.5 litres,
(B) 1.5 litres. Comment on the reliability of each of these predictions.
In the equation of the regression line found in part (i), explain the role of the coefficient of $v$ in the relationship between time taken and volume of water.
Calculate the values of the residuals for $v = 0.8$ and $v = 1.0$.
Explain how, on a scatter diagram with the regression line drawn accurately on it, a residual could be measured and its sign determined.
(a) A farmer grows Brussels sprouts. The diameter of sprouts in a particular batch, measured in mm , is Normally distributed with mean 28 and variance 16. Sprouts that are between 24 mm and 33 mm in diameter are sold to a supermarket.
Find the probability that the diameter of a randomly selected sprout will be within this range.
The farmer sells the sprouts in this range to the supermarket for 10 pence per kilogram. The farmer sells sprouts under 24 mm in diameter to a frozen food factory for 5 pence per kilogram. Sprouts over 33 mm in diameter are thrown away. Estimate the total income received by the farmer for the batch, which weighs 25000 kg .
By harvesting sprouts earlier, the mean diameter for another batch can be reduced to $k \mathrm {~mm}$. Find the value of $k$ for which only $5 \%$ of the sprouts will be above 33 mm in diameter. You may assume that the variance is still 16 .
(b) The farmer also grows onions. The weight in kilograms of the onions is Normally distributed with mean 0.155 and variance 0.005 . He is trying out a new variety, which he hopes will yield a higher mean weight. In order to test this, he takes a random sample of 25 onions of the new variety and finds that their total weight is 4.77 kg . You should assume that the weight in kilograms of the new variety is Normally distributed with variance 0.005 .
Write down suitable null and alternative hypotheses for the test in terms of $\mu$. State the meaning of $\mu$ in this case.
Carry out the test at the $1 \%$ level.

OCR MEI S2 2007 January Q3

3 An electrical retailer gives customers extended guarantees on washing machines. Under this guarantee all repairs in the first 3 years are free. The retailer records the numbers of free repairs made to 80 machines.

Number of repairs	0	1	2	3	$> 3$
Frequency	53	20	6	1	0

Show that the sample mean is 0.4375 .
The sample standard deviation $s$ is 0.6907 . Explain why this supports a suggestion that a Poisson distribution may be a suitable model for the distribution of the number of free repairs required by a randomly chosen washing machine. The random variable $X$ denotes the number of free repairs required by a randomly chosen washing machine. For the remainder of this question you should assume that $X$ may be modelled by a Poisson distribution with mean 0.4375.
Find $\mathrm { P } ( X = 1 )$. Comment on your answer in relation to the data in the table.
The manager decides to monitor 8 washing machines sold on one day. Find the probability that there are at least 12 free repairs in total on these 8 machines. You may assume that the 8 machines form an independent random sample.
A launderette with 8 washing machines has needed 12 free repairs. Why does your answer to part (iv) suggest that the Poisson model with mean 0.4375 is unlikely to be a suitable model for free repairs on the machines in the launderette? Give a reason why the model may not be appropriate for the launderette. The retailer also sells tumble driers with the same guarantee. The number of free repairs on a tumble drier in three years can be modelled by a Poisson distribution with mean 0.15 . A customer buys a tumble drier and a washing machine.
Assuming that free repairs are required independently, find the probability that
(A) the two appliances need a total of 3 free repairs between them,
(B) each appliance needs exactly one free repair.

OCR MEI S2 2007 January Q4

4 Two educational researchers are investigating the relationship between personal ambitions and home location of students. The researchers classify students into those whose main personal ambition is good academic results and those who have some other ambition. A random sample of 480 students is selected.

One researcher summarises the data as follows.
Observed Home location
\cline { 3 - 4 } City Non-city
\multirow{2}{*}{Ambition} Good results 102 147
\cline { 2 - 4 } Other 75 156
Carry out a test at the $5 \%$ significance level to examine whether there is any association between home location and ambition. State carefully your null and alternative hypotheses. Your working should include a table showing the contributions of each cell to the test statistic.

The other researcher summarises the same data in a different way as follows.

Observed		Home location
\cline { 3 - 5 }	City	Town	Country
\multirow{2}{*}{Ambition}	Good results	102	83	64
\cline { 2 - 5 }	Other	75	64	92

(A) Calculate the expected frequencies for both 'Country' cells.
(B) The test statistic for these data is 10.94 . Carry out a test at the $5 \%$ level based on this table, using the same hypotheses as in part (i).
(C) The table below gives the contribution of each cell to the test statistic. Discuss briefly how personal ambitions are related to home location.

	Home location
\cline { 2 - 5 }	City	Town	Country
\multirow{2}{*}{Ambition}	Good results	1.129	0.596	3.540
\cline { 2 - 5 }	Other	1.217	0.643	3.816

Comment briefly on whether the analysis in part (ii) means that the conclusion in part (i) is invalid.

OCR MEI S2 2008 January Q1

1 A biology student is carrying out an experiment to study the effect of a hormone on the growth of plant shoots. The student applies the hormone at various concentrations to a random sample of twelve shoots and measures the growth of each shoot. The data are illustrated on the scatter diagram below, together with the summary statistics for these data. The variables $x$ and $y$, measured in suitable units, represent concentration and growth respectively.
\includegraphics[max width=\textwidth, alt={}, center]{20fc4222-95c6-4b59-8e89-913dd988eb44-2_693_897_534_625} $$n = 12 , \Sigma x = 30 , \Sigma y = 967.6 , \Sigma x ^ { 2 } = 90 , \Sigma y ^ { 2 } = 78926 , \Sigma x y = 2530.3 .$$

State which of the two variables $x$ and $y$ is the independent variable and which is the dependent variable. Briefly explain your answers.
Calculate the equation of the regression line of $y$ on $x$.
Use the equation of the regression line to calculate estimates of shoot growth for concentrations of
(A) 1.2,
(B) 4.3. Comment on the reliability of each of these estimates.
Calculate the value of the residual for the data point where $x = 3$ and $y = 80$.
In further experiments, the student finds that using concentration $x = 6$ results in shoot growths of around $y = 20$. In the light of all the available information, what can be said about the relationship between $x$ and $y$ ?

OCR MEI S2 2008 January Q2

2 A large hotel has 90 bedrooms. Sometimes a guest makes a booking for a room, but then does not arrive. This is called a 'no-show'. On average $10 \%$ of bookings are no-shows. The hotel manager accepts up to 94 bookings before saying that the hotel is full. If at least 4 of these bookings are no-shows then there will be enough rooms for all of the guests. 94 bookings have been made for each night in August. You should assume that all bookings are independent.

State the distribution of the number of no-shows on one night in August.
State the conditions under which the use of a Poisson distribution is appropriate as an approximation to a binomial distribution.
Use a Poisson approximating distribution to find the probability that, on one night in August,
(A) there are exactly 4 no-shows,
(B) there are enough rooms for all of the guests who do arrive.
Find the probability that, on all of the 31 nights in August, there are enough rooms for all of the guests who arrive.
(A) In August there are $31 \times 94 = 2914$ bookings altogether. State the exact distribution of the total number of no-shows during August.
(B) Use a suitable approximating distribution to find the probability that there are at most 300 no-shows altogether during August.

OCR MEI S2 2008 January Q3

3 In a large population, the diastolic blood pressure (DBP) of 5-year-old children is Normally distributed with mean 56 and standard deviation 6.5.

Find the probability that the DBP of a randomly selected 5-year-old child is between 52.5 and 57.5. The DBP of young adults is Normally distributed with mean 68 and standard deviation 10.
A 5-year-old child and a young adult are selected at random. Find the probability that the DBP of one of them is over 62 and the other is under 62.
Sketch both distributions on a single diagram.
For another age group, the DBP is Normally distributed with mean 82. The DBP of $12 \%$ of people in this age group is below 62. Find the standard deviation for this age group.

OCR MEI S2 2008 January Q4

12 marks

A researcher believes that there may be some association between a student's sex and choice of certain subjects at A-level. A random sample of 250 A -level students is selected. The table below shows, for each sex, how many study either or both of the two subjects, Mathematics and English.
Mathematics only English only Both Neither Row totals
Male 38 19 6 32 95
Female 42 55 9 49 155
Column totals 80 74 15 81 250
Carry out a test at the $5 \%$ significance level to examine whether there is any association between a student's sex and choice of subjects. State carefully your null and alternative hypotheses. Your working should include a table showing the contributions of each cell to the test statistic. [12]
Over a long period it has been determined that the mean score of students in a particular English module is 67.4 and the standard deviation is 8.9. A new teaching method is introduced with the aim of improving the results. A random sample of 12 students taught by the new method is selected. Their mean score is found to be 68.3. Carry out a test at the $10 \%$ level to investigate whether the new method appears to have been successful. State carefully your null and alternative hypotheses. You should assume that the scores are Normally distributed and that the standard deviation is unchanged.

OCR MEI S2 2005 June Q1

1 A student is collecting data on traffic arriving at a motorway service station during weekday lunchtimes. The random variable $X$ denotes the number of cars arriving in a randomly chosen period of ten seconds.

State two assumptions necessary if a Poisson distribution is to provide a suitable model for the distribution of $X$. Comment briefly on whether these assumptions are likely to be valid. The student counts the number of arrivals, $x$, in each of 100 ten-second periods. The data are shown in the table below.
$x$ 0 1 2 3 4 5 $> 5$
Frequency, $f$ 18 39 20 12 8 3 0
Show that the sample mean is 1.62 and calculate the sample variance.
Do your calculations in part (ii) support the suggestion that a Poisson distribution is a suitable model for the distribution of $X$ ? Explain your answer. For the remainder of this question you should assume that $X$ may be modelled by a Poisson distribution with mean 1.62 .
Find $\mathrm { P } ( X = 2 )$. Comment on your answer in relation to the data in the table.
Find the probability that at least ten cars arrive in a period of 50 seconds during weekday lunchtimes.
Use a suitable approximating distribution to find the probability that no more than 550 cars arrive in a randomly chosen period of one hour during weekday lunchtimes.

OCR MEI S2 2005 June Q2

2 The fuel economy of a car varies from day to day according to weather and driving conditions. Fuel economy is measured in miles per gallon (mpg). The fuel economy of a particular petrol-fuelled type of car is known to be Normally distributed with mean 38.5 mpg and standard deviation 4.0 mpg .

Find the probability that on a randomly selected day the fuel economy of a car of this type will be above 45.0 mpg .
The manufacturer wishes to quote a fuel economy figure which will be exceeded on $90 \%$ of days. What figure should be quoted? The daily fuel economy of a similar type of car which is diesel-fuelled is known to be Normally distributed with mean 51.2 mpg and unknown standard deviation $\sigma \mathrm { mpg }$.
Given that on 75\% of days the fuel economy of this type of car is below 55.0 mpg , show that $\sigma = 5.63$.
Draw a sketch to illustrate both distributions on a single diagram.
Find the probability that the fuel economy of either the petrol or the diesel model (or both) will be above 45.0 mpg on a randomly selected day. You may assume that the fuel economies of the two models are independent.

OCR MEI S2 2005 June Q3

3 In a triathlon, competitors have to swim 600 metres, cycle 40 kilometres and run 10 kilometres. To improve her strength, a triathlete undertakes a training programme in which she carries weights in a rucksack whilst running. She runs a specific course and notes the total time taken for each run. Her coach is investigating the relationship between time taken and weight carried. The times taken with eight different weights are illustrated on the scatter diagram below, together with the summary statistics for these data. The variables $x$ and $y$ represent weight carried in kilograms and time taken in minutes respectively.
\includegraphics[max width=\textwidth, alt={}, center]{be463718-caf7-4bc8-b838-143ab4681d6e-4_627_1536_630_281} Summary statistics: $n = 8 , \Sigma x = 36 , \Sigma y = 214.8 , \Sigma x ^ { 2 } = 204 , \Sigma y ^ { 2 } = 5775.28 , \Sigma x y = 983.6$.

Calculate the equation of the regression line of $y$ on $x$. On one of the eight runs, the triathlete was carrying 4 kilograms and took 27.5 minutes. On this run she was delayed when she tripped and fell over.
Calculate the value of the residual for this weight.
The coach decides to recalculate the equation of the regression line without the data for this run. Would it be preferable to use this recalculated equation or the equation found in part (i) to estimate the delay when the triathlete tripped and fell over? Explain your answer. The triathlete's coach claims that there is positive correlation between cycling and swimming times in triathlons. The product moment correlation coefficient of the times of twenty randomly selected competitors in these two sections is 0.209 .
Carry out a hypothesis test at the $5 \%$ level to examine the coach's claim, explaining your conclusions clearly.
What distributional assumption is necessary for this test to be valid? How can you use a scatter diagram to decide whether this assumption is likely to be true?

OCR MEI S2 2005 June Q9

9 JUNE 2005
Morning
1 hour 30 minutes
Additional materials:
Answer booklet
Graph paper
MEI Examination Formulae and Tables (MF2) TIME 1 hour 30 minutes

Write your name, centre number and candidate number in the spaces provided on the answer booklet.
Answer all the questions.
You are permitted to use a graphical calculator in this paper.

The number of marks is given in brackets [ ] at the end of each question or part question.
You are advised that an answer may receive no marks unless you show sufficient detail of the working to indicate that a correct method is being used.
Final answers should be given to a degree of accuracy appropriate to the context.
The total number of marks for this paper is 72.

State two assumptions necessary if a Poisson distribution is to provide a suitable model for the distribution of $X$. Comment briefly on whether these assumptions are likely to be valid. The student counts the number of arrivals, $x$, in each of 100 ten-second periods. The data are shown in the table below. Carry out a test at the $5 \%$ level of significance to examine whether there is any association between type of customer and type of drink. State carefully your null and alternative hypotheses.

OCR MEI S2 2006 June Q1

1 A low-cost airline charges for breakfasts on its early morning flights. On average, $10 \%$ of passengers order breakfast.

Find the probability that, out of 8 randomly selected passengers, exactly 1 orders breakfast.
Use a suitable Poisson approximating distribution to find the probability that the number of breakfasts ordered by 30 randomly selected passengers is
(A) exactly 6,
(B) at least 8 .
State the conditions under which the use of a Poisson distribution is appropriate as an approximation to a binomial distribution.
The aircraft carries 120 passengers and the flight is always full. Find the mean $\mu$ and variance $\sigma ^ { 2 }$ of a Normal approximating distribution suitable for modelling the total number of passengers on the flight who order breakfast.
Use your Normal approximating distribution to calculate the probability that more than 15 breakfasts are ordered on a particular flight.
The airline wishes to be at least $99 \%$ certain that the plane will have sufficient breakfasts for all passengers who order them. Find the minimum number of breakfasts which should be carried on each flight.

OCR MEI S2 2006 June Q2

2 The head circumference of 3-year-old boys is known to be Normally distributed with mean 49.7 cm and standard deviation 1.6 cm .

Find the probability that the head circumference of a randomly selected 3 -year-old boy will be
(A) over 51.5 cm ,
(B) between 48.0 and 51.5 cm .
Four 3-year-old boys are selected at random. Find the probability that exactly one of them has head circumference between 48.0 and 51.5 cm .
The head circumference of 3-year-old girls is known to be Normally distributed with mean $\mu$ and standard deviation $\sigma$. Given that $60 \%$ of 3-year-old girls have head circumference below 49.0 cm and $30 \%$ have head circumference below 47.5 cm , find the values of $\mu$ and $\sigma$. A nutritionist claims that boys who have been fed on a special organic diet will have a larger mean head circumference than other boys. A random sample of ten 3 -year-old boys who have been fed on this organic diet is selected. It is found that their mean head circumference is 50.45 cm .
Using the null and alternative hypotheses $\mathrm { H } _ { 0 } : \mu = 49.7 \mathrm {~cm} , \mathrm { H } _ { 1 } : \mu > 49.7 \mathrm {~cm}$, carry out a test at the $10 \%$ significance level to examine the nutritionist's claim. Explain the meaning of $\mu$ in these hypotheses. You may assume that the standard deviation of the head circumference of organically fed 3 -year-old boys is 1.6 cm .

OCR MEI S2 2006 June Q3

3 A student is investigating the relationship between the length $x \mathrm {~mm}$ and circumference $y \mathrm {~mm}$ of plums from a large crop. The student measures the dimensions of a random sample of 10 plums from this crop. Summary statistics for these dimensions are as follows. $$\begin{aligned} & \sum x = 4715 \quad \sum y = 13175 \quad \sum x ^ { 2 } = 2237725
& \sum y ^ { 2 } = 17455825 \quad \sum x y = 6235575 \quad n = 10 \end{aligned}$$

Calculate the sample product moment correlation coefficient.
Carry out a hypothesis test at the $5 \%$ significance level to determine whether there is any correlation between length and circumference of plums from this crop. State your hypotheses clearly, defining any symbols which you use.
(A) Explain the meaning of a 5\% significance level.
(B) State one advantage and one disadvantage of using a $1 \%$ significance level rather than a $5 \%$ significance level in a hypothesis test. The student decides to take another random sample of 10 plums. Using the same hypotheses as in part (ii), the correlation coefficient for this second sample is significant at the $5 \%$ level. The student decides to ignore the first result and concludes that there is correlation between the length and circumference of plums in the crop.
Comment on the student's decision to ignore the first result. Suggest a better way in which the student could proceed.

OCR MEI S2 2006 June Q4

4 A survey of a random sample of 250 people is carried out. Their musical preferences are categorized as pop, classical or jazz. Their ages are categorized as under 25, 25 to 50, or over 50. The results are as follows.

\multirow{2}{*}{}		Musical preference			\multirow{2}{*}{Row totals}
		Pop	Classical	Jazz
\multirow{3}{*}{Age group}	Under 25	57	15	12	84
	25-50	43	21	21	85
	Over 50	22	32	27	81
Column totals		122	68	60	250

Carry out a test at the $5 \%$ significance level to examine whether there is any association between musical preference and age group. State carefully your null and alternative hypotheses. Your working should include a table showing the contributions of each cell to the test statistic.
Discuss briefly how musical preferences vary between the age groups, as shown by the contributions to the test statistic.

OCR MEI S2 2007 June Q1

1 The random variable $X$ represents the time taken in minutes for a haircut at a barber's shop. $X$ is Normally distributed with mean 11 and standard deviation 3 .

Find $\mathrm { P } ( X < 10 )$.
Find the probability that exactly 3 out of 8 randomly selected haircuts take less than 10 minutes.
Use a suitable approximating distribution to find the probability that at least 50 out of 100 randomly selected haircuts take less than 10 minutes. A new hairdresser joins the shop. The shop manager suspects that she takes longer on average than the other staff to do a haircut. In order to test this, the manager records the time taken for 25 randomly selected cuts by the new hairdresser. The mean time for these cuts is 12.34 minutes. You should assume that the time taken by the new hairdresser is Normally distributed with standard deviation 3 minutes.
Write down suitable null and alternative hypotheses for the test.
Carry out the test at the $5 \%$ level.

OCR MEI S2 2007 June Q2

2 A medical student is trying to estimate the birth weight of babies using pre-natal scan images. The actual weights, $x \mathrm {~kg}$, and the estimated weights, $y \mathrm {~kg}$, of ten randomly selected babies are given in the table below.

$x$	2.61	2.73	2.87	2.96	3.05	3.14	3.17	3.24	3.76	4.10
$y$	3.2	2.6	3.5	3.1	2.8	2.7	3.4	3.3	4.4	4.1

Calculate the value of Spearman's rank correlation coefficient.
Carry out a hypothesis test at the $5 \%$ level to determine whether there is positive association between the student's estimates and the actual birth weights of babies in the underlying population.
Calculate the value of the product moment correlation coefficient of the sample. You may use the following summary statistics in your calculations: $$\Sigma x = 31.63 , \quad \Sigma y = 33.1 , \quad \Sigma x ^ { 2 } = 101.92 , \quad \Sigma y ^ { 2 } = 112.61 , \quad \Sigma x y = 106.51 .$$
Explain why, if the underlying population has a bivariate Normal distribution, it would be preferable to carry out a hypothesis test based on the product moment correlation coefficient. Comment briefly on the significance of the product moment correlation coefficient in relation to that of Spearman’s rank correlation coefficient.

OCR MEI S2 2007 June Q3

3 The number of calls received at an office per 5 minutes is modelled by a Poisson distribution with mean 3.2.

Find the probability of
(A) exactly one call in a 5 -minute period,
(B) at least 6 calls in a 5 -minute period.
Find the probability of
(A) exactly one call in a 1 -minute period,
(B) exactly one call in each of five successive 1-minute periods.
Use a suitable approximating distribution to find the probability of at most 45 calls in a period of 1 hour. Two assumptions required for a Poisson distribution to be a suitable model are that calls arrive
- at a uniform average rate,
- independently of each other.
- Comment briefly on the validity of each of these assumptions if the office is
  (A) the enquiry department of a bank,
  (B) a police emergency control room.

OCR MEI S2 2007 June Q4

4 The sexes and ages of a random sample of 300 runners taking part in marathons are classified as follows.

Observed

Sex

\multirow{2}{*}{Row totals}

Carry out a test at the $5 \%$ significance level to examine whether there is any association between age group and sex. State carefully your null and alternative hypotheses. Your working should include a table showing the contributions of each cell to the test statistic.
Does your analysis support the suggestion that women are less likely than men to enter marathons as they get older? Justify your answer. For marathons in general, on average $3 \%$ of runners are 'Female, 50 and over'. The random variable $X$ represents the number of 'Female, 50 and over' runners in a random sample of size 300.
Use a suitable approximating distribution to find $\mathrm { P } ( X \geqslant 12 )$.

OCR MEI S2 2008 June Q1

1 A researcher believes that there is a negative correlation between money spent by the government on education and population growth in various countries. A random sample of 48 countries is selected to investigate this belief. The level of government spending on education $x$, measured in suitable units, and the annual percentage population growth rate $y$, are recorded for these countries. Summary statistics for these data are as follows. $$\Sigma x = 781.3 \quad \Sigma y = 57.8 \quad \Sigma x ^ { 2 } = 14055 \quad \Sigma y ^ { 2 } = 106.3 \quad \Sigma x y = 880.1 \quad n = 48$$

Calculate the sample product moment correlation coefficient.
Carry out a hypothesis test at the $5 \%$ significance level to investigate the researcher's belief. State your hypotheses clearly, defining any symbols which you use.
State the distributional assumption which is necessary for this test to be valid. Explain briefly how a scatter diagram may be used to check whether this assumption is likely to be valid.
A student suggests that if the variables are negatively correlated then population growth rates can be reduced by increasing spending on education. Explain why the student may be wrong. Discuss an alternative explanation for the correlation.
State briefly one advantage and one disadvantage of using a smaller sample size in this investigation.

OCR MEI S2 2008 June Q2

2 A public water supply contains bacteria. Each day an analyst checks the water quality by counting the number of bacteria in a random sample of 5 ml of water. Throughout this question, you should assume that the bacteria occur randomly at a mean rate of 0.37 bacteria per 5 ml of water.

Use a Poisson distribution to
(A) find the probability that a 5 ml sample contains exactly 2 bacteria,
(B) show that the probability that a 5 ml sample contains more than 2 bacteria is 0.0064 .
The month of September has 30 days. Find the probability that during September there is at most one day when a 5 ml sample contains more than 2 bacteria. The daily 5 ml sample is the first stage of the quality control process. The remainder of the process is as follows.
- If the 5 ml sample contains more than 2 bacteria, then a 50 ml sample is taken.
- If this 50 ml sample contains more than 8 bacteria, then a sample of 1000 ml is taken.
- If this 1000 ml sample contains more than 90 bacteria, then the supply is declared to be 'questionable'.
- Find the probability that a random sample of 50 ml contains more than 8 bacteria.
- Use a suitable approximating distribution to find the probability that a random sample of 1000 ml contains more than 90 bacteria.
- Find the probability that the supply is declared to be questionable.

\(x\)	13.3	17.2	16.9	18.7	18.4	19.3	23.1	15.0	20.6	14.4
\(y\)	9	11	14	26	43	25	52	15	10	7

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