Questions — OCR MEI (4456 questions)

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OCR MEI S1 2012 June Q5
8 marks Moderate -0.3
5 A manufacturer produces titanium bicycle frames. The bicycle frames are tested before use and on average \(5 \%\) of them are found to be faulty. A cheaper manufacturing process is introduced and the manufacturer wishes to check whether the proportion of faulty bicycle frames has increased. A random sample of 18 bicycle frames is selected and it is found that 4 of them are faulty. Carry out a hypothesis test at the \(5 \%\) significance level to investigate whether the proportion of faulty bicycle frames has increased.
OCR MEI S1 2012 June Q6
18 marks Moderate -0.3
6 The engine sizes \(x \mathrm {~cm} ^ { 3 }\) of a sample of 80 cars are summarised in the table below.
Engine size \(x\)\(500 \leqslant x \leqslant 1000\)\(1000 < x \leqslant 1500\)\(1500 < x \leqslant 2000\)\(2000 < x \leqslant 3000\)\(3000 < x \leqslant 5000\)
Frequency72226187
  1. Draw a histogram to illustrate the distribution.
  2. A student claims that the midrange is \(2750 \mathrm {~cm} ^ { 3 }\). Discuss briefly whether he is likely to be correct.
  3. Calculate estimates of the mean and standard deviation of the engine sizes. Explain why your answers are only estimates.
  4. Hence investigate whether there are any outliers in the sample.
  5. A vehicle duty of \(\pounds 1000\) is proposed for all new cars with engine size greater than \(2000 \mathrm {~cm} ^ { 3 }\). Assuming that this sample of cars is representative of all new cars in Britain and that there are 2.5 million new cars registered in Britain each year, calculate an estimate of the total amount of money that this vehicle duty would raise in one year.
  6. Why in practice might your estimate in part (v) turn out to be too high?
OCR MEI S1 2012 June Q7
18 marks Standard +0.3
7 Yasmin has 5 coins. One of these coins is biased with P (heads) \(= 0.6\). The other 4 coins are fair. She tosses all 5 coins once and records the number of heads, \(X\).
  1. Show that \(\mathrm { P } ( X = 0 ) = 0.025\).
  2. Show that \(\mathrm { P } ( X = 1 ) = 0.1375\). The table shows the probability distribution of \(X\).
    \(r\)012345
    \(\mathrm { P } ( X = r )\)0.0250.13750.30.3250.1750.0375
  3. Draw a vertical line chart to illustrate the probability distribution.
  4. Comment on the skewness of the distribution.
  5. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
  6. Yasmin tosses the 5 coins three times. Find the probability that the total number of heads is 3 . \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}
OCR MEI S1 2013 June Q1
6 marks Moderate -0.8
1 The weights, \(x\) grams, of 100 potatoes are summarised as follows. $$n = 100 \quad \sum x = 24940 \quad \sum x ^ { 2 } = 6240780$$
  1. Calculate the mean and standard deviation of \(x\).
  2. The weights, \(y\) grams, of the potatoes after they have been peeled are given by the formula \(y = 0.9 x - 15\). Deduce the mean and standard deviation of the weights of the potatoes after they have been peeled.
OCR MEI S1 2013 June Q2
8 marks Standard +0.3
2 Every evening, 5 men and 5 women are chosen to take part in a phone-in competition. Of these 10 people, exactly 3 will win a prize. These 3 prize-winners are chosen at random.
  1. Find the probability that, on a particular evening, 2 of the prize-winners are women and the other is a man. Give your answer as a fraction in its simplest form.
  2. Four evenings are selected at random. Find the probability that, on at least three of the four evenings, 2 of the prize-winners are women and the other is a man.
OCR MEI S1 2013 June Q3
7 marks Standard +0.3
3 The weights of bags of a particular brand of flour are quoted as 1.5 kg . In fact, on average \(10 \%\) of bags are underweight.
  1. Find the probability that, in a random sample of 50 bags, there are exactly 5 bags which are underweight.
  2. Bags are randomly chosen and packed into boxes of 20 . Find the probability that there is at least one underweight bag in a box.
  3. A crate contains 48 boxes. Find the expected number of boxes in the crate which contain at least one underweight bag.
OCR MEI S1 2013 June Q4
7 marks Moderate -0.8
4 Martin has won a competition. For his prize he is given six sealed envelopes, of which he is allowed to open exactly three and keep their contents. Three of the envelopes each contain \(\pounds 5\) and the other three each contain \(\pounds 1000\). Since the envelopes are identical on the outside, he chooses three of them at random. Let \(\pounds X\) be the total amount of money that he receives in prize money.
  1. Show that \(\mathrm { P } ( X = 15 ) = 0.05\). The probability distribution of \(X\) is given in the table below.
    \(r\)15101020053000
    \(\mathrm { P } ( X = r )\)0.050.450.450.05
  2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
OCR MEI S1 2013 June Q5
8 marks Moderate -0.3
5 A researcher is investigating whether people can identify whether a glass of water they are given is bottled water or tap water. She suspects that people do no better than they would by guessing. Twenty people are selected at random; thirteen make a correct identification. She carries out a hypothesis test.
  1. Explain why the null hypothesis should be \(p = 0.5\), where \(p\) represents the probability that a randomly selected person makes a correct identification.
  2. Briefly explain why she uses an alternative hypothesis of \(p > 0.5\).
  3. Complete the test at the \(5 \%\) significance level.
OCR MEI S1 2013 June Q6
18 marks Easy -1.2
6 The birth weights in kilograms of 25 female babies are shown below, in ascending order.
1.392.502.682.762.822.822.843.033.063.163.163.243.32
3.363.403.543.563.563.703.723.723.844.024.244.34
  1. Find the median and interquartile range of these data.
  2. Draw a box and whisker plot to illustrate the data.
  3. Show that there is exactly one outlier. Discuss whether this outlier should be removed from the data. The cumulative frequency curve below illustrates the birth weights of 200 male babies. \includegraphics[max width=\textwidth, alt={}, center]{6b886da6-3fb8-4b4c-b572-f4b770ae5a4c-3_929_1569_1450_248}
  4. Find the median and interquartile range of the birth weights of the male babies.
  5. Compare the weights of the female and male babies.
  6. Two of these male babies are chosen at random. Calculate an estimate of the probability that both of these babies weigh more than any of the female babies.
OCR MEI S1 2013 June Q7
18 marks Standard +0.3
7 Jenny has six darts. She throws darts, one at a time, aiming each at the bull's-eye. The probability that she hits the bull's-eye with her first dart is 0.1 . For any subsequent throw, the probability of hitting the bull's-eye is 0.2 if the previous dart hit the bull's-eye and 0.05 otherwise.
  1. Illustrate the possible outcomes for her first, second and third darts on a probability tree diagram.
  2. Find the probability that
    (A) she hits the bull's-eye with at least one of her first three darts,
    (B) she hits the bull's-eye with exactly one of her first three darts.
  3. Given that she hits the bull's-eye with at least one of her first three darts, find the probability that she hits the bull's-eye with exactly one of them. Jenny decides that, if she hits the bull's-eye with any of her first three darts, she will stop after throwing three darts. Otherwise she will throw all six darts.
  4. Find the probability that she hits the bull's-eye three times in total.
OCR MEI S1 2015 June Q2
5 marks Easy -1.3
2 A survey is being carried out into the sports viewing habits of people in a particular area. As part of the survey, 250 people are asked which of the following sports they have watched on television in the past month.
  • Football
  • Cycling
  • Rugby
The numbers of people who have watched these sports are shown in the Venn diagram. \includegraphics[max width=\textwidth, alt={}, center]{6015ae6c-bf76-4a0c-af0f-5c53f9c5ed2a-2_723_917_1183_575} One of the people is selected at random.
  1. Find the probability that this person has in the past month
    (A) watched cycling but not football,
    (B) watched either one or two of the three sports.
  2. Given that this person has watched cycling, find the probability that this person has not watched football.
OCR MEI S1 2015 June Q3
3 marks Easy -1.2
3 A normal pack of 52 playing cards contains 4 aces. A card is drawn at random from the pack. It is then replaced and the pack is shuffled, after which another card is drawn at random.
  1. Find the probability that neither card is an ace.
  2. This process is repeated 10 times. Find the expected number of times for which neither card is an ace.
OCR MEI S1 2015 June Q4
6 marks Moderate -0.8
4 A rugby team of 15 people is to be selected from a squad of 25 players.
  1. How many different teams are possible?
  2. In fact the team has to consist of 8 forwards and 7 backs. If 13 of the squad are forwards and the other 12 are backs, how many different teams are now possible?
  3. Find the probability that, if the team is selected at random from the squad of 25 players, it contains the correct numbers of forwards and backs.
OCR MEI S1 2015 June Q5
8 marks Easy -1.2
5 At a tourist information office the numbers of people seeking information each hour over the course of a 12-hour day are shown below. $$\begin{array} { l l l l l l l l l l l l } 6 & 25 & 38 & 39 & 31 & 18 & 35 & 31 & 33 & 15 & 21 & 28 \end{array}$$
  1. Construct a sorted stem and leaf diagram to represent these data.
  2. State the type of skewness suggested by your stem and leaf diagram.
  3. For these data find the median, the mean and the mode. Comment on the usefulness of the mode in this case.
OCR MEI S1 2015 June Q6
8 marks Standard +0.3
6 Three fair six-sided dice are thrown. The random variable \(X\) represents the highest of the three scores on the dice.
  1. Show that \(\mathrm { P } ( X = 6 ) = \frac { 91 } { 216 }\). The table shows the probability distribution of \(X\).
    \(r\)123456
    \(\mathrm { P } ( X = r )\)\(\frac { 1 } { 216 }\)\(\frac { 7 } { 216 }\)\(\frac { 19 } { 216 }\)\(\frac { 37 } { 216 }\)\(\frac { 61 } { 216 }\)\(\frac { 91 } { 216 }\)
  2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
OCR MEI S1 2015 June Q7
17 marks Standard +0.3
7 A drug for treating a particular minor illness cures, on average, \(78 \%\) of patients. Twenty people with this minor illness are selected at random and treated with the drug.
  1. \(( A )\) Find the probability that exactly 19 patients are cured.
    (B) Find the probability that at most 18 patients are cured. \(( C )\) Find the expected number of patients who are cured.
  2. A pharmaceutical company is trialling a new drug to treat this illness. Researchers at the company hope that a higher percentage of patients will be cured when given this new drug. Twenty patients are selected at random, and given the new drug. Of these, 19 are cured. Carry out a hypothesis test at the \(1 \%\) significance level to investigate whether there is any evidence to suggest that the new drug is more effective than the old one.
  3. If the researchers had chosen to carry out the hypothesis test at the \(5 \%\) significance level, what would the result have been? Justify your answer.
OCR MEI S1 2015 June Q8
19 marks Standard +0.3
8 The box and whisker plot below summarises the weights in grams of the 20 chocolates in a box. \includegraphics[max width=\textwidth, alt={}, center]{6015ae6c-bf76-4a0c-af0f-5c53f9c5ed2a-4_287_1177_319_427}
  1. Find the interquartile range of the data and hence determine whether there are any outliers at either end of the distribution. Ben buys a box of these chocolates each weekend. The chocolates all look the same on the outside, but 7 of them have orange centres, 6 have cherry centres, 4 have coffee centres and 3 have lemon centres. One weekend, each of Ben's 3 children eats one of the chocolates, chosen at random.
  2. Calculate the probabilities of the following events. A: all 3 chocolates have orange centres \(B\) : all 3 chocolates have the same centres
  3. Find \(\mathrm { P } ( A \mid B )\) and \(\mathrm { P } ( B \mid A )\). The following weekend, Ben buys an identical box of chocolates and again each of his 3 children eats one of the chocolates, chosen at random.
  4. Find the probability that, on both weekends, the 3 chocolates that they eat all have orange centres.
  5. Ben likes all of the chocolates except those with cherry centres. On another weekend he is the first of his family to eat some of the chocolates. Find the probability that he has to select more than 2 chocolates before he finds one that he likes. \section*{END OF QUESTION PAPER} \section*{OCR
    Oxford Cambridge and RSA}
OCR MEI S2 2009 January Q1
20 marks Moderate -0.3
1 A researcher is investigating whether there is a relationship between the population size of cities and the average walking speed of pedestrians in the city centres. Data for the population size, \(x\) thousands, and the average walking speed of pedestrians, \(y \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of eight randomly selected cities are given in the table below.
\(x\)18435294982067841530
\(y\)1.150.971.261.351.281.421.321.64
  1. Calculate the value of Spearman's rank correlation coefficient.
  2. Carry out a hypothesis test at the \(5 \%\) significance level to determine whether there is any association between population size and average walking speed. In another investigation, the researcher selects a random sample of six adult males of particular ages and measures their maximum walking speeds. The data are shown in the table below, where \(t\) years is the age of the adult and \(w \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the maximum walking speed. Also shown are summary statistics and a scatter diagram on which the regression line of \(w\) on \(t\) is drawn.
    \(t\)203040506070
    \(w\)2.492.412.382.141.972.03
    $$n = 6 \quad \Sigma t = 270 \quad \Sigma w = 13.42 \quad \Sigma t ^ { 2 } = 13900 \quad \Sigma w ^ { 2 } = 30.254 \quad \Sigma t w = 584.6$$ \includegraphics[max width=\textwidth, alt={}, center]{77b97142-afb6-41d6-8fec-e982b7a7501b-2_728_1091_1379_529}
  3. Calculate the equation of the regression line of \(w\) on \(t\).
  4. (A) Use this equation to calculate an estimate of maximum walking speed of an 80 -year-old male.
    (B) Explain why it might not be appropriate to use the equation to calculate an estimate of maximum walking speed of a 10 -year-old male.
OCR MEI S2 2009 January Q2
18 marks Moderate -0.3
2 Clover stems usually have three leaves. Occasionally a clover stem has four leaves. This is considered by some to be lucky and is known as a four-leaf clover. On average 1 in 10000 clover stems is a four-leaf clover. You may assume that four-leaf clovers occur randomly and independently. A random sample of 5000 clover stems is selected.
  1. State the exact distribution of \(X\), the number of four-leaf clovers in the sample.
  2. Explain why \(X\) may be approximated by a Poisson distribution. Write down the mean of this Poisson distribution.
  3. Use this Poisson distribution to find the probability that the sample contains at least one four-leaf clover.
  4. Find the probability that in 20 samples, each of 5000 clover stems, there are exactly 9 samples which contain at least one four-leaf clover.
  5. Find the expected number of these 20 samples which contain at least one four-leaf clover. The table shows the numbers of four-leaf clovers in these 20 samples.
    Number of four-leaf clovers012\(> 2\)
    Number of samples11720
  6. Calculate the mean and variance of the data in the table.
  7. Briefly comment on whether your answers to parts (v) and (vi) support the use of the Poisson approximating distribution in part (iii).
OCR MEI S2 2009 January Q3
17 marks Moderate -0.3
3 The number of minutes, \(X\), for which a particular model of laptop computer will run on battery power is Normally distributed with mean 115.3 and standard deviation 21.9.
  1. (A) Find \(\mathrm { P } ( X < 120 )\).
    (B) Find \(\mathrm { P } ( 100 < X < 110 )\).
    (C) Find the value of \(k\) for which \(\mathrm { P } ( X > k ) = 0.9\). The number of minutes, \(Y\), for which a different model of laptop computer will run on battery power is known to be Normally distributed with mean \(\mu\) and standard deviation \(\sigma\).
  2. Given that \(\mathrm { P } ( Y < 180 ) = 0.7\) and \(\mathrm { P } ( Y < 140 ) = 0.15\), find the values of \(\mu\) and \(\sigma\).
  3. Find values of \(a\) and \(b\) for which \(\mathrm { P } ( a < Y < b ) = 0.95\).
OCR MEI S2 2009 January Q4
17 marks Standard +0.3
4 A gardening research organisation is running a trial to examine the growth and the size of flowers of various plants.
  1. In the trial, seeds of three types of plant are sown. The growth of each plant is classified as good, average or poor. The results are shown in the table.
    \multirow{2}{*}{}Growth\multirow[t]{2}{*}{Row totals}
    GoodAveragePoor
    \multirow{3}{*}{Type of plant}Coriander12281555
    Aster7182348
    Fennel14221147
    Column totals336849150
    Carry out a test at the \(5 \%\) significance level to examine whether there is any association between growth and type of plant. State carefully your null and alternative hypotheses. Include a table of the contributions of each cell to the test statistic.
  2. It is known that the diameter of marigold flowers is Normally distributed with mean 47 mm and standard deviation 8.5 mm . A certain fertiliser is expected to cause flowers to have a larger mean diameter, but without affecting the standard deviation. A large number of marigolds are grown using this fertiliser. The diameters of a random sample of 50 of the flowers are measured and the mean diameter is found to be 49.2 mm . Carry out a hypothesis test at the \(1 \%\) significance level to check whether flowers grown with this fertiliser appear to be larger on average. Use hypotheses \(\mathrm { H } _ { 0 } : \mu = 47 , \mathrm { H } _ { 1 } : \mu > 47\), where \(\mu \mathrm { mm }\) represents the mean diameter of all marigold flowers grown with this fertiliser.
OCR MEI S2 2010 January Q1
19 marks Moderate -0.3
1 A pilot records the take-off distance for his light aircraft on runways at various altitudes. The data are shown in the table below, where \(a\) metres is the altitude and \(t\) metres is the take-off distance. Also shown are summary statistics for these data.
\(a\)0300600900120015001800
\(t\)63570477683692310081105
$$n = 7 \quad \Sigma a = 6300 \quad \Sigma t = 5987 \quad \Sigma a ^ { 2 } = 8190000 \quad \Sigma t ^ { 2 } = 5288931 \quad \Sigma a t = 6037800$$
  1. Draw a scatter diagram to illustrate these data.
  2. State which of the two variables \(a\) and \(t\) is the independent variable and which is the dependent variable. Briefly explain your answer.
  3. Calculate the equation of the regression line of \(t\) on \(a\).
  4. Use the equation of the regression line to calculate estimates of the take-off distance for altitudes
    (A) 800 metres,
    (B) 2500 metres. Comment on the reliability of each of these estimates.
  5. Calculate the value of the residual for the data point where \(a = 1200\) and \(t = 923\), and comment on its sign.
OCR MEI S2 2010 January Q2
18 marks Moderate -0.5
2 On average 2\% of a particular model of laptop computer are faulty. Faults occur independently and randomly.
  1. Find the probability that exactly 1 of a batch of 10 laptops is faulty.
  2. State the conditions under which the use of a Poisson distribution is appropriate as an approximation to a binomial distribution.
  3. A school buys a batch of 150 of these laptops. Use a Poisson approximating distribution to find the probability that
    (A) there are no faulty laptops in the batch,
    (B) there are more than the expected number of faulty laptops in the batch.
  4. A large company buys a batch of 2000 of these laptops for its staff.
    (A) State the exact distribution of the number of faulty laptops in this batch.
    (B) Use a suitable approximating distribution to find the probability that there are at most 50 faulty laptops in this batch.
OCR MEI S2 2010 January Q3
17 marks Standard +0.3
3 In an English language test for 12-year-old children, the raw scores, \(X\), are Normally distributed with mean 45.3 and standard deviation 11.5.
  1. Find
    (A) \(\mathrm { P } ( X < 50 )\),
    (B) \(\mathrm { P } ( 45.3 < X < 50 )\).
  2. Find the least raw score which would be obtained by the highest scoring \(10 \%\) of children.
  3. The raw score is then scaled so that the scaled score is Normally distributed with mean 100 and standard deviation 15. This scaled score is then rounded to the nearest integer. Find the probability that a randomly selected child gets a rounded score of exactly 111 .
  4. In a Mathematics test for 12-year-old children, the raw scores, \(Y\), are Normally distributed with mean \(\mu\) and standard deviation \(\sigma\). Given that \(\mathrm { P } ( Y < 15 ) = 0.3\) and \(\mathrm { P } ( Y < 22 ) = 0.8\), find the values of \(\mu\) and \(\sigma\).
OCR MEI S2 2010 January Q4
18 marks Moderate -0.3
4 A council provides waste paper recycling services for local businesses. Some businesses use the standard service for recycling paper, others use a special service for dealing with confidential documents, and others use both. Businesses are classified as small or large. A survey of a random sample of 285 businesses gives the following data for size of business and recycling service.
Recycling Service
\cline { 3 - 5 } \multicolumn{2}{|c|}{}StandardSpecialBoth
Size of
business
Small352644
Large555273
  1. Write down null and alternative hypotheses for a test to examine whether there is any association between size of business and recycling service used. The contributions to the test statistic for the usual \(\chi ^ { 2 }\) test are shown in the table below.
    Recycling Service
    \cline { 3 - 5 } \multicolumn{2}{|c|}{}StandardSpecialBoth
    Size of
    business
    		Small0.10230.26070.0186
    Large0.05970.15200.0108
    The sum of these contributions is 0.6041 .
  2. Calculate the expected frequency for large businesses using the special service. Verify the corresponding contribution 0.1520 to the test statistic.
  3. Carry out the test at the \(5 \%\) level of significance, stating your conclusion clearly. The council is also investigating the weight of rubbish in domestic dustbins. In 2008 the average weight of rubbish in bins was 32.8 kg . The council has now started a recycling initiative and wishes to determine whether there has been a reduction in the weight of rubbish in bins. A random sample of 50 domestic dustbins is selected and it is found that the mean weight of rubbish per bin is now 30.9 kg , and the standard deviation is 3.4 kg .
  4. Carry out a test at the \(5 \%\) level to investigate whether the mean weight of rubbish has been reduced in comparison with 2008 . State carefully your null and alternative hypotheses. www.ocr.org.uk after the live examination series.
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