Questions — OCR MEI (4333 questions)

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OCR MEI S1 2012 June Q1
6 marks Moderate -0.8
1 At a garden centre there is a box containing 50 hyacinth bulbs. Of these, 30 will produce a blue flower and the remaining 20 will produce a red flower. Unfortunately they have become mixed together so that it is not known which of the bulbs will produce a blue flower and which will produce a red flower. Karen buys 3 of these bulbs.
  1. Find the probability that all 3 of these bulbs will produce blue flowers.
  2. Find the probability that Karen will have at least one flower of each colour from her 3 bulbs.
OCR MEI S1 2012 June Q2
6 marks Moderate -0.8
2 An examination paper consists of two sections. Section A has 5 questions and Section B has 9 questions. Candidates are required to answer 6 questions.
  1. In how many different ways can a candidate choose 6 questions, if 3 are from Section A and 3 are from Section B?
  2. Another candidate randomly chooses 6 questions to answer. Find the probability that this candidate chooses 3 questions from each section.
OCR MEI S1 2012 June Q3
8 marks Moderate -0.8
3 At a call centre, \(85 \%\) of callers are put on hold before being connected to an operator. A random sample of 30 callers is selected.
  1. Find the probability that exactly 29 of these callers are put on hold.
  2. Find the probability that at least 29 of these callers are put on hold.
  3. If 10 random samples, each of 30 callers, are selected, find the expected number of samples in which at least 29 callers are put on hold.
OCR MEI S1 2012 June Q4
8 marks Moderate -0.8
4 It is known that \(8 \%\) of the population of a large city use a particular web browser. A researcher wishes to interview some people from the city who use this browser. He selects people at random, one at a time.
  1. Find the probability that the first person that he finds who uses this browser is
    (A) the third person selected,
    (B) the second or third person selected.
  2. Find the probability that at least one of the first 20 people selected uses this browser.
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 2014 June Q1
8 marks Easy -1.3
1 The ages, \(x\) years, of the senior members of a running club are summarised in the table below.
Age \(( x )\)\(20 \leqslant x < 30\)\(30 \leqslant x < 40\)\(40 \leqslant x < 50\)\(50 \leqslant x < 60\)\(60 \leqslant x < 70\)\(70 \leqslant x < 80\)\(80 \leqslant x < 90\)
Frequency10304223951
  1. Draw a cumulative frequency diagram to illustrate the data.
  2. Use your diagram to estimate the median and interquartile range of the data.
OCR MEI S1 2014 June Q2
8 marks Standard +0.3
2 Candidates applying for jobs in a large company take an aptitude test, as a result of which they are either accepted, rejected or retested, with probabilities \(0.2,0.5\) and 0.3 respectively. When a candidate is retested for the first time, the three possible outcomes and their probabilities remain the same as for the original test. When a candidate is retested for the second time there are just two possible outcomes, accepted or rejected, with probabilities 0.4 and 0.6 respectively.
  1. Draw a probability tree diagram to illustrate the outcomes.
  2. Find the probability that a randomly selected candidate is accepted.
  3. Find the probability that a randomly selected candidate is retested at least once, given that this candidate is accepted.
OCR MEI S1 2014 June Q3
6 marks Moderate -0.8
3 Each weekday, Marta travels to school by bus. Sometimes she arrives late.
  • \(L\) is the event that Marta arrives late.
  • \(R\) is the event that it is raining.
You are given that \(\mathrm { P } ( L ) = 0.15 , \mathrm { P } ( R ) = 0.22\) and \(\mathrm { P } ( L \mid R ) = 0.45\).
  1. Use this information to show that the events \(L\) and \(R\) are not independent.
  2. Find \(\mathrm { P } ( L \cap R )\).
  3. Draw a Venn diagram showing the events \(L\) and \(R\), and fill in the probability corresponding to each of the four regions of your diagram.
OCR MEI S1 2014 June Q4
6 marks Moderate -0.3
4 There are 16 girls and 14 boys in a class. Four of them are to be selected to form a quiz team. The team is to be selected at random.
  1. Find the probability that all 4 members of the team will be girls.
  2. Find the probability that the team will contain at least one girl and at least one boy.
OCR MEI S1 2014 June Q5
8 marks Moderate -0.8
5 The probability distribution of the random variable \(X\) is given by the formula $$\mathrm { P } ( X = r ) = k + 0.01 r ^ { 2 } \text { for } r = 1,2,3,4,5 \text {. }$$
  1. Show that \(k = 0.09\). Using this value of \(k\), display the probability distribution of \(X\) in a table.
  2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
OCR MEI S1 2014 June Q7
19 marks Challenging +1.2
7 It is known that on average \(85 \%\) of seeds of a particular variety of tomato will germinate. Ramesh selects 15 of these seeds at random and sows them.
  1. (A) Find the probability that exactly 12 germinate.
    (B) Find the probability that fewer than 12 germinate. The following year Ramesh finds that he still has many seeds left. Because the seeds are now one year old, he suspects that the germination rate will be lower. He conducts a trial by randomly selecting \(n\) of these seeds and sowing them. He then carries out a hypothesis test at the \(1 \%\) significance level to investigate whether he is correct.
  2. Write down suitable null and alternative hypotheses for the test. Give a reason for your choice of alternative hypothesis.
  3. In a trial with \(n = 20\), Ramesh finds that 13 seeds germinate. Carry out the test.
  4. Suppose instead that Ramesh conducts the trial with \(n = 50\), and finds that 33 seeds germinate. Given that the critical value for the test in this case is 35 , complete the test.
  5. If \(n\) is small, there is no point in carrying out the test at the \(1 \%\) significance level, as the null hypothesis cannot be rejected however many seeds germinate. Find the least value of \(n\) for which the null hypothesis can be rejected, quoting appropriate probabilities to justify your answer.
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 \(\operatorname { Var } ( X )\).