Questions — OCR MEI S1 (300 questions)

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OCR MEI S1 2009 June Q5
8 marks Moderate -0.8
5 The frequency table below shows the distance travelled by 1200 visitors to a particular UK tourist destination in August 2008.
Distance \(( d\) miles \()\)\(0 \leqslant d < 50\)\(50 \leqslant d < 100\)\(100 \leqslant d < 200\)\(200 \leqslant d < 400\)
Frequency360400307133
  1. Draw a histogram on graph paper to illustrate these data.
  2. Calculate an estimate of the median distance.
OCR MEI S1 2009 June Q6
6 marks Moderate -0.8
6 Whitefly, blight and mosaic virus are three problems which can affect tomato plants. 100 tomato plants are examined for these problems. The numbers of plants with each type of problem are shown in the Venn diagram. 47 of the plants have none of the problems. \includegraphics[max width=\textwidth, alt={}, center]{3a5d18f5-b1fe-4513-ae4e-f37c20f172b5-3_668_812_998_664}
  1. One of the 100 plants is selected at random. Find the probability that this plant has
    (A) at most one of the problems,
    (B) exactly two of the problems.
  2. Three of the 100 plants are selected at random. Find the probability that all of them have at least one of the problems. Section B (36 marks)
OCR MEI S1 2009 June Q7
18 marks Easy -1.2
7 Laura frequently flies to business meetings and often finds that her flights are delayed. A flight may be delayed due to technical problems, weather problems or congestion problems, with probabilities \(0.2,0.15\) and 0.1 respectively. The tree diagram shows this information. \includegraphics[max width=\textwidth, alt={}, center]{3a5d18f5-b1fe-4513-ae4e-f37c20f172b5-4_608_1651_532_248}
  1. Write down the values of the probabilities \(a , b\) and \(c\) shown in the tree diagram. One of Laura's flights is selected at random.
  2. Find the probability that Laura's flight is not delayed and hence write down the probability that it is delayed.
  3. Find the probability that Laura's flight is delayed due to just one of the three problems.
  4. Given that Laura's flight is delayed, find the probability that the delay is due to just one of the three problems.
  5. Given that Laura's flight has no technical problems, find the probability that it is delayed.
  6. In a particular year, Laura has 110 flights. Find the expected number of flights that are delayed.
OCR MEI S1 2009 June Q8
18 marks Standard +0.3
8 The Department of Health 'eat five a day' advice recommends that people should eat at least five portions of fruit and vegetables per day. In a particular school, \(20 \%\) of pupils eat at least five a day.
  1. 15 children are selected at random.
    (A) Find the probability that exactly 3 of them eat at least five a day.
    (B) Find the probability that at least 3 of them eat at least five a day.
    (C) Find the expected number who eat at least five a day. A programme is introduced to encourage children to eat more portions of fruit and vegetables per day. At the end of this programme, the diets of a random sample of 15 children are analysed. A hypothesis test is carried out to examine whether the proportion of children in the school who eat at least five a day has increased.
  2. (A) Write down suitable null and alternative hypotheses for the test.
    (B) Give a reason for your choice of the alternative hypothesis.
  3. Find the critical region for the test at the \(10 \%\) significance level, showing all of your calculations. Hence complete the test, given that 7 of the 15 children eat at least five a day.
OCR MEI S1 2010 June Q1
8 marks Moderate -0.8
1 A business analyst collects data about the distribution of hourly wages, in \(\pounds\), of shop-floor workers at a factory. These data are illustrated in the box and whisker plot. \includegraphics[max width=\textwidth, alt={}, center]{091d6f43-ad01-4849-9f3c-3e58349aa169-2_204_1422_484_363}
  1. Name the type of skewness of the distribution.
  2. Find the interquartile range and hence show that there are no outliers at the lower end of the distribution, but there is at least one outlier at the upper end.
  3. Suggest possible reasons why this may be the case.
OCR MEI S1 2010 June Q2
7 marks Moderate -0.8
2 The probability distribution of the random variable \(X\) is given by the formula $$\mathrm { P } ( X = r ) = k r ( 5 - r ) \text { for } r = 1,2,3,4 .$$
  1. Show that \(k = 0.05\).
  2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
OCR MEI S1 2010 June Q3
7 marks Moderate -0.8
3 The lifetimes in hours of 90 components are summarised in the table.
Lifetime \(( x\) hours \()\)\(0 < x \leqslant 20\)\(20 < x \leqslant 30\)\(30 < x \leqslant 50\)\(50 < x \leqslant 65\)\(65 < x \leqslant 100\)
Frequency2413142118
  1. Draw a histogram to illustrate these data.
  2. In which class interval does the median lie? Justify your answer.
OCR MEI S1 2010 June Q4
6 marks Standard +0.3
4 Each packet of Cruncho cereal contains one free fridge magnet. There are five different types of fridge magnet to collect. They are distributed, with equal probability, randomly and independently in the packets. Keith is about to start collecting these fridge magnets.
  1. Find the probability that the first 2 packets that Keith buys contain the same type of fridge magnet.
  2. Find the probability that Keith collects all five types of fridge magnet by buying just 5 packets.
  3. Hence find the probability that Keith has to buy more than 5 packets to acquire a complete set.
OCR MEI S1 2010 June Q5
8 marks Moderate -0.8
5 A retail analyst records the numbers of loaves of bread of a particular type bought by a sample of shoppers in a supermarket.
Number of loaves012345
Frequency372311301
  1. Calculate the mean and standard deviation of the numbers of loaves bought per person.
  2. Each loaf costs \(\pounds 1.04\). Calculate the mean and standard deviation of the amount spent on loaves per person.
OCR MEI S1 2010 June Q6
18 marks Standard +0.3
6 A manufacturer produces tiles. On average 10\% of the tiles produced are faulty. Faulty tiles occur randomly and independently. A random sample of 18 tiles is selected.
  1. (A) Find the probability that there are exactly 2 faulty tiles in the sample.
    (B) Find the probability that there are more than 2 faulty tiles in the sample.
    (C) Find the expected number of faulty tiles in the sample. A cheaper way of producing the tiles is introduced. The manufacturer believes that this may increase the proportion of faulty tiles. In order to check this, a random sample of 18 tiles produced using the cheaper process is selected and a hypothesis test is carried out.
  2. (A) Write down suitable null and alternative hypotheses for the test.
    (B) Explain why the alternative hypothesis has the form that it does.
  3. Find the critical region for the test at the \(5 \%\) level, showing all of your calculations.
  4. In fact there are 4 faulty tiles in the sample. Complete the test, stating your conclusion clearly.
OCR MEI S1 2010 June Q7
18 marks Moderate -0.3
7 One train leaves a station each hour. The train is either on time or late. If the train is on time, the probability that the next train is on time is 0.95 . If the train is late, the probability that the next train is on time is 0.6 . On a particular day, the 0900 train is on time.
  1. Illustrate the possible outcomes for the 1000,1100 and 1200 trains on a probability tree diagram.
  2. Find the probability that
    (A) all three of these trains are on time,
    (B) just one of these three trains is on time,
    (C) the 1200 train is on time.
  3. Given that the 1200 train is on time, find the probability that the 1000 train is also on time. 3
  4. Write any calculations on page 5. \includegraphics[max width=\textwidth, alt={}, center]{091d6f43-ad01-4849-9f3c-3e58349aa169-4_2276_1490_324_363}
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.