Questions — OCR MEI (4333 questions)

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OCR MEI S1 2013 January Q4
7 marks Moderate -0.8
4 At a dog show, three out of eleven dogs are to be selected for a national competition.
  1. Find the number of possible selections.
  2. Five of the eleven dogs are terriers. Assuming that the dogs are selected at random, find the probability that at least two of the three dogs selected for the national competition are terriers.
OCR MEI S1 2013 January Q5
5 marks Easy -1.2
5 Malik is playing a game in which he has to throw a 6 on a fair six-sided die to start the game. Find the probability that
  1. Malik throws a 6 for the first time on his third attempt,
  2. Malik needs at most ten attempts to throw a 6.
OCR MEI S1 2013 January Q6
18 marks Standard +0.3
6 The heights \(x \mathrm {~cm}\) of 100 boys in Year 7 at a school are summarised in the table below.
Height\(125 \leqslant x \leqslant 140\)\(140 < x \leqslant 145\)\(145 < x \leqslant 150\)\(150 < x \leqslant 160\)\(160 < x \leqslant 170\)
Frequency252924184
  1. Estimate the number of boys who have heights of at least 155 cm .
  2. Calculate an estimate of the median height of the 100 boys.
  3. Draw a histogram to illustrate the data. The histogram below shows the heights of 100 girls in Year 7 at the same school. \includegraphics[max width=\textwidth, alt={}, center]{76283206-687f-45d6-9204-952d60843cf1-3_865_1349_1297_349}
  4. How many more girls than boys had heights exceeding 160 cm ?
  5. Calculate an estimate of the mean height of the 100 girls.
OCR MEI S1 2013 January Q7
18 marks Standard +0.3
7 A coffee shop provides free internet access for its customers. It is known that the probability that a randomly selected customer is accessing the internet is 0.35 , independently of all other customers.
  1. 10 customers are selected at random.
    (A) Find the probability that exactly 5 of them are accessing the internet.
    (B) Find the probability that at least 5 of them are accessing the internet.
    (C) Find the expected number of these customers who are accessing the internet. Another coffee shop also provides free internet access. It is suspected that the probability that a randomly selected customer at this coffee shop is accessing the internet may be different from 0.35 . A random sample of 20 customers at this coffee shop is selected. Of these, 10 are accessing the internet.
  2. Carry out a hypothesis test at the \(5 \%\) significance level to investigate whether the probability for this coffee shop is different from 0.35 . Give a reason for your choice of alternative hypothesis.
  3. To get a more reliable result, a much larger random sample of 200 customers is selected over a period of time, and another hypothesis test is carried out. You are given that 90 of the 200 customers were accessing the internet. You are also given that, if \(X\) has the binomial distribution with parameters \(n = 200\) and \(p = 0.35\), then \(\mathrm { P } ( X \geqslant 90 ) = 0.0022\). Using the same hypotheses and significance level which you used in part (ii), complete this test.
OCR MEI S1 2009 June Q1
5 marks Easy -1.8
1 In a traffic survey, the number of people in each car passing the survey point is recorded. The results are given in the following frequency table.
Number of people1234
Frequency5031165
  1. Write down the median and mode of these data.
  2. Draw a vertical line diagram for these data.
  3. State the type of skewness of the distribution.
OCR MEI S1 2009 June Q2
5 marks Moderate -0.8
2 There are 14 girls and 11 boys in a class. A quiz team of 5 students is to be chosen from the class.
  1. How many different teams are possible?
  2. If the team must include 3 girls and 2 boys, find how many different teams are possible.
OCR MEI S1 2009 June Q3
8 marks Moderate -0.8
3 Dwayne is a car salesman. The numbers of cars, \(x\), sold by Dwayne each month during the year 2008 are summarised by $$n = 12 , \quad \Sigma x = 126 , \quad \Sigma x ^ { 2 } = 1582 .$$
  1. Calculate the mean and standard deviation of the monthly numbers of cars sold.
  2. Dwayne earns \(\pounds 500\) each month plus \(\pounds 100\) commission for each car sold. Show that the mean of Dwayne's monthly earnings is \(\pounds 1550\). Find the standard deviation of Dwayne's monthly earnings.
  3. Marlene is a car saleswoman and is paid in the same way as Dwayne. During 2008 her monthly earnings have mean \(\pounds 1625\) and standard deviation \(\pounds 280\). Briefly compare the monthly numbers of cars sold by Marlene and Dwayne during 2008.
OCR MEI S1 2009 June Q4
4 marks Easy -1.2
4 The table shows the probability distribution of the random variable \(X\).
\(r\)10203040
\(\mathrm { P } ( X = r )\)0.20.30.30.2
  1. Explain why \(\mathrm { E } ( X ) = 25\).
  2. Calculate \(\operatorname { Var } ( X )\).
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 2011 June Q1
5 marks Easy -1.2
1 In the Paris-Roubaix cycling race, there are a number of sections of cobbled road. The lengths of these sections, measured in metres, are illustrated in the histogram. \includegraphics[max width=\textwidth, alt={}, center]{854cb8fb-d75d-4854-b3ec-d7edbb21ea7e-2_899_1397_477_372}
  1. Find the number of sections which are between 1000 and 2000 metres in length.
  2. Name the type of skewness suggested by the histogram.
  3. State the minimum and maximum possible values of the midrange.
OCR MEI S1 2011 June Q2
5 marks Easy -1.2
2 I have 5 books, each by a different author. The authors are Austen, Brontë, Clarke, Dickens and Eliot.
  1. If I arrange the books in a random order on my bookshelf, find the probability that the authors are in alphabetical order with Austen on the left.
  2. If I choose two of the books at random, find the probability that I choose the books written by Austen and Brontë. \(325 \%\) of the plants of a particular species have red flowers. A random sample of 6 plants is selected.
  3. Find the probability that there are no plants with red flowers in the sample.
  4. If 50 random samples of 6 plants are selected, find the expected number of samples in which there are no plants with red flowers.
OCR MEI S1 2011 June Q4
7 marks Easy -1.3
4 Two fair six-sided dice are thrown. The random variable \(X\) denotes the difference between the scores on the two dice. The table shows the probability distribution of \(X\).
\(r\)012345
\(\mathrm { P } ( X = r )\)\(\frac { 1 } { 6 }\)\(\frac { 5 } { 18 }\)\(\frac { 2 } { 9 }\)\(\frac { 1 } { 6 }\)\(\frac { 1 } { 9 }\)\(\frac { 1 } { 18 }\)
  1. Draw a vertical line chart to illustrate the probability distribution.
  2. Use a probability argument to show that
    (A) \(\mathrm { P } ( X = 1 ) = \frac { 5 } { 18 }\),
    (B) \(\mathrm { P } ( X = 0 ) = \frac { 1 } { 6 }\).
  3. Find the mean value of \(X\).
OCR MEI S1 2011 June Q6
7 marks Easy -1.3
6 The numbers of eggs laid by a sample of 70 female herring gulls are shown in the table.
Number of eggs1234
Frequency1040155
  1. Find the mean and standard deviation of the number of eggs laid per gull.
  2. The sample did not include female herring gulls that laid no eggs. How would the mean and standard deviation change if these gulls were included?
OCR MEI S1 2011 June Q7
18 marks Standard +0.3
7 Any patient who fails to turn up for an outpatient appointment at a hospital is described as a 'no-show'. At a particular hospital, on average \(15 \%\) of patients are no-shows. A random sample of 20 patients who have outpatient appointments is selected.
  1. Find the probability that
    (A) there is exactly 1 no-show in the sample,
    (B) there are at least 2 no-shows in the sample. The hospital management introduces a policy of telephoning patients before appointments. It is hoped that this will reduce the proportion of no-shows. In order to check this, a random sample of \(n\) patients is selected. The number of no-shows in the sample is recorded and a hypothesis test is carried out at the 5\% level.
  2. Write down suitable null and alternative hypotheses for the test. Give a reason for your choice of alternative hypothesis.
  3. In the case that \(n = 20\) and the number of no-shows in the sample is 1 , carry out the test.
  4. In another case, where \(n\) is large, the number of no-shows in the sample is 6 and the critical value for the test is 8 . Complete the test.
  5. In the case that \(n \leqslant 18\), explain why there is no point in carrying out the test at the \(5 \%\) level.
OCR MEI S1 2011 June Q8
18 marks Easy -1.3
8 The heating quality of the coal in a sample of 50 sacks is measured in suitable units. The data are summarised below.
Heating quality \(( x )\)\(9.1 \leqslant x \leqslant 9.3\)\(9.3 < x \leqslant 9.5\)\(9.5 < x \leqslant 9.7\)\(9.7 < x \leqslant 9.9\)\(9.9 < x \leqslant 10.1\)
Frequency5715167
  1. Draw a cumulative frequency diagram to illustrate these data.
  2. Use the diagram to estimate the median and interquartile range of the data.
  3. Show that there are no outliers in the sample.
  4. Three of these 50 sacks are selected at random. Find the probability that
    (A) in all three, the heating quality \(x\) is more than 9.5, \(( B )\) in at least two, the heating quality \(x\) is more than 9.5. OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download from our public website (\href{http://www.ocr.org.uk}{www.ocr.org.uk}) after the live examination series.
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