Questions — OCR MEI S1 (292 questions)

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OCR MEI S1 2008 June Q1
1 In a survey, a sample of 44 fields is selected. Their areas ( \(x\) hectares) are summarised in the grouped frequency table.
Area \(( x )\)\(0 < x \leqslant 3\)\(3 < x \leqslant 5\)\(5 < x \leqslant 7\)\(7 < x \leqslant 10\)\(10 < x \leqslant 20\)
Frequency3813146
  1. Calculate an estimate of the sample mean and the sample standard deviation.
  2. Determine whether there could be any outliers at the upper end of the distribution.
OCR MEI S1 2008 June Q2
2 In the 2001 census, people living in Wales were asked whether or not they could speak Welsh. A resident of Wales is selected at random.
  • \(W\) is the event that this person speaks Welsh.
  • \(C\) is the event that this person is a child.
You are given that \(\mathrm { P } ( W ) = 0.20 , \mathrm { P } ( C ) = 0.17\) and \(\mathrm { P } ( W \cap C ) = 0.06\).
  1. Determine whether the events \(W\) and \(C\) are independent.
  2. Draw a Venn diagram, showing the events \(W\) and \(C\), and fill in the probability corresponding to each region of your diagram.
  3. Find \(\mathrm { P } ( W \mid C )\).
  4. Given that \(\mathrm { P } \left( W \mid C ^ { \prime } \right) = 0.169\), use this information and your answer to part (iii) to comment very briefly on how the ability to speak Welsh differs between children and adults.
OCR MEI S1 2008 June Q3
3 In a game of darts, a player throws three darts. Let \(X\) represent the number of darts which hit the bull's-eye. The probability distribution of \(X\) is shown in the table.
\(r\)0123
\(\mathrm { P } ( X = r )\)0.50.35\(p\)\(q\)
  1. (A) Show that \(p + q = 0.15\).
    (B) Given that the expectation of \(X\) is 0.67 , show that \(2 p + 3 q = 0.32\).
    (C) Find the values of \(p\) and \(q\).
  2. Find the variance of \(X\).
OCR MEI S1 2008 June Q4
4 A small business has 8 workers. On a given day, the probability that any particular worker is off sick is 0.05 , independently of the other workers.
  1. A day is selected at random. Find the probability that
    (A) no workers are off sick,
    (B) more than one worker is off sick.
  2. There are 250 working days in a year. Find the expected number of days in the year on which more than one worker is off sick.
OCR MEI S1 2008 June Q5
5 A psychology student is investigating memory. In an experiment, volunteers are given 30 seconds to try to memorise a number of items. The items are then removed and the volunteers have to try to name all of them. It has been found that the probability that a volunteer names all of the items is 0.35 . The student believes that this probability may be increased if the volunteers listen to the same piece of music while memorising the items and while trying to name them. The student selects 15 volunteers at random to do the experiment while listening to music. Of these volunteers, 8 name all of the items.
  1. Write down suitable hypotheses for a test to determine whether there is any evidence to support the student's belief, giving a reason for your choice of alternative hypothesis.
  2. Carry out the test at the \(5 \%\) significance level.
OCR MEI S1 2008 June Q6
6 In a large town, 79\% of the population were born in England, 20\% in the rest of the UK and the remaining 1\% overseas. Two people are selected at random. You may use the tree diagram below in answering this question.
\includegraphics[max width=\textwidth, alt={}, center]{be764df3-ff20-415d-9c5c-10edabf350de-4_946_1119_580_513}
  1. Find the probability that
    (A) both of these people were born in the rest of the UK,
    (B) at least one of these people was born in England,
    (C) neither of these people was born overseas.
  2. Find the probability that both of these people were born in the rest of the UK given that neither was born overseas.
  3. (A) Five people are selected at random. Find the probability that at least one of them was not born in England.
    (B) An interviewer selects \(n\) people at random. The interviewer wishes to ensure that the probability that at least one of them was not born in England is more than \(90 \%\). Find the least possible value of \(n\). You must show working to justify your answer.
OCR MEI S1 2008 June Q7
7 The histogram shows the age distribution of people living in Inner London in 2001.
\includegraphics[max width=\textwidth, alt={}, center]{be764df3-ff20-415d-9c5c-10edabf350de-5_814_1383_349_379} Data sourced from the 2001 Census, \href{http://www.statistics.gov.uk}{www.statistics.gov.uk}
  1. State the type of skewness shown by the distribution.
  2. Use the histogram to estimate the number of people aged under 25.
  3. The table below shows the cumulative frequency distribution.
    Age2030405065100
    Cumulative frequency (thousands)66012401810\(a\)24902770
    (A) Use the histogram to find the value of \(a\).
    (B) Use the table to calculate an estimate of the median age of these people. The ages of people living in Outer London in 2001 are summarised below.
    Age ( \(x\) years)\(0 \leqslant x < 20\)\(20 \leqslant x < 30\)\(30 \leqslant x < 40\)\(40 \leqslant x < 50\)\(50 \leqslant x < 65\)\(65 \leqslant x < 100\)
    Frequency (thousands)1120650770590680610
  4. Illustrate these data by means of a histogram.
  5. Make two brief comments on the differences between the age distributions of the populations of Inner London and Outer London.
  6. The data given in the table for Outer London are used to calculate the following estimates. Mean 38.5, median 35.7, midrange 50, standard deviation 23.7, interquartile range 34.4.
    The final group in the table assumes that the maximum age of any resident is 100 years. These estimates are to be recalculated, based on a maximum age of 105, rather than 100. For each of the five estimates, state whether it would increase, decrease or be unchanged.
OCR MEI S1 Q1
1 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 Q2
2 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 Q3
3 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 Q1
1 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 Q2
2 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 Q3
3 It is known that \(25 \%\) of students in a particular city are smokers. A random sample of 20 of the students is selected.
  1. (A) Find the probability that there are exactly 4 smokers in the sample.
    (B) Find the probability that there are at least 3 but no more than 6 smokers in the sample
    (C) Write down the expected number of smokers in the sample. A new health education programme is introduced. This programme aims to reduce the percentage of students in this city who are smokers. After the programme has been running for a year, it is decided to carry out a hypothesis test to assess the effectiveness of the programme. A random sample of 20 students is selected.
  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 3 smokers in the sample. Complete the test, stating your conclusion clearly.
OCR MEI S1 Q1
1 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 Q2
2 Mark is playing solitaire on his computer. The probability that he wins a game is 0.2 , independently of all other games that he plays.
  1. Find the expected number of wins in 12 games.
  2. Find the probability that
    (A) he wins exactly 2 out of the next 12 games that he plays,
    (B) he wins at least 2 out of the next 12 games that he plays.
  3. Mark's friend Ali also plays solitaire. Ali claims that he is better at winning games than Mark. In a random sample of 20 games played by Ali, he wins 7 of them. Write down suitable hypotheses for a test at the \(5 \%\) level to investigate whether Ali is correct. Give a reason for your choice of alternative hypothesis. Carry out the test.
OCR MEI S1 Q3
3 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 Q1
1 In a multiple-choice test there are 30 questions. For each question, there is a \(60 \%\) chance that a randomly selected student answers correctly, independently of all other questions.
  1. Find the probability that a randomly selected student gets a total of exactly 20 questions correct.
  2. If 100 randomly selected students take the test, find the expected number of students who get exactly 20 questions correct.
OCR MEI S1 Q2
2 An environmental health officer monitors the air pollution level in a city street. Each day the level of pollution is classified as low, medium or high. The probabilities of each level of pollution on a randomly chosen day are as given in the table.
Pollution levelLowMediumHigh
Probability0.50.350.15
  1. Three days are chosen at random. Find the probability that the pollution level is
    (A) low on all 3 days,
    (B) low on at least one day,
    (C) low on one day, medium on another day, and high on the other day.
  2. Ten days are chosen at random. Find the probability that
    (A) there are no days when the pollution level is high,
    (B) there is exactly one day when the pollution level is high. The environmental health officer believes that pollution levels will be low more frequently in a different street. On 20 randomly selected days she monitors the pollution level in this street and finds that it is low on 15 occasions.
  3. Carry out a test at the \(5 \%\) level to determine if there is evidence to suggest that she is correct. Use hypotheses \(\mathrm { H } _ { 0 } : p = 0.5 , \mathrm { H } _ { 1 } : p > 0.5\), where \(p\) represents the probability that the pollution level in this street is low. Explain why \(\mathrm { H } _ { 1 }\) has this form.
OCR MEI S1 Q3
3 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 Q1
1 An online shopping company takes orders through its website. On average \(80 \%\) of orders from the website are delivered within 24 hours. The quality controller selects 10 orders at random to check when they are delivered.
  1. Find the probability that
    (A) exactly 8 of these orders are delivered within 24 hours,
    (B) at least 8 of these orders are delivered within 24 hours. The company changes its delivery method. The quality controller suspects that the changes will mean that fewer than \(80 \%\) of orders will be delivered within 24 hours. A random sample of 18 orders is checked and it is found that 12 of them arrive within 24 hours.
  2. Write down suitable hypotheses and carry out a test at the \(5 \%\) significance level to determine whether there is any evidence to support the quality controller's suspicion.
  3. A statistician argues that it is possible that the new method could result in either better or worse delivery times. Therefore it would be better to carry out a 2 -tail test at the \(5 \%\) significance level. State the alternative hypothesis for this test. Assuming that the sample size is still 18, find the critical region for this test, showing all of your calculations.
OCR MEI S1 Q2
2 In a game of darts, a player throws three darts. Let \(X\) represent the number of darts which hit the bull's-eye. The probability distribution of \(X\) is shown in the table.
\(r\)0123
\(\mathrm { P } ( X = r )\)0.50.35\(p\)\(q\)
  1. (A) Show that \(p + q = 0.15\).
    (B) Given that the expectation of \(X\) is 0.67 , show that \(2 p + 3 q = 0.32\).
    (C) Find the values of \(p\) and \(q\).
  2. Find the variance of \(X\).
OCR MEI S1 Q3
3 A psychology student is investigating memory. In an experiment, volunteers are given 30 seconds to try to memorise a number of items. The items are then removed and the volunteers have to try to name all of them. It has been found that the probability that a volunteer names all of the items is 0.35 . The student believes that this probability may be increased if the volunteers listen to the same piece of music while memorising the items and while trying to name them. The student selects 15 volunteers at random to do the experiment while listening to music. Of these volunteers, 8 name all of the items.
  1. Write down suitable hypotheses for a test to determine whether there is any evidence to support the student's belief, giving a reason for your choice of alternative hypothesis.
  2. Carry out the test at the \(5 \%\) significance level.
OCR MEI S1 Q4
4 A particular product is made from human blood given by donors. The product is stored in bags. The production process is such that, on average, \(5 \%\) of bags are faulty. Each bag is carefully tested before use.
  1. 12 bags are selected at random.
    (A) Find the probability that exactly one bag is faulty.
    (B) Find the probability that at least two bags are faulty.
    (C) Find the expected number of faulty bags in the sample.
  2. A random sample of \(n\) bags is selected. The production manager wishes there to be a probability of one third or less of finding any faulty bags in the sample. Find the maximum possible value of \(n\), showing your working clearly.
  3. A scientist believes that a new production process will reduce the proportion of faulty bags. A random sample of 60 bags made using the new process is checked and one bag is found to be faulty. Write down suitable hypotheses and carry out a hypothesis test at the \(10 \%\) level to determine whether there is evidence to suggest that the scientist is correct.
OCR MEI S1 Q1
1 A multinational accountancy firm receives a large number of job applications from graduates each year. On average \(20 \%\) of applicants are successful. A researcher in the human resources department of the firm selects a random sample of 17 graduate applicants.
  1. Find the probability that at least 4 of the 17 applicants are successful.
  2. Find the expected number of successful applicants in the sample.
  3. Find the most likely number of successful applicants in the sample, justifying your answer. It is suggested that mathematics graduates are more likely to be successful than those from other fields. In order to test this suggestion, the researcher decides to select a new random sample of 17 mathematics graduate applicants. The researcher then carries out a hypothesis test at the \(5 \%\) significance level.
  4. (A) Write down suitable null and alternative hypotheses for the test.
    (B) Give a reason for your choice of the alternative hypothesis.
  5. Find the critical region for the test at the \(5 \%\) level, showing all of your calculations.
  6. Explain why the critical region found in part (v) would be unaltered if a \(10 \%\) significance level were used.
OCR MEI S1 Q2
2 When onion seeds are sown outdoors, on average two-thirds of them germinate. A gardener sows seeds in pairs, in the hope that at least one will germinate.
  1. Assuming that germination of one of the seeds in a pair is independent of germination of the other seed, find the probability that, if a pair of seeds is selected at random,
    (A) both seeds germinate,
    (B) just one seed germinates,
    (C) neither seed germinates.
  2. Explain why the assumption of independence is necessary in order to calculate the above probabilities. Comment on whether the assumption is likely to be valid.
  3. A pair of seeds is sown. Find the expectation and variance of the number of seeds in the pair which germinate.
  4. The gardener plants 200 pairs of seeds. If both seeds in a pair germinate, the gardener destroys one of the two plants so that only one is left to grow. Of the plants that remain after this, only \(85 \%\) successfully grow to form an onion. Find the expected number of onions grown from the 200 pairs of seeds. If the seeds are sown in a greenhouse, the germination rate is higher. The seed manufacturing company claims that the germination rate is \(90 \%\). The gardener suspects that the rate will not be as high as this, and carries out a trial to investigate. 18 randomly selected seeds are sown in the greenhouse and it is found that 14 germinate.
  5. Write down suitable hypotheses and carry out a test at the \(5 \%\) level to determine whether there is any evidence to support the gardener's suspicions.