Questions — OCR MEI S1 (300 questions)

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OCR MEI S1 2011 June Q5
8 marks Moderate -0.8
In a recent survey, a large number of working people were asked whether they worked full-time or part-time, with part-time being defined as less than 25 hours per week. One of the respondents is selected at random. • \(W\) is the event that this person works part-time. • \(F\) is the event that this person is female. You are given that P(\(W\)) = 0.14, P(\(F\)) = 0.41 and P(\(W \cap F\)) = 0.11.
  1. Draw a Venn diagram showing the events \(W\) and \(F\), and fill in the probability corresponding to each of the four regions of your diagram. [3]
  2. Determine whether the events \(W\) and \(F\) are independent. [2]
  3. Find P(\(W\) | \(F\)) and explain what this probability represents. [3]
OCR MEI S1 2011 June Q6
7 marks Moderate -0.8
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. [5]
  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? [2]
OCR MEI S1 2011 June Q7
18 marks Standard +0.3
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
    1. there is exactly 1 no-show in the sample, [3]
    2. there are at least 2 no-shows in the sample. [2]
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.
  1. Write down suitable null and alternative hypotheses for the test. Give a reason for your choice of alternative hypothesis. [4]
  2. In the case that \(n = 20\) and the number of no-shows in the sample is 1, carry out the test. [4]
  3. 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. [3]
  4. In the case that \(n \leqslant 18\), explain why there is no point in carrying out the test at the 5% level. [2]
OCR MEI S1 2011 June Q8
18 marks Moderate -0.3
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 \(\leqslant x <\) 9.39.3 \(< x \leqslant\) 9.59.5 \(< x \leqslant\) 9.79.7 \(< x \leqslant\) 9.99.9 \(< x \leqslant\) 10.1
Frequency5715167
  1. Draw a cumulative frequency diagram to illustrate these data. [5]
  2. Use the diagram to estimate the median and interquartile range of the data. [3]
  3. Show that there are no outliers in the sample. [3]
  4. Three of these 50 sacks are selected at random. Find the probability that
    1. in all three, the heating quality \(x\) is more than 9.5, [3]
    2. in at least two, the heating quality \(x\) is more than 9.5. [4]
OCR MEI S1 2014 June Q1
8 marks Easy -1.3
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. [5]
  2. Use your diagram to estimate the median and interquartile range of the data. [3]
OCR MEI S1 2014 June Q2
8 marks Moderate -0.8
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. [3]
  2. Find the probability that a randomly selected candidate is accepted. [2]
  3. Find the probability that a randomly selected candidate is retested at least once, given that this candidate is accepted. [3]
OCR MEI S1 2014 June Q3
6 marks Easy -1.2
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. [1]
  2. Find \(\mathrm{P}(L \cap R)\). [2]
  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. [3]
OCR MEI S1 2014 June Q4
6 marks Moderate -0.8
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. [3]
  2. Find the probability that the team will contain at least one girl and at least one boy. [3]
OCR MEI S1 2014 June Q5
8 marks Moderate -0.8
The probability distribution of the random variable \(X\) is given by the formula $$\mathrm{P}(X = r) = k + 0.01r^2 \text{ for } r = 1, 2, 3, 4, 5.$$
  1. Show that \(k = 0.09\). Using this value of \(k\), display the probability distribution of \(X\) in a table. [3]
  2. Find \(\mathrm{E}(X)\) and \(\mathrm{Var}(X)\). [5]
OCR MEI S1 2014 June Q6
17 marks Moderate -0.8
The weights, \(w\) grams, of a random sample of 60 carrots of variety A are summarised in the table below.
Weight\(30 \leqslant w < 50\)\(50 \leqslant w < 60\)\(60 \leqslant w < 70\)\(70 \leqslant w < 80\)\(80 \leqslant w < 90\)
Frequency111018147
  1. Draw a histogram to illustrate these data. [5]
  2. Calculate estimates of the mean and standard deviation of \(w\). [4]
  3. Use your answers to part (ii) to investigate whether there are any outliers. [3]
The weights, \(x\) grams, of a random sample of 50 carrots of variety B are summarised as follows. $$n = 50 \quad \sum x = 3624.5 \quad \sum x^2 = 265416$$
  1. Calculate the mean and standard deviation of \(x\). [3]
  2. Compare the central tendency and variation of the weights of varieties A and B. [2]
OCR MEI S1 2014 June Q7
19 marks Standard +0.3
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. Find the probability that exactly 12 germinate. [3]
    2. Find the probability that fewer than 12 germinate. [2]
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.
  1. Write down suitable null and alternative hypotheses for the test. Give a reason for your choice of alternative hypothesis. [4]
  2. In a trial with \(n = 20\), Ramesh finds that 13 seeds germinate. Carry out the test. [4]
  3. 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. [3]
  4. 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. [3]
OCR MEI S1 Q1
8 marks Moderate -0.8
Four letters are taken out of their envelopes for signing. Unfortunately they are replaced randomly, one in each envelope. The probability distribution for the number of letters, \(X\), which are now in the correct envelope is given in the following table.
\(r\)01
P(X = r)\(\frac{3}{8}\)\(\frac{1}{3}\)\(\frac{1}{4}\)0\(\frac{1}{24}\)
  1. Explain why the case \(X = 3\) is impossible. [1]
  2. Explain why P(\(X = 4\)) = \(\frac{1}{24}\). [2]
  3. Calculate E(\(X\)) and Var(\(X\)). [5]
OCR MEI S1 Q2
5 marks Easy -1.2
A company sells sugar in bags which are labelled as containing 450 grams. Although the mean weight of sugar in a bag is more than 450 grams, there is concern that too many bags are underweight. The company can adjust the mean or the standard deviation of the weight of sugar in a bag.
  1. State two adjustments the company could make. [2]
The weights, \(x\) grams, of a random sample of 25 bags are now recorded.
  1. Given that \(\sum x = 11409\) and \(\sum x^2 = 5206937\), calculate the sample mean and sample standard deviation of these weights. [3]
OCR MEI S1 Q3
8 marks Moderate -0.8
Jeremy is a computing consultant who sometimes works at home. The number, \(X\), of days that Jeremy works at home in any given week is modelled by the probability distribution P(\(X = r\)) = \(\frac{1}{40}r(r + 1)\) for \(r = 1, 2, 3, 4\).
  1. Verify that P(\(X = 4\)) = \(\frac{1}{2}\). [1]
  2. Calculate E(\(X\)) and Var(\(X\)). [5]
  3. Jeremy works for 45 weeks each year. Find the expected number of weeks during which he works at home for exactly 2 days. [2]
OCR MEI S1 Q4
7 marks Moderate -0.8
A sprinter runs many 100-metre trials, and the time, \(x\) seconds, for each is recorded. A sample of eight of these times is taken, as follows. 10.53 \quad 10.61 \quad 10.04 \quad 10.49 \quad 10.63 \quad 10.55 \quad 10.47 \quad 10.63
  1. Calculate the sample mean, \(\bar{x}\), and sample standard deviation, \(s\), of these times. [3]
  2. Show that the time of 10.04 seconds may be regarded as an outlier. [2]
  3. Discuss briefly whether or not the time of 10.04 seconds should be discarded. [2]
OCR MEI S1 Q5
6 marks Moderate -0.8
The number, \(X\), of children per family in a certain city is modelled by the probability distribution P(\(X = r\)) = \(k(6 - r)(1 + r)\) for \(r = 0, 1, 2, 3, 4\).
  1. Copy and complete the following table and hence show that the value of \(k\) is \(\frac{1}{50}\). [3]
    \(r\)01234
    P(\(X = r\))\(6k\)\(10k\)
  2. Calculate E(\(X\)). [2]
  3. Hence write down the probability that a randomly selected family in this city has more than the mean number of children. [1]
OCR MEI S1 Q6
17 marks Moderate -0.8
The weights, \(w\) grams, of a random sample of 60 carrots of variety A are summarised in the table below.
Weight\(30 \leqslant w < 50\)\(50 \leqslant w < 60\)\(60 \leqslant w < 70\)\(70 \leqslant w < 80\)\(80 \leqslant w < 90\)
Frequency111018147
  1. Draw a histogram to illustrate these data. [5]
  2. Calculate estimates of the mean and standard deviation of \(w\). [4]
  3. Use your answers to part (ii) to investigate whether there are any outliers. [3]
The weights, \(x\) grams, of a random sample of 50 carrots of variety B are summarised as follows. \(n = 50\) \quad \(\sum x = 3624.5\) \quad \(\sum x^2 = 265416\)
  1. Calculate the mean and standard deviation of \(x\). [3]
  2. Compare the central tendency and variation of the weights of varieties A and B. [2]
OCR MEI S1 Q7
7 marks Moderate -0.8
A supermarket chain buys a batch of 10000 scratchcard draw tickets for sale in its stores. 50 of these tickets have a £10 prize, 20 of them have a £100 prize, one of them has a £5000 prize and all of the rest have no prize. This information is summarised in the frequency table below.
Prize money£0£10£100£5000
Frequency992950201
  1. Find the mean and standard deviation of the prize money per ticket. [4]
  2. I buy two of these tickets at random. Find the probability that I win either two £10 prizes or two £100 prizes. [3]
OCR MEI S1 Q1
8 marks Moderate -0.8
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
    1. the third person selected, [3]
    2. the second or third person selected. [2]
  2. Find the probability that at least one of the first 20 people selected uses this browser. [3]
OCR MEI S1 Q2
8 marks Standard +0.3
Jimmy and Alan are playing a tennis match against each other. The winner of the match is the first player to win three sets. Jimmy won the first set and Alan won the second set. For each of the remaining sets, the probability that Jimmy wins a set is • 0.7 if he won the previous set, • 0.4 if Alan won the previous set. It is not possible to draw a set.
  1. Draw a probability tree diagram to illustrate the possible outcomes for each of the remaining sets. [3]
  2. Find the probability that Alan wins the match. [3]
  3. Find the probability that the match ends after exactly four sets have been played. [2]
OCR MEI S1 Q3
6 marks Moderate -0.8
In a food survey, a large number of people are asked whether they like tomato soup, mushroom soup, both or neither. One of these people is selected at random. • \(T\) is the event that this person likes tomato soup. • \(M\) is the event that this person likes mushroom soup. You are given that \(\text{P}(T) = 0.55\), \(\text{P}(M) = 0.33\) and \(\text{P}(T|M) = 0.80\).
  1. Use this information to show that the events \(T\) and \(M\) are not independent. [1]
  2. Find \(\text{P}(T \cap M)\). [2]
  3. Draw a Venn diagram showing the events \(T\) and \(M\), and fill in the probability corresponding to each of the four regions of your diagram. [3]
OCR MEI S1 Q4
4 marks Easy -1.2
25% of the plants of a particular species have red flowers. A random sample of 6 plants is selected.
  1. Find the probability that there are no plants with red flowers in the sample. [2]
  2. If 50 random samples of 6 plants are selected, find the expected number of samples in which there are no plants with red flowers. [2]
OCR MEI S1 Q5
8 marks Moderate -0.8
In a recent survey, a large number of working people were asked whether they worked full-time or part-time, with part-time being defined as less than 25 hours per week. One of the respondents is selected at random. • \(W\) is the event that this person works part-time. • \(F\) is the event that this person is female. You are given that \(\text{P}(W) = 0.14\), \(\text{P}(F) = 0.41\) and \(\text{P}(W \cap F) = 0.11\).
  1. Draw a Venn diagram showing the events \(W\) and \(F\), and fill in the probability corresponding to each of the four regions of your diagram. [3]
  2. Determine whether the events \(W\) and \(F\) are independent. [2]
  3. Find \(\text{P}(W|F)\) and explain what this probability represents. [3]
OCR MEI S1 Q6
4 marks Moderate -0.8
The table shows all the possible products of the scores on two fair four-sided dice. \includegraphics{figure_6}
  1. Find the probability that the product of the two scores is less than 10. [1]
  2. Show that the events 'the score on the first die is even' and 'the product of the scores on the two dice is less than 10' are not independent. [3]
OCR MEI S1 Q7
8 marks Moderate -0.8
Andy can walk to work, travel by bike or travel by bus. The tree diagram shows the probabilities of any day being dry or wet and the corresponding probabilities for each of Andy's methods of travel. \includegraphics{figure_7} A day is selected at random. Find the probability that
  1. the weather is wet and Andy travels by bus, [2]
  2. Andy walks or travels by bike, [3]
  3. the weather is dry given that Andy walks or travels by bike. [3]