Questions — OCR MEI S1 (300 questions)

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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 \(\operatorname { Var } ( X )\).
OCR MEI S1 2015 June Q7
17 marks Standard +0.3
7 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 2015 June Q8
19 marks Standard +0.3
8 The box and whisker plot below summarises the weights in grams of the 20 chocolates in a box. \includegraphics[max width=\textwidth, alt={}, center]{6015ae6c-bf76-4a0c-af0f-5c53f9c5ed2a-4_287_1177_319_427}
  1. Find the interquartile range of the data and hence determine whether there are any outliers at either end of the distribution. Ben buys a box of these chocolates each weekend. The chocolates all look the same on the outside, but 7 of them have orange centres, 6 have cherry centres, 4 have coffee centres and 3 have lemon centres. One weekend, each of Ben's 3 children eats one of the chocolates, chosen at random.
  2. Calculate the probabilities of the following events. A: all 3 chocolates have orange centres \(B\) : all 3 chocolates have the same centres
  3. Find \(\mathrm { P } ( A \mid B )\) and \(\mathrm { P } ( B \mid A )\). The following weekend, Ben buys an identical box of chocolates and again each of his 3 children eats one of the chocolates, chosen at random.
  4. Find the probability that, on both weekends, the 3 chocolates that they eat all have orange centres.
  5. Ben likes all of the chocolates except those with cherry centres. On another weekend he is the first of his family to eat some of the chocolates. Find the probability that he has to select more than 2 chocolates before he finds one that he likes. \section*{END OF QUESTION PAPER} \section*{OCR
    Oxford Cambridge and RSA}
OCR MEI S1 2009 January Q1
7 marks Easy -1.2
1 A supermarket chain buys a batch of 10000 scratchcard draw tickets for sale in its stores. 50 of these tickets have a \(\pounds 10\) prize, 20 of them have a \(\pounds 100\) prize, one of them has a \(\pounds 5000\) prize and all of the rest have no prize. This information is summarised in the frequency table below.
Prize money\(\pounds 0\)\(\pounds 10\)\(\pounds 100\)\(\pounds 5000\)
Frequency992950201
  1. Find the mean and standard deviation of the prize money per ticket.
  2. I buy two of these tickets at random. Find the probability that I win either two \(\pounds 10\) prizes or two \(\pounds 100\) prizes.
OCR MEI S1 2009 January Q2
5 marks Moderate -0.8
2 Thomas has six tiles, each with a different letter of his name on it.
  1. Thomas arranges these letters in a random order. Find the probability that he arranges them in the correct order to spell his name.
  2. On another occasion, Thomas picks three of the six letters at random. Find the probability that he picks the letters T, O and M (in any order).
OCR MEI S1 2009 January Q3
8 marks Easy -1.3
3 A zoologist is studying the feeding behaviour of a group of 4 gorillas. The random variable \(X\) represents the number of gorillas that are feeding at a randomly chosen moment. The probability distribution of \(X\) is shown in the table below.
\(r\)01234
\(\mathrm { P } ( X = r )\)\(p\)0.10.050.050.25
  1. Find the value of \(p\).
  2. Find the expectation and variance of \(X\).
  3. The zoologist observes the gorillas on two further occasions. Find the probability that there are at least two gorillas feeding on both occasions.
OCR MEI S1 2009 January Q4
8 marks Moderate -0.8
4 A pottery manufacturer makes teapots in batches of 50. On average 3\% of teapots are faulty.
  1. Find the probability that in a batch of 50 there is
    (A) exactly one faulty teapot,
    (B) more than one faulty teapot.
  2. The manufacturer produces 240 batches of 50 teapots during one month. Find the expected number of batches which contain exactly one faulty teapot.
OCR MEI S1 2009 January Q5
8 marks Moderate -0.8
5 Each day Anna drives to work.
  • \(R\) is the event that it is raining.
  • \(L\) is the event that Anna arrives at work late.
You are given that \(\mathrm { P } ( R ) = 0.36 , \mathrm { P } ( L ) = 0.25\) and \(\mathrm { P } ( R \cap L ) = 0.2\).
  1. Determine whether the events \(R\) and \(L\) are independent.
  2. Draw a Venn diagram showing the events \(R\) and \(L\). Fill in the probability corresponding to each of the four regions of your diagram.
  3. Find \(\mathrm { P } ( L \mid R )\). State what this probability represents.
OCR MEI S1 2009 January Q6
17 marks Easy -1.2
6 The temperature of a supermarket fridge is regularly checked to ensure that it is working correctly. Over a period of three months the temperature (measured in degrees Celsius) is checked 600 times. These temperatures are displayed in the cumulative frequency diagram below. \includegraphics[max width=\textwidth, alt={}, center]{7b92607f-1bf9-45f0-997b-fe76c88b5fcd-4_1054_1649_539_248}
  1. Use the diagram to estimate the median and interquartile range of the data.
  2. Use your answers to part (i) to show that there are very few, if any, outliers in the sample.
  3. Suppose that an outlier is identified in these data. Discuss whether it should be excluded from any further analysis.
  4. Copy and complete the frequency table below for these data.
    Temperature
    \(( t\) degrees Celsius \()\)
    		\(3.0 \leqslant t \leqslant 3.4\)\(3.4 < t \leqslant 3.8\)\(3.8 < t \leqslant 4.2\)\(4.2 < t \leqslant 4.6\)\(4.6 < t \leqslant 5.0\)
    Frequency243157
  5. Use your table to calculate an estimate of the mean.
  6. The standard deviation of the temperatures in degrees Celsius is 0.379 . The temperatures are converted from degrees Celsius into degrees Fahrenheit using the formula \(F = 1.8 C + 32\). Hence estimate the mean and find the standard deviation of the temperatures in degrees Fahrenheit.
OCR MEI S1 2009 January Q7
19 marks Standard +0.3
7 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 2016 June Q1
7 marks Moderate -0.8
1 The stem and leaf diagram illustrates the weights in grams of 20 house sparrows.
250
26058
2779
28145
29002
3077
316
32047
3333
Key: \(\quad 27 \quad \mid \quad 7 \quad\) represents 27.7 grams
  1. Find the median and interquartile range of the data.
  2. Determine whether there are any outliers.
OCR MEI S1 2016 June Q2
7 marks Moderate -0.8
2 In a hockey league, each team plays every other team 3 times. The probabilities that Team A wins, draws and loses to Team B are given below.
  • \(\mathrm { P } (\) Wins \() = 0.5\)
  • \(\mathrm { P } (\) Draws \() = 0.3\)
  • \(\mathrm { P } (\) Loses \() = 0.2\)
The outcomes of the 3 matches are independent.
  1. Find the probability that Team A does not lose in any of the 3 matches.
  2. Find the probability that Team A either wins all 3 matches or draws all 3 matches or loses all 3 matches.
  3. Find the probability that, in the 3 matches, exactly two of the outcomes, 'Wins', 'Draws' and 'Loses' occur for Team A.
OCR MEI S1 2016 June Q3
6 marks Easy -1.3
3
  1. There are 5 runners in a race. How many different finishing orders are possible? [You should assume that there are no 'dead heats', where two runners are given the same position.] For the remainder of this question you should assume that all finishing orders are equally likely.
  2. The runners are denoted by \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E }\). Find the probability that they either finish in the order ABCDE or in the order EDCBA.
  3. Find the probability that the first 3 runners to finish are \(\mathrm { A } , \mathrm { B }\) and C , in that order.
  4. Find the probability that the first 3 runners to finish are \(\mathrm { A } , \mathrm { B }\) and C , in any order.
OCR MEI S1 2016 June Q4
8 marks Moderate -0.3
4 The probability distribution of the random variable \(X\) is given by the formula $$\mathrm { P } ( X = r ) = \frac { k } { r ( r - 1 ) } \text { for } r = 2,3,4,5,6 .$$
  1. Show that the value of \(k\) is 1.2 . Using this value of \(k\), show the probability distribution of \(X\) in a table.
  2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
OCR MEI S1 2016 June Q5
8 marks Easy -1.3
5 Measurements of sunshine and rainfall are made each day at a particular weather station. For a randomly chosen day,
  • \(R\) is the event that at least 1 mm of rainfall is recorded,
  • \(S\) is the event that at least 1 hour of sunshine is recorded.
You are given that \(\mathrm { P } ( R ) = 0.28 , \mathrm { P } ( S ) = 0.87\) and \(\mathrm { P } ( R \cup S ) = 0.94\).
  1. Find \(\mathrm { P } ( R \cap S )\).
  2. Draw a Venn diagram showing the events \(R\) and \(S\), and fill in the probability corresponding to each of the four regions of your diagram.
  3. Find \(\mathrm { P } ( R \mid S )\) and state what this probability represents in this context.
OCR MEI S1 2016 June Q6
18 marks Moderate -0.8
6 An online store has a total of 930 different types of women's running shoe on sale. The prices in pounds of the types of women's running shoe are summarised in the table below.
Price \(( \pounds x )\)\(10 \leqslant x \leqslant 40\)\(40 < x \leqslant 50\)\(50 < x \leqslant 60\)\(60 < x \leqslant 80\)\(80 < x \leqslant 200\)
Frequency147109182317175
  1. Calculate estimates of the mean and standard deviation of the shoe prices.
  2. Calculate an estimate of the percentage of types of shoe that cost at least \(\pounds 100\).
  3. Draw a histogram to illustrate the data. The corresponding histogram below shows the prices in pounds of the 990 types of men's running shoe on sale at the same online store. \includegraphics[max width=\textwidth, alt={}, center]{aff0c5b2-011b-49a0-bf05-6d905f890eba-4_643_1192_340_440}
  4. State the type of skewness shown by the histogram for men's running shoes.
  5. Martin is investigating the percentage of types of shoe on sale at the store that cost more than \(\pounds 100\). He believes that this percentage is greater for men's shoes than for women's shoes. Estimate the percentage for men's shoes and comment on whether you can be certain which percentage is higher.
  6. You are given that the mean and standard deviation of the prices of men's running shoes are \(\pounds 68.83\) and \(\pounds 42.93\) respectively. Compare the central tendency and variation of the prices of men's and women's running shoes at the store.
OCR MEI S1 2016 June Q7
18 marks Moderate -0.3
7 To withdraw money from a cash machine, the user has to enter a 4-digit PIN (personal identification number). There are several thousand possible 4-digit PINs, but a survey found that \(10 \%\) of cash machine users use the PIN '1234'.
  1. 16 cash machine users are selected at random.
    (A) Find the probability that exactly 3 of them use 1234 as their PIN.
    (B) Find the probability that at least 3 of them use 1234 as their PIN.
    (C) Find the expected number of them who use 1234 as their PIN. An advertising campaign aims to reduce the number of people who use 1234 as their PIN. A hypothesis test is to be carried out to investigate whether the advertising campaign has been successful.
  2. Write down suitable null and alternative hypotheses for the test. Give a reason for your choice of alternative hypothesis.
  3. A random sample of 20 cash machine users is selected.
    (A) Explain why the test could not be carried out at the \(10 \%\) significance level.
    (B) The test is to be carried out at the \(k \%\) significance level. State the lowest integer value of \(k\) for which the test could result in the rejection of the null hypothesis.
  4. A new random sample of 60 cash machine users is selected. It is found that 2 of them use 1234 as their PIN. You are given that, if \(X \sim \mathrm {~B} ( 60,0.1 )\), then (to 4 decimal places) $$\mathrm { P } ( X = 2 ) = 0.0393 , \quad \mathrm { P } ( X < 2 ) = 0.0138 , \quad \mathrm { P } ( X \leqslant 2 ) = 0.0530 .$$ Using the same hypotheses as in part (ii), carry out the test at the \(5 \%\) significance level. \section*{END OF QUESTION PAPER}
OCR MEI S1 Q3
8 marks Standard +0.3
3 The Venn diagram illustrates the occurrence of two events \(A\) and \(B\). \includegraphics[max width=\textwidth, alt={}, center]{1ad9c390-b42f-47d8-86c5-f73a42d97721-02_513_826_1713_658} You are given that \(\mathrm { P } ( A \cap B ) = 0.3\) and that the probability that neither \(A\) nor \(B\) occurs is 0.1 . You are also given that \(\mathrm { P } ( A ) = 2 \mathrm { P } ( B )\). Find \(\mathrm { P } ( B )\).
OCR MEI S1 Q7
18 marks Easy -1.3
7 The cumulative frequency graph below illustrates the distances that 176 children live from their primary school. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Distance from school} \includegraphics[alt={},max width=\textwidth]{1ad9c390-b42f-47d8-86c5-f73a42d97721-04_1073_1571_580_340}
\end{figure}
  1. Use the graph to estimate, to the nearest 10 metres,
    (A) the median distance from school,
    (B) the lower quartile, upper quartile and interquartile range.
  2. Draw a box and whisker plot to illustrate the data. The graph on page 4 used the following grouped data.
    Distance (metres)20040060080010001200
    Cumulative frequency2064118150169176
  3. Copy and complete the grouped frequency table below describing the same data.
    Distance ( \(d\) metres)Frequency
    \(0 < d \leqslant 200\)20
    \(200 < d \leqslant 400\)
  4. Hence estimate the mean distance these children live from school. It is subsequently found that none of the 176 children lives within 100 metres of the school.
  5. Calculate the revised estimate of the mean distance.
  6. Describe what change needs to be made to the cumulative frequency graph.
OCR MEI S1 2005 June Q5
6 marks Moderate -0.8
  1. On the insert, complete the table giving the lowest common multiples of all pairs of integers between 1 and 6 .
    [0pt] [1]
    \multirow{2}{*}{}Second integer
    123456
    \multirow{6}{*}{First integer}1123456
    22264106
    336312156
    4441212
    551015
    666612
    Two fair dice are thrown and the lowest common multiple of the two scores is found.
  2. Use the table to find the probabilities of the following events.
    (A) The lowest common multiple is greater than 6 .
    (B) The lowest common multiple is a multiple of 5 .
    (C) The lowest common multiple is both greater than 6 and a multiple of 5 .
  3. Use your answers to part (ii) to show that the events "the lowest common multiple is greater than 6 " and "the lowest common multiple is a multiple of 5 " are not independent.
OCR MEI S1 Q3
6 marks Easy -1.8
  1. On die insert, complete the lable giving due lowest common multiples of all pairs of integers between 1 and 6 .
    Second integer
    \cline { 2 - 8 } \multicolumn{2}{|c|}{}123456
    \multirow{5}{*}{
    First
    integer
    }    		1123456
    \cline { 2 - 8 }22264106
    \cline { 2 - 8 }336312156
    \cline { 2 - 8 }4441212
    \cline { 2 - 8 }551015
    \cline { 2 - 8 }666612
    Two fair dice are thrown and the lowest common multiple of the two scores is found.
  2. Use the table to find the probabilities of the following events.
    (A) The lowest common multiple is greater than 6 .
    (B) The lowest common multiple is a multiple of 5 .
    (C) The lowest common multiple is both greater than 6 and a multiple of 5.
  3. Use your answers to part (ii) to show that the events "the lowest common multiple is greater than 6 " and "the lowest common multiple is a multiple of 5 " are not independent.