Questions S1 (1967 questions)

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OCR S1 2013 June Q9
8 marks Standard +0.3
9 A game is played with a token on a board with a grid printed on it. The token starts at the point \(( 0,0 )\) and moves in steps. Each step is either 1 unit in the positive \(x\)-direction with probability 0.8 , or 1 unit in the positive \(y\)-direction with probability 0.2 . The token stops when it reaches a point with a \(y\)-coordinate of 1 . It is given that the token stops at \(( X , 1 )\).
  1. (a) Find the probability that \(X = 10\).
    (b) Find the probability that \(X < 10\).
  2. Find the expected number of steps taken by the token.
  3. Hence, write down the value of \(\mathrm { E } ( X )\).
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.
Edexcel S1 2016 June Q1
12 marks Moderate -0.8
  1. The percentage oil content, \(p\), and the weight, \(w\) milligrams, of each of 10 randomly selected sunflower seeds were recorded. These data are summarised below.
$$\sum w ^ { 2 } = 41252 \quad \sum w p = 27557.8 \quad \sum w = 640 \quad \sum p = 431 \quad \mathrm {~S} _ { p p } = 2.72$$
  1. Find the value of \(\mathrm { S } _ { w w }\) and the value of \(\mathrm { S } _ { w p }\)
  2. Calculate the product moment correlation coefficient between \(p\) and \(w\)
  3. Give an interpretation of your product moment correlation coefficient. The equation of the regression line of \(p\) on \(w\) is given in the form \(p = a + b w\)
  4. Find the equation of the regression line of \(p\) on \(w\)
  5. Hence estimate the percentage oil content of a sunflower seed which weighs 60 milligrams.
Edexcel S1 2016 June Q2
3 marks Easy -1.2
2. The time taken to complete a puzzle, in minutes, is recorded for each person in a club. The times are summarised in a grouped frequency distribution and represented by a histogram. One of the class intervals has a frequency of 20 and is shown by a bar of width 1.5 cm and height 12 cm on the histogram. The total area under the histogram is \(94.5 \mathrm {~cm} ^ { 2 }\) Find the number of people in the club.
(3)
VILM SIMI NI JIIIM I ON OC
VILV SIHI NI JAHM ION OC
VJ4V SIHI NI JIIYM ION OC
Edexcel S1 2016 June Q3
14 marks Easy -1.2
3. The discrete random variable \(X\) has probability distribution $$\mathrm { P } ( X = x ) = \frac { 1 } { 5 } \quad x = 1,2,3,4,5$$
  1. Write down the name given to this distribution. Find
  2. \(\mathrm { P } ( X = 4 )\)
  3. \(\mathrm { F } ( 3 )\)
  4. \(\mathrm { P } ( 3 X - 3 > X + 4 )\)
  5. Write down \(\mathrm { E } ( X )\)
  6. Find \(\mathrm { E } \left( X ^ { 2 } \right)\)
  7. Hence find \(\operatorname { Var } ( X )\) Given that \(\mathrm { E } ( a X - 3 ) = 11.4\)
  8. find \(\operatorname { Var } ( a X - 3 )\)
Edexcel S1 2016 June Q4
12 marks Moderate -0.8
4. A researcher recorded the time, \(t\) minutes, spent using a mobile phone during a particular afternoon, for each child in a club. The researcher coded the data using \(v = \frac { t - 5 } { 10 }\) and the results are summarised in the table below.
Coded Time (v)Frequency ( \(\boldsymbol { f }\) )Coded Time Midpoint (m)
\(0 \leqslant v < 5\)202.5
\(5 \leqslant v < 10\)24\(a\)
\(10 \leqslant v < 15\)1612.5
\(15 \leqslant v < 20\)1417.5
\(20 \leqslant v < 30\)6\(b\)
$$\text { (You may use } \sum f m = 825 \text { and } \sum f m ^ { 2 } = 12012.5 \text { ) }$$
  1. Write down the value of \(a\) and the value of \(b\).
  2. Calculate an estimate of the mean of \(v\).
  3. Calculate an estimate of the standard deviation of \(v\).
  4. Use linear interpolation to estimate the median of \(v\).
  5. Hence describe the skewness of the distribution. Give a reason for your answer.
  6. Calculate estimates of the mean and the standard deviation of the time spent using a mobile phone during the afternoon by the children in this club.
Edexcel S1 2016 June Q5
8 marks Moderate -0.3
5. A biased tetrahedral die has faces numbered \(0,1,2\) and 3 . The die is rolled and the number face down on the die, \(X\), is recorded. The probability distribution of \(X\) is
\(x\)0123
\(\mathrm { P } ( X = x )\)\(\frac { 1 } { 6 }\)\(\frac { 1 } { 6 }\)\(\frac { 1 } { 6 }\)\(\frac { 1 } { 2 }\)
If \(X = 3\) then the final score is 3
If \(X \neq 3\) then the die is rolled again and the final score is the sum of the two numbers. The random variable \(T\) is the final score.
  1. Find \(\mathrm { P } ( T = 2 )\)
  2. Find \(\mathrm { P } ( T = 3 )\)
  3. Given that the die is rolled twice, find the probability that the final score is 3
Edexcel S1 2016 June Q6
11 marks Moderate -0.3
6. Three events \(A , B\) and \(C\) are such that $$\mathrm { P } ( A ) = \frac { 2 } { 5 } \quad \mathrm { P } ( C ) = \frac { 1 } { 2 } \quad \mathrm { P } ( A \cup B ) = \frac { 5 } { 8 }$$ Given that \(A\) and \(C\) are mutually exclusive find
  1. \(\mathrm { P } ( A \cup C )\) Given that \(A\) and \(B\) are independent
  2. show that \(\mathrm { P } ( B ) = \frac { 3 } { 8 }\)
  3. Find \(\mathrm { P } ( A \mid B )\) Given that \(\mathrm { P } \left( C ^ { \prime } \cap B ^ { \prime } \right) = 0.3\)
  4. draw a Venn diagram to represent the events \(A , B\) and \(C\)
Edexcel S1 2016 June Q7
15 marks Standard +0.3
7. A machine fills bottles with water. The volume of water delivered by the machine to a bottle is \(X \mathrm { ml }\) where \(X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\) One of these bottles of water is selected at random. Given that \(\mu = 503\) and \(\sigma = 1.6\)
  1. find
    1. \(\mathrm { P } ( X > 505 )\)
    2. \(\mathrm { P } ( 501 < X < 505 )\)
  2. Find \(w\) such that \(\mathrm { P } ( 1006 - w < X < w ) = 0.9426\) Following adjustments to the machine, the volume of water delivered by the machine to a bottle is such that \(\mu = 503\) and \(\sigma = q\) Given that \(\mathrm { P } ( X < r ) = 0.01\) and \(\mathrm { P } ( X > r + 6 ) = 0.05\)
  3. find the value of \(r\) and the value of \(q\)
Edexcel S1 2018 June Q1
13 marks Moderate -0.3
  1. A random sample of 10 cars of different makes and sizes is taken and the published miles per gallon, \(p\), and the actual miles per gallon, \(m\), are recorded. The data are coded using variables \(x = \frac { p } { 10 }\) and \(y = m - 25\)
The results for the coded data are summarised below.
\(\boldsymbol { x }\)6.893.675.925.044.873.924.715.143.655.23
\(\boldsymbol { y }\)30322151381513.5319
(You may use \(\sum y ^ { 2 } = 2628.25 \quad \sum x y = 768.58 \quad \mathrm {~S} _ { x x } = 9.25924 \quad \mathrm {~S} _ { x y } = 74.664\) )
  1. Show that \(\mathrm { S } _ { y y } = 626.025\)
  2. Find the product moment correlation coefficient between \(x\) and \(y\).
  3. Give a reason to support fitting a regression model of the form \(y = a + b x\) to these data.
  4. Find the equation of the regression line of \(y\) on \(x\), giving your answer in the form \(y = a + b x\).
    Give the value of \(a\) and the value of \(b\) to 3 significant figures. A car's published miles per gallon is 44
  5. Estimate the actual miles per gallon for this particular car.
  6. Comment on the reliability of your estimate in part (e). Give a reason for your answer.