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Edexcel S1 2024 October Q1
Easy -1.2
  1. The back-to-back stem and leaf diagram on page 3 shows information about the running times of 31 Action films and 31 Comedy films.
    The running times are given to the nearest minute.
    1. Write down the modal running time for these Action films.
    Some of the quartiles for these two distributions are shown in the table below.
    Action filmsComedy films
    Lower quartile121\(a\)
    Median\(b\)117
    Upper quartile138\(c\)
  2. Find the value of \(a\), the value of \(b\) and the value of \(c\)
  3. For these Action films find, to one decimal place,
    1. the mean running time,
    2. the standard deviation of the running times.
      (You may use \(\sum x = 4016\) and \(\sum x ^ { 2 } = 525056\) where \(x\) is the running time, in minutes, of an Action film.) One measure of skewness is found using $$\frac { \text { mean - mode } } { \text { standard deviation } }$$
  4. Evaluate this measure and describe the skewness for the running times of these Action films.
  5. Comment on one difference between the distribution of the running times of these Action films and the distribution of the running times of these Comedy films. State the values of any statistics you have used to support your comment.
    TotalsAction filmsComedy filmsTotals
    (1)092235(5)
    (0)10356689(6)
    (5)986421102467999(8)
    (10)99876543101212466777789(11)
    (8)87775421131(1)
    (7)776643114(0)
    Key: \(0 | 9 | 2\) means 90 minutes for an Action film and 92 minutes for a Comedy film
Edexcel S1 2024 October Q2
Moderate -0.8
  1. A biologist records the length, \(y \mathrm {~cm}\), and the weight, \(w \mathrm {~kg}\), of 50 rabbits. The following summary statistics are calculated from these data.
$$\sum y = 2015 \quad \sum y ^ { 2 } = 81938.5 \quad \sum w = 125 \quad \mathrm {~S} _ { w w } = 72.25 \quad \mathrm {~S} _ { y w } = 219.55$$
    1. Show that \(\mathrm { S } _ { y y } = 734\)
    2. Calculate the product moment correlation coefficient for these data. Give your answer to 3 decimal places.
  2. Interpret your value of the product moment correlation coefficient. The biologist believes that a linear regression model may be appropriate to describe these data.
  3. State, with a reason, whether or not your value of the product moment correlation coefficient is consistent with the biologist’s belief.
  4. Find the equation of the regression line of \(w\) on \(y\), giving your answer in the form \(w = a + b y\) Jeff has a pet rabbit of length 45 cm .
  5. Use your regression equation to estimate the weight of Jeff's rabbit.
Edexcel S1 2024 October Q3
Moderate -0.8
  1. A group of 200 adults were asked whether they read cooking magazines, travel magazines or sport magazines.
    Their replies showed that
  • 29 read only cooking magazines
  • 33 read only travel magazines
  • 42 read only sport magazines
  • 17 read cooking magazines and sport magazines but not travel magazines
  • 11 read travel magazines and sport magazines but not cooking magazines
  • 22 read cooking magazines and travel magazines but not sport magazines
  • 32 do not read cooking magazines, travel magazines or sport magazines
    1. Using this information, complete the Venn diagram on page 11
One of these adults was chosen at random.
  • Find the probability that this adult,
    1. reads cooking magazines and travel magazines and sport magazines,
    2. does not read cooking magazines. Given that this adult reads travel magazines,
  • find the probability that this adult also reads sport magazines.
    \includegraphics[max width=\textwidth, alt={}]{fe416f2e-bc81-444b-a0ca-f0eae9a8b149-11_851_1086_296_493}
    •
    Edexcel S1 2024 October Q4
    Moderate -0.8
    1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
    The distances, \(m\) miles, a motorbike travels on a full tank of petrol can be modelled by a normal distribution with mean 170 miles and standard deviation 16 miles.
    1. Find the probability that, on a randomly selected journey, the motorbike could travel at least 190 miles on a full tank of petrol. The probability that, on a randomly selected journey, the motorbike could travel at least \(d\) miles on a full tank of petrol is 0.9
    2. Find the value of \(d\)
    Edexcel S1 2024 October Q5
    Moderate -0.3
    5.
    \includegraphics[max width=\textwidth, alt={}]{fe416f2e-bc81-444b-a0ca-f0eae9a8b149-16_990_1473_246_296}
    The histogram shows the number of hours worked in a given week by a group of 64 freelance photographers.
    1. Give a reason to justify the use of a histogram to represent these data. Given that 16 of these freelance photographers spent between 10 and 20 hours working in this week,
    2. estimate the number that spent between 12 and 24 hours working in this week.
    3. Find an estimate for the median time spent working in this week by these 64 freelance photographers. Charlie decides to model these data using a normal distribution. Charlie calculates an estimate of the mean to be 23.9 hours to one decimal place.
    4. Comment on Charlie's decision to use a normal distribution. Give a justification for your answer.
    Edexcel S1 2024 October Q6
    Moderate -0.3
    1. A biased die with six faces is rolled. The discrete random variable \(X\) represents the score which is uppermost. The cumulative distribution function of \(X\) is shown in the table below.
    \(x\)123456
    \(\mathrm {~F} ( x )\)0.10.2\(3 k\)\(5 k\)\(7 k\)\(10 k\)
    1. Find the value of the constant \(k\)
    2. Find the probability distribution of \(X\) A biased die with eight faces is rolled. The discrete random variable \(Y\) represents the score which is uppermost. The probability distribution of \(Y\) is shown in the table below, where \(a\) and \(b\) are constants.
      \(y\)12345678
      \(\mathrm { P } ( Y = y )\)\(a\)\(a\)\(a\)\(b\)\(b\)\(b\)0.110.05
      Given that \(\mathrm { E } ( Y ) = 4.02\)
    3. form and solve two equations in \(a\) and \(b\) to show that \(a = 0.15\) You must show your working.
      (Solutions relying on calculator technology are not acceptable.)
    4. Show that \(\mathrm { E } \left( Y ^ { 2 } \right) = 20.7\)
    5. Find \(\operatorname { Var } ( 5 - 2 Y )\) These dice are each rolled once. The scores on the two dice are independent.
    6. Find the probability that the sum of these two scores is 3
    Edexcel S1 2024 October Q7
    Moderate -0.3
    1. A box contains only red counters and black counters.
    There are \(n\) red counters and \(n + 1\) black counters.
    Two counters are selected at random, one at a time without replacement, from the box.
    1. Complete the tree diagram for this information. Give your probabilities in terms of \(n\) where necessary. \includegraphics[max width=\textwidth, alt={}, center]{fe416f2e-bc81-444b-a0ca-f0eae9a8b149-24_940_1180_591_413}
    2. Show that the probability that the two counters selected are different colours is $$\frac { n + 1 } { 2 n + 1 }$$ The probability that the two counters selected are different colours is \(\frac { 25 } { 49 }\)
    3. Find the total number of counters in the box before any counters were selected. Given that the two counters selected are different colours,
    4. find the probability that the 1st counter is black. You must show your working.
    Edexcel S1 2024 October Q8
    Standard +0.8
    1. An orchard produces apples.
    The weights, \(A\) grams, of its apples are normally distributed with mean \(\mu\) grams and standard deviation \(\sigma\) grams. It is known that $$\mathrm { P } ( A < 162 ) = 0.1 \text { and } \mathrm { P } ( 162 < A < 175 ) = 0.7508$$
    1. Calculate the value of \(\mu\) and the value of \(\sigma\) A second orchard also produces apples.
      The weights, \(B\) grams, of its apples have distribution \(B \sim N \left( 215,10 ^ { 2 } \right)\) An outlier is a value that is
      greater than \(\mathrm { Q } _ { 3 } + 1.5 \times \left( \mathrm { Q } _ { 3 } - \mathrm { Q } _ { 1 } \right)\) or smaller than \(\mathrm { Q } _ { 1 } - 1.5 \times \left( \mathrm { Q } _ { 3 } - \mathrm { Q } _ { 1 } \right)\) An apple is selected at random from this second orchard.
      Using \(\mathrm { Q } _ { 3 } = 221.74\) grams,
    2. find the probability that this apple is an outlier.
    CAIE S1 2023 March Q1
    8 marks Moderate -0.8
    Each year the total number of hours, \(x\), of sunshine in Kintoo is recorded during the month of June. The results for the last 60 years are summarised in the table.
    \(x\)\(30 \leqslant x < 60\)\(60 \leqslant x < 90\)\(90 \leqslant x < 110\)\(110 \leqslant x < 140\)\(140 \leqslant x < 180\)\(180 \leqslant x \leqslant 240\)
    Number of years48142572
    1. Draw a cumulative frequency graph to illustrate the data. [3]
    2. Use your graph to estimate the 70th percentile of the data. [2]
    3. Calculate an estimate for the mean number of hours of sunshine in Kintoo during June over the last 60 years. [3]
    CAIE S1 2023 March Q2
    7 marks Moderate -0.3
    Alisha has four coins. One of these coins is biased so that the probability of obtaining a head is 0.6. The other three coins are fair. Alisha throws the four coins at the same time. The random variable \(X\) denotes the number of heads obtained.
    1. Show that the probability of obtaining exactly one head is 0.225. [3]
    2. Complete the following probability distribution table for \(X\). [2]
      \(x\)01234
      P(\(X = x\))0.050.2250.075
    3. Given that E(\(X\)) = 2.1, find the value of Var(\(X\)). [2]
    CAIE S1 2023 March Q3
    6 marks Moderate -0.8
    80\% of the residents of Kinwawa are in favour of a leisure centre being built in the town. 20 residents of Kinwawa are chosen at random and asked, in turn, whether they are in favour of the leisure centre.
    1. Find the probability that more than 17 of these residents are in favour of the leisure centre. [3]
    2. Find the probability that the 5th person asked is the first person who is not in favour of the leisure centre. [1]
    3. Find the probability that the 7th person asked is the second person who is not in favour of the leisure centre. [2]
    CAIE S1 2023 March Q4
    3 marks Standard +0.3
    The probability that it will rain on any given day is \(x\). If it is raining, the probability that Aran wears a hat is 0.8 and if it is not raining, the probability that he wears a hat is 0.3. Whether it is raining or not, if Aran wears a hat, the probability that he wears a scarf is 0.4. If he does not wear a hat, the probability that he wears a scarf is 0.1. The probability that on a randomly chosen day it is not raining and Aran is not wearing a hat or a scarf is 0.36. Find the value of \(x\). [3]
    CAIE S1 2023 March Q5
    3 marks Standard +0.8
    Marco has four boxes labelled \(K\), \(L\), \(M\) and \(N\). He places them in a straight line in the order \(K\), \(L\), \(M\), \(N\) with \(K\) on the left. Marco also has four coloured marbles: one is red, one is green, one is white and one is yellow. He places a single marble in each box, at random. Events \(A\) and \(B\) are defined as follows. \(A\): The white marble is in either box \(L\) or box \(M\). \(B\): The red marble is to the left of both the green marble and the yellow marble. Determine whether or not events \(A\) and \(B\) are independent. [3]
    CAIE S1 2023 March Q6
    11 marks Standard +0.3
    In a cycling event the times taken to complete a course are modelled by a normal distribution with mean 62.3 minutes and standard deviation 8.4 minutes.
    1. Find the probability that a randomly chosen cyclist has a time less than 74 minutes. [2]
    2. Find the probability that 4 randomly chosen cyclists all have times between 50 and 74 minutes. [4]
    In a different cycling event, the times can also be modelled by a normal distribution. 23\% of the cyclists have times less than 36 minutes and 10\% of the cyclists have times greater than 54 minutes.
    1. Find estimates for the mean and standard deviation of this distribution. [5]
    CAIE S1 2023 March Q7
    12 marks Standard +0.3
    1. Find the number of different arrangements of the 9 letters in the word DELIVERED in which the three Es are together and the two Ds are not next to each other. [4]
    2. Find the probability that a randomly chosen arrangement of the 9 letters in the word DELIVERED has exactly 4 letters between the two Ds. [5]
    Five letters are selected from the 9 letters in the word DELIVERED.
    1. Find the number of different selections if the 5 letters include at least one D and at least one E. [3]
    CAIE S1 2002 June Q1
    4 marks Easy -1.2
    Events \(A\) and \(B\) are such that \(\text{P}(A) = 0.3\), \(\text{P}(B) = 0.8\) and \(\text{P}(A \text{ and } B) = 0.4\). State, giving a reason in each case, whether events \(A\) and \(B\) are
    1. independent, [2]
    2. mutually exclusive. [2]
    CAIE S1 2002 June Q2
    6 marks Easy -1.2
    The manager of a company noted the times spent in 80 meetings. The results were as follows.
    Time (\(t\) minutes)\(0 < t \leq 15\)\(15 < t \leq 30\)\(30 < t \leq 60\)\(60 < t \leq 90\)\(90 < t \leq 120\)
    Number of meetings4724387
    Draw a cumulative frequency graph and use this to estimate the median time and the interquartile range. [6]
    CAIE S1 2002 June Q3
    7 marks Moderate -0.8
    A fair cubical die with faces numbered 1, 1, 1, 2, 3, 4 is thrown and the score noted. The area \(A\) of a square of side equal to the score is calculated, so, for example, when the score on the die is 3, the value of \(A\) is 9.
    1. Draw up a table to show the probability distribution of \(A\). [3]
    2. Find \(\text{E}(A)\) and \(\text{Var}(A)\). [4]
    CAIE S1 2002 June Q4
    7 marks Moderate -0.8
    1. In a spot check of the speeds \(x \text{ km h}^{-1}\) of 30 cars on a motorway, the data were summarised by \(\Sigma(x - 110) = -47.2\) and \(\Sigma(x - 110)^2 = 5460\). Calculate the mean and standard deviation of these speeds. [4]
    2. On another day the mean speed of cars on the motorway was found to be \(107.6 \text{ km h}^{-1}\) and the standard deviation was \(13.8 \text{ km h}^{-1}\). Assuming these speeds follow a normal distribution and that the speed limit is \(110 \text{ km h}^{-1}\), find what proportion of cars exceed the speed limit. [3]
    CAIE S1 2002 June Q5
    8 marks Moderate -0.3
    The digits of the number 1223678 can be rearranged to give many different 7-digit numbers. Find how many different 7-digit numbers can be made if
    1. there are no restrictions on the order of the digits, [2]
    2. the digits 1, 3, 7 (in any order) are next to each other, [3]
    3. these 7-digit numbers are even. [3]
    CAIE S1 2002 June Q6
    8 marks Standard +0.3
    1. In a normal distribution with mean \(\mu\) and standard deviation \(\sigma\), \(\text{P}(X > 3.6) = 0.5\) and \(\text{P}(X > 2.8) = 0.6554\). Write down the value of \(\mu\), and calculate the value of \(\sigma\). [4]
    2. If four observations are taken at random from this distribution, find the probability that at least two observations are greater than 2.8. [4]
    CAIE S1 2002 June Q7
    10 marks Moderate -0.3
    1. A garden shop sells polyanthus plants in boxes, each box containing the same number of plants. The number of plants per box which produce yellow flowers has a binomial distribution with mean 11 and variance 4.95.
      1. Find the number of plants per box. [4]
      2. Find the probability that a box contains exactly 12 plants which produce yellow flowers. [2]
    2. Another garden shop sells polyanthus plants in boxes of 100. The shop's advertisement states that the probability of any polyanthus plant producing a pink flower is 0.3. Use a suitable approximation to find the probability that a box contains fewer than 35 plants which produce pink flowers. [4]
    CAIE S1 2010 June Q1
    5 marks Moderate -0.8
    The times in minutes for seven students to become proficient at a new computer game were measured. The results are shown below. $$15 \quad 10 \quad 48 \quad 10 \quad 19 \quad 14 \quad 16$$
    1. Find the mean and standard deviation of these times. [2]
    2. State which of the mean, median or mode you consider would be most appropriate to use as a measure of central tendency to represent the data in this case. [1]
    3. For each of the two measures of average you did not choose in part (ii), give a reason why you consider it inappropriate. [2]
    CAIE S1 2010 June Q2
    5 marks Moderate -0.8
    The lengths of new pencils are normally distributed with mean 11 cm and standard deviation 0.095 cm.
    1. Find the probability that a pencil chosen at random has a length greater than 10.9 cm. [2]
    2. Find the probability that, in a random sample of 6 pencils, at least two have lengths less than 10.9 cm. [3]
    CAIE S1 2010 June Q3
    6 marks Moderate -0.8
    \includegraphics{figure_3} The birth weights of random samples of 900 babies born in country \(A\) and 900 babies born in country \(B\) are illustrated in the cumulative frequency graphs. Use suitable data from these graphs to compare the central tendency and spread of the birth weights of the two sets of babies. [6]