Questions — Edexcel S1 (606 questions)

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Edexcel S1 Q7
Easy -1.8
7. In a school there are 148 students in Years 12 and 13 studying Science, Humanities or Arts subjects. Of these students, 89 wear glasses and the others do not. There are 30 Science students of whom 18 wear glasses. The corresponding figures for the Humanities students are 68 and 44 respectively. A student is chosen at random. Find the probability that this student
  1. is studying Arts subjects,
  2. does not wear glasses, given that the student is studying Arts subjects. Amongst the Science students, \(80 \%\) are right-handed. Corresponding percentages for Humanities and Arts students are 75\% and 70\% respectively. A student is again chosen at random.
  3. Find the probability that this student is right-handed.
  4. Given that this student is right-handed, find the probability that the student is studying Science subjects.
    1. (a) Describe the main features and uses of a box plot.
    Children from schools \(A\) and \(B\) took part in a fun run for charity. The times, to the nearest minute, taken by the children from school \(A\) are summarised in Figure 1. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{3d4f7bfb-b235-418a-9411-a4d0b3188254-015_398_1045_946_461}
    \end{figure}
    1. Write down the time by which \(75 \%\) of the children in school \(A\) had completed the run.
    2. State the name given to this value.
  6. Explain what you understand by the two crosses ( X ) on Figure 1.
Edexcel S1 Q8
Moderate -0.8
8. The lifetimes of bulbs used in a lamp are normally distributed. A company \(X\) sells bulbs with a mean lifetime of 850 hours and a standard deviation of 50 hours.
  1. Find the probability of a bulb, from company \(X\), having a lifetime of less than 830 hours.
  2. In a box of 500 bulbs, from company \(X\), find the expected number having a lifetime of less than 830 hours. A rival company \(Y\) sells bulbs with a mean lifetime of 860 hours and \(20 \%\) of these bulbs have a lifetime of less than 818 hours.
  3. Find the standard deviation of the lifetimes of bulbs from company \(Y\). Both companies sell the bulbs for the same price.
  4. State which company you would recommend. Give reasons for your answer.
    \begin{table}[h]
    \captionsetup{labelformat=empty} \caption{}
    \end{table}
Edexcel S1 2003 June Q1
5 marks Easy -1.8
  1. In a particular week, a dentist treats 100 patients. The length of time, to the nearest minute, for each patient's treatment is summarised in the table below.
Time
(minutes)
\(4 - 7\)8\(9 - 10\)11\(12 - 16\)\(17 - 20\)
Number
of
patients
122018221513
Draw a histogram to illustrate these data.
Edexcel S1 2003 June Q2
6 marks Moderate -0.5
2. The lifetimes of batteries used for a computer game have a mean of 12 hours and a standard deviation of 3 hours. Battery lifetimes may be assumed to be normally distributed. Find the lifetime, \(t\) hours, of a battery such that 1 battery in 5 will have a lifetime longer than \(t\).
Edexcel S1 2003 June Q3
10 marks Moderate -0.8
3. A company owns two petrol stations \(P\) and \(Q\) along a main road. Total daily sales in the same week for \(P ( \pounds p )\) and for \(Q ( \pounds q )\) are summarised in the table below.
\(p\)\(q\)
Monday47605380
Tuesday53954460
Wednesday58404640
Thursday46505450
Friday53654340
Saturday49905550
Sunday43655840
When these data are coded using \(x = \frac { p - 4365 } { 100 }\) and \(y = \frac { q - 4340 } { 100 }\), $$\Sigma x = 48.1 , \Sigma y = 52.8 , \Sigma x ^ { 2 } = 486.44 , \Sigma y ^ { 2 } = 613.22 \text { and } \Sigma x y = 204.95 .$$
  1. Calculate \(S _ { x y } , S _ { x x }\) and \(S _ { y y }\).
  2. Calculate, to 3 significant figures, the value of the product moment correlation coefficient between \(x\) and \(y\).
    1. Write down the value of the product moment correlation coefficient between \(p\) and \(q\).
    2. Give an interpretation of this value.
Edexcel S1 2003 June Q4
11 marks Moderate -0.3
4. The discrete random variable \(X\) has probability function $$\mathrm { P } ( X = x ) = \begin{array} { l l } k \left( x ^ { 2 } - 9 \right) , & x = 4,5,6 \\ 0 , & \text { otherwise } \end{array}$$ where \(k\) is a positive constant.
  1. Show that \(k = \frac { 1 } { 50 }\).
  2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
  3. Find \(\operatorname { Var } ( 2 X - 3 )\).
Edexcel S1 2003 June Q5
12 marks Easy -1.2
5. The random variable \(X\) represents the number on the uppermost face when a fair die is thrown.
  1. Write down the name of the probability distribution of \(X\).
  2. Calculate the mean and the variance of \(X\). Three fair dice are thrown and the numbers on the uppermost faces are recorded.
  3. Find the probability that all three numbers are 6 .
  4. Write down all the different ways of scoring a total of 16 when the three numbers are added together.
  5. Find the probability of scoring a total of 16 .
Edexcel S1 2003 June Q6
16 marks Moderate -0.8
6. The number of bags of potato crisps sold per day in a bar was recorded over a two-week period. The results are shown below. $$20,15,10,30,33,40,5,11,13,20,25,42,31,17$$
  1. Calculate the mean of these data.
  2. Draw a stem and leaf diagram to represent these data.
  3. Find the median and the quartiles of these data. An outlier is an observation that falls either \(1.5 \times\) (interquartile range) above the upper quartile or \(1.5 \times\) (interquartile range) below the lower quartile.
  4. Determine whether or not any items of data are outliers.
  5. On graph paper draw a box plot to represent these data. Show your scale clearly.
  6. Comment on the skewness of the distribution of bags of crisps sold per day. Justify your answer.
Edexcel S1 2003 June Q7
16 marks Moderate -0.8
  1. Eight students took tests in mathematics and physics. The marks for each student are given in the table below where \(m\) represents the mathematics mark and \(p\) the physics mark.
\multirow{2}{*}{}Student
\(A\)B\(C\)D\(E\)\(F\)G\(H\)
\multirow{2}{*}{Mark}\(m\)9141310782017
\(p\)1123211519103126
A science teacher believes that students' marks in physics depend upon their mathematical ability. The teacher decides to investigate this relationship using the test marks.
  1. Write down which is the explanatory variable in this investigation.
  2. Draw a scatter diagram to illustrate these data.
  3. Showing your working, find the equation of the regression line of \(p\) on \(m\).
  4. Draw the regression line on your scatter diagram. A ninth student was absent for the physics test, but she sat the mathematics test and scored 15 .
  5. Using this model, estimate the mark she would have scored in the physics test.
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.
    Edexcel S1 2023 June Q1
    8 marks Moderate -0.8
    The histogram shows the distances, in km, that 274 people travel to work. \includegraphics{figure_1} Given that 60 of these people travel between 10km and 20km to work, estimate
    1. the number of people who travel between 22km and 45km to work, [3]
    2. the median distance travelled to work by these 274 people, [2]
    3. the mean distance travelled to work by these 274 people. [3]
    Edexcel S1 2023 June Q2
    13 marks Moderate -0.3
    Two students, Olive and Shan, collect data on the weight, \(w\) grams, and the tail length, \(t\) cm, of 15 mice. Olive summarised the data as follows \(S_tt = 5.3173\) \quad \(\sum w^2 = 6089.12\) \quad \(\sum tw = 2304.53\) \quad \(\sum w = 297.8\) \quad \(\sum t = 114.8\)
    1. Calculate the value of \(S_{ww}\) and the value of \(S_{tw}\) [3]
    2. Calculate the value of the product moment correlation coefficient between \(w\) and \(t\) [2]
    3. Show that the equation of the regression line of \(w\) on \(t\) can be written as $$w = -16.7 + 4.77t$$ [3]
    4. Give an interpretation of the gradient of the regression line. [1]
    5. Explain why it would not be appropriate to use the regression line in part (c) to estimate the weight of a mouse with a tail length of 2cm. [2]
    Shan decided to code the data using \(x = t - 6\) and \(y = \frac{w}{2} - 5\)
    1. Write down the value of the product moment correlation coefficient between \(x\) and \(y\) [1]
    2. Write down an equation of the regression line of \(y\) on \(x\) You do not need to simplify your equation. [1]
    Edexcel S1 2023 June Q3
    9 marks Moderate -0.8
    Jim records the length, \(l\) mm, of 81 salmon. The data are coded using \(x = l - 600\) and the following summary statistics are obtained. $$n = 81 \quad \sum x = 3711 \quad \sum x^2 = 475181$$
    1. Find the mean length of these salmon. [3]
    2. Find the variance of the lengths of these salmon. [2]
    The weight, \(w\) grams, of each of the 81 salmon is recorded to the nearest gram. The recorded results for the 81 salmon are summarised in the box plot below. \includegraphics{figure_2}
    1. Find the maximum number of salmon that have weights in the interval $$4600 < w \leqslant 7700$$ [1]
    Raj says that the box plot is incorrect as Jim has not included outliers. For these data an outlier is defined as a value that is more than \(1.5 \times\) IQR above the upper quartile \quad or \quad \(1.5 \times\) IQR below the lower quartile
    1. Show that there are no outliers. [3]
    Edexcel S1 2023 June Q4
    9 marks Moderate -0.8
    A bag contains a large number of coloured counters. Each counter is labelled A, B or C 30% of the counters are labelled A 45% of the counters are labelled B The rest of the counters are labelled C It is known that 2% of the counters labelled A are red 4% of the counters labelled B are red 6% of the counters labelled C are red One counter is selected at random from the bag.
    1. Complete the tree diagram on the opposite page to illustrate this information. [2]
    2. Calculate the probability that the counter is labelled A and is not red. [2]
    3. Calculate the probability that the counter is red. [2]
    4. Given that the counter is red, find the probability that it is labelled C [3]
    \includegraphics{figure_3}
    Edexcel S1 2023 June Q5
    13 marks Standard +0.3
    A discrete random variable \(Y\) has probability function $$\mathrm{P}(Y = y) = \begin{cases} k(3 - y) & y = 1, 2 \\ k(y^2 - 8) & y = 3, 4, 5 \\ k & y = 6 \\ 0 & \text{otherwise} \end{cases}$$ where \(k\) is a constant.
    1. Show that \(k = \frac{1}{30}\) [2]
    Find the exact value of
    1. P\((1 < Y \leqslant 4)\) [2]
    2. E\((Y)\) [2]
    The random variable \(X = 15 - 2Y\)
    1. Calculate P\((Y \geqslant X)\) [3]
    2. Calculate Var\((X)\) [4]
    Edexcel S1 2023 June Q6
    9 marks Moderate -0.3
    Three events \(A\), \(B\) and \(C\) are such that $$\mathrm{P}(A) = 0.1 \quad \mathrm{P}(B|A) = 0.3 \quad \mathrm{P}(A \cup B) = 0.25 \quad \mathrm{P}(C) = 0.5$$ Given that \(A\) and \(C\) are mutually exclusive
    1. find P\((A \cup C)\) [1]
    2. Show that P\((B) = 0.18\) [3]
    Given also that \(B\) and \(C\) are independent,
    1. draw a Venn diagram to represent the events \(A\), \(B\) and \(C\) and the probabilities associated with each region. [5]
    Edexcel S1 2023 June Q7
    14 marks Standard +0.3
    A machine squeezes apples to extract their juice. The volume of juice, \(J\) ml, extracted from 1 kg of apples is modelled by a normal distribution with mean \(\mu\) and standard deviation \(\sigma\) Given that \(\mu = 500\) and \(\sigma = 25\) use standardisation to
      1. show that P\((J > 510) = 0.3446\) [2]
      2. calculate the value of \(d\) such that P\((J > d) = 0.9192\) [3]
    Zen randomly selects 5 bags each containing 1 kg of apples and records the volume of juice extracted from each bag of apples.
    1. Calculate the probability that each of the 5 bags of apples produce less than 510ml of juice. [2]
    Following adjustments to the machine, the volume of juice, \(R\) ml, extracted from 1 kg of apples is such that \(\mu = 520\) and \(\sigma = k\) Given that P\((R < r) = 0.15\) and P\((R > 3r - 800) = 0.005\)
    1. find the value of \(r\) and the value of \(k\) [7]
    Edexcel S1 2002 January Q1
    4 marks Easy -1.8
    1. Explain briefly what you understand by
      1. a statistical experiment, [1]
      2. an event. [1]
    2. State one advantage and one disadvantage of a statistical model. [2]