Questions S1 (2020 questions)

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Edexcel S1 2022 October Q3
10 marks Moderate -0.5
  1. Morgan is investigating the body length, \(b\) centimetres, of squirrels.
A random sample of 8 squirrels is taken and the data for each squirrel is coded using $$x = \frac { b - 21 } { 2 }$$ The results for the coded data are summarised below $$\sum x = - 1.2 \quad \sum x ^ { 2 } = 5.1$$
  1. Find the mean of \(b\)
  2. Find the standard deviation of \(b\) A 9th squirrel is added to the sample. Given that for all 9 squirrels \(\sum x = 0\)
  3. find
    1. the body length of the 9th squirrel,
    2. the standard deviation of \(x\) for all 9 squirrels.
Edexcel S1 2022 October Q4
4 marks Moderate -0.8
  1. The cumulative distribution function of the discrete random variable \(W\), which takes only the values 6,7 and 8 , is given by
$$F ( W ) = \frac { ( w + 3 ) ( w - 1 ) } { 77 } \text { for } w = 6,7,8$$ Find \(\mathrm { E } ( W )\)
VIAV SIHI NI III IM IONOOCVIIIV SIHI NI III IM I I N O OVI4V SIHI NI III IM I ON OC
Edexcel S1 2022 October Q5
12 marks Standard +0.3
  1. The weights, \(W\) grams, of kiwi fruit grown on a farm are normally distributed with mean 80 grams and standard deviation 8 grams.
The table shows the classifications of the kiwi fruit by their weight, where \(k\) is a positive constant.
SmallLarge
TinyPetiteExtraJumboMega
\(w < 66\)\(66 \leqslant w < 70\)\(70 \leqslant w < 80\)\(80 \leqslant w < k\)\(w \geqslant k\)
One kiwi fruit is selected at random from those grown on the farm.
  1. Find the probability that this kiwi fruit is Large. 35\% of the kiwi fruit are Jumbo.
  2. Find the value of \(k\) to one decimal place. 75\% of Tiny kiwi fruit weigh more than \(y\) grams.
  3. Find the value of \(y\) giving your answer to one decimal place.
Edexcel S1 2022 October Q6
11 marks Standard +0.3
The Venn diagram shows the events \(A , B , C\) and \(D\), where \(p , q , r\) and \(s\) are probabilities. \includegraphics[max width=\textwidth, alt={}, center]{1fda59cb-059e-4850-810f-cc3e69bc058e-20_504_826_296_621}
  1. Write down the value of
    1. \(\mathrm { P } ( A )\)
    2. \(\mathrm { P } ( A \mid B )\)
    3. \(\mathrm { P } ( A \mid C )\) Given that \(\mathrm { P } \left( B ^ { \prime } \cap D ^ { \prime } \right) = \frac { 7 } { 10 }\) and \(\mathrm { P } ( C \mid D ) = \frac { 3 } { 5 }\)
  2. find the exact value of \(q\) and the exact value of \(r\) Given also that \(\mathrm { P } \left( B \cup C ^ { \prime } \right) = \frac { 5 } { 8 }\)
  3. find the exact value of \(s\)
Edexcel S1 2022 October Q7
14 marks Standard +0.3
  1. Adana selects one number at random from the distribution of \(X\) which has the following probability distribution.
\(x\)0510
\(\mathrm { P } ( X = x )\)0.10.20.7
  1. Given that the number selected by Adana is not 5 , write down the probability it is 0
  2. Show that \(\mathrm { E } \left( X ^ { 2 } \right) = 75\)
  3. Find \(\operatorname { Var } ( X )\)
  4. Find \(\operatorname { Var } ( 4 - 3 X )\) Bruno and Charlie each independently select one number at random from the distribution of \(X\)
  5. Find the probability that the number Bruno selects is greater than the number Charlie selects. Devika multiplies Bruno's number by Charlie's number to obtain a product, \(D\)
  6. Determine the probability distribution of \(D\)
Edexcel S1 2023 October Q1
11 marks Moderate -0.8
  1. Sally plays a game in which she can either win or lose.
A turn consists of up to 3 games. On each turn Sally plays the game up to 3 times. If she wins the first 2 games or loses the first 2 games, then she will not play the 3rd game.
  • The probability that Sally wins the first game in a turn is 0.7
  • If Sally wins a game the probability that she wins the next game is 0.6
  • If Sally loses a game the probability that she wins the next game is 0.2
    1. Use this information to complete the tree diagram on page 3
    2. Find the probability that Sally wins the first 2 games in a turn.
    3. Find the probability that Sally wins exactly 2 games in a turn.
Given that Sally wins 2 games in a turn,
  • find the probability that she won the first 2 games. Given that Sally won the first game in a turn,
  • find the probability that she won 2 games. 1st game 2nd game win
    •
    Edexcel S1 2023 October Q2
    13 marks Easy -1.2
    1. The weights, to the nearest kilogram, of a sample of 33 red kangaroos taken in December are summarised in the stem and leaf diagram below.
    Weight (kg)Totals
    16(1)
    236(2)
    3246(3)
    42556678(7)
    534777899(8)
    6022338(7)
    728(2)
    826(2)
    94(1)
    Key: 3 | 2 represents 32 kg
    1. Find
      1. the value of the median
      2. the value of \(Q _ { 1 }\) and the value of \(Q _ { 3 }\) for the weights of these red kangaroos. For these data an outlier is defined as a value that is
        greater than \(Q _ { 3 } + 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)\) or smaller than \(Q _ { 1 } - 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)\)
    2. Show that there are 2 outliers for these data. Figure 1 on page 7 shows a box plot for the weights of the same 33 red kangaroos taken in February, earlier in the year.
    3. In the space on Figure 1, draw a box plot to represent the weights of these red kangaroos in December.
    4. Compare the distribution of the weights of red kangaroos taken in February with the distribution of the weights of red kangaroos taken in December of the same year. You should interpret your comparisons in the context of the question.
      \includegraphics[max width=\textwidth, alt={}]{f94b29e0-081f-45e8-99a7-ac835eec91e5-07_2267_51_307_36}
      \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{f94b29e0-081f-45e8-99a7-ac835eec91e5-07_766_1803_1777_132} \captionsetup{labelformat=empty} \caption{Figure 1}
      \end{figure} Turn over for a spare grid if you need to redraw your box plot. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{f94b29e0-081f-45e8-99a7-ac835eec91e5-09_901_1833_1653_114} \captionsetup{labelformat=empty} \caption{Figure 1}
      \end{figure} \begin{verbatim} (Total for Question 2 is 13 marks) \end{verbatim}
    Edexcel S1 2023 October Q3
    12 marks Easy -1.2
    1. Bob shops at a market each week. The event that Bob buys carrots is denoted by \(C\) Bob buys onions is denoted by \(O\) At each visit, Bob may buy neither, or one, or both of these items. The probability that Bob buys carrots is 0.65
      Bob does not buy onions is 0.3
      Bob buys onions but not carrots is 0.15
      The Venn diagram below represents the events \(C\) and \(O\)
      [diagram]

      where \(w , x , y\) and \(z\) are probabilities.
      1. Find the value of \(w\), the value of \(x\), the value of \(y\) and the value of \(z\) For one visit to the market,
      2. find the probability that Bob buys either carrots or onions but not both.
      3. Show that the events \(C\) and \(O\) are not independent.
      4. \(F , G\) and \(H\) are 3 events. \(F\) and \(H\) are mutually exclusive. \(F\) and \(G\) are independent. Given that $$\mathrm { P } ( F ) = \frac { 2 } { 7 } \quad \mathrm { P } ( H ) = \frac { 1 } { 4 } \quad \mathrm { P } ( F \cup G ) = \frac { 5 } { 8 }$$
      (a) find \(\operatorname { P } ( F \cup H )\)
    2. find \(\mathrm { P } ( G )\)
    3. find \(\operatorname { P } ( F \cap G )\)
    Edexcel S1 2023 October Q4
    12 marks Moderate -0.3
    1. The discrete random variable \(X\) has the following probability distribution.
    \(x\)1234
    \(\mathrm { P } ( X = x )\)\(\frac { 1 } { 10 }\)\(\frac { 1 } { 5 }\)\(\frac { 3 } { 10 }\)\(\frac { 2 } { 5 }\)
    1. Show that \(\mathrm { E } \left( \frac { 1 } { X } \right) = \frac { 2 } { 5 }\)
    2. Find \(\operatorname { Var } \left( \frac { 1 } { X } \right)\) The random variable \(Y = \frac { 30 } { X }\)
    3. Find
      1. \(\mathrm { E } ( Y )\)
      2. \(\operatorname { Var } ( Y )\)
    4. Find \(\mathrm { P } ( X < 3 \mid Y < 20 )\)
    Edexcel S1 2023 October Q5
    15 marks Standard +0.3
    1. The weights, \(X\) grams, of a particular variety of fruit are normally distributed with
    $$X \sim \mathrm {~N} \left( 210,25 ^ { 2 } \right)$$ A fruit of this variety is selected at random.
    1. Show that the probability that the weight of this fruit is less than 240 grams is 0.8849
    2. Find the probability that the weight of this fruit is between 190 grams and 240 grams.
    3. Find the value of \(k\) such that \(\mathrm { P } ( 210 - k < X < 210 + k ) = 0.95\) A wholesaler buys large numbers of this variety of fruit and classifies the lightest \(15 \%\) as small.
    4. Find the maximum weight of a fruit that is classified as small. You must show your working clearly. The weights, \(Y\) grams, of a second variety of this fruit are normally distributed with $$Y \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)$$ Given that \(5 \%\) of these fruit weigh less than 152 grams and \(40 \%\) weigh more than 180 grams,
    5. calculate the mean and standard deviation of the weights of this variety of fruit.
    Edexcel S1 2023 October Q6
    12 marks Moderate -0.3
    1. The variables \(x\) and \(y\) have the following regression equations based on the same 12 observations.
    \cline { 2 - 2 } \multicolumn{1}{c|}{}Regression equation
    \(y\) on \(x\)\(y = 1.4 x + 1.5\)
    \(x\) on \(y\)\(x = 1.2 + 0.2 y\)
      1. Find the point of intersection of these lines.
      2. Hence show that \(\sum x = 25\) Given that $$\sum x y = \frac { 6961 } { 60 }$$
      (b) Find \(S _ { x y }\)
    2. Find the product moment correlation coefficient between \(x\) and \(y\)
    Edexcel S1 2018 Specimen 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. \(\_\_\_\_\) VAYV SIHI NI JIIIM ION OC
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    Edexcel S1 2018 Specimen Q2
    3 marks Moderate -0.8
    1. 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.
    Edexcel S1 2018 Specimen 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 \(\operatorname { Var } ( a X - 3 )\) \includegraphics[max width=\textwidth, alt={}, center]{b7500cc1-caa6-4767-bb2e-e3d70474e805-09_2261_54_312_34} \(\_\_\_\_\) VAYV SIHI NI JIIIM ION OC
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    Edexcel S1 2018 Specimen Q4
    12 marks Moderate -0.8
    1. 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. \(\_\_\_\_\) VAYV SIHI NI JIIIM ION OC
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    Edexcel S1 2018 Specimen Q5
    8 marks Moderate -0.3
    1. 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 \(\_\_\_\_\) VAYV SIHI NI JIIIM ION OC
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    Edexcel S1 2018 Specimen 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 \(C\) \includegraphics[max width=\textwidth, alt={}, center]{b7500cc1-caa6-4767-bb2e-e3d70474e805-21_2260_53_312_33} \(\_\_\_\_\) VAYV SIHI NI JIIIM ION OC
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    Edexcel S1 2018 Specimen Q7
    15 marks Moderate -0.3
    1. 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 \(q\) \(\_\_\_\_\) VAYV SIHI NI JIIIM ION OC
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    Edexcel S1 Specimen Q1
    4 marks Easy -1.2
    1. Gary compared the total attendance, \(x\), at home matches and the total number of goals, \(y\), scored at home during a season for each of 12 football teams playing in a league. He correctly calculated:
    $$S _ { x x } = 1022500 \quad S _ { y y } = 130.9 \quad S _ { x y } = 8825$$
    1. Calculate the product moment correlation coefficient for these data.
    2. Interpret the value of the correlation coefficient. Helen was given the same data to analyse. In view of the large numbers involved she decided to divide the attendance figures by 100 . She then calculated the product moment correlation coefficient between \(\frac { x } { 100 }\) and \(y\).
    3. Write down the value Helen should have obtained.
    Edexcel S1 Specimen Q2
    10 marks Easy -1.2
    2. An experiment consists of selecting a ball from a bag and spinning a coin. The bag contains 5 red balls and 7 blue balls. A ball is selected at random from the bag, its colour is noted and then the ball is returned to the bag. When a red ball is selected, a biased coin with probability \(\frac { 2 } { 3 }\) of landing heads is spun.
    When a blue ball is selected a fair coin is spun.
    1. Complete the tree diagram below to show the possible outcomes and associated probabilities. \includegraphics[max width=\textwidth, alt={}, center]{61983561-79f7-4883-8ae7-ab1f4955d444-04_785_385_744_568} \includegraphics[max width=\textwidth, alt={}, center]{61983561-79f7-4883-8ae7-ab1f4955d444-04_1054_483_760_954} Shivani selects a ball and spins the appropriate coin.
    2. Find the probability that she obtains a head. Given that Tom selected a ball at random and obtained a head when he spun the appropriate coin,
    3. find the probability that Tom selected a red ball. Shivani and Tom each repeat this experiment.
    4. Find the probability that the colour of the ball Shivani selects is the same as the colour of the ball Tom selects.
    Edexcel S1 Specimen Q3
    11 marks Moderate -0.8
    1. The discrete random variable \(X\) has probability distribution given by
    \(x\)- 10123
    \(\mathrm { P } ( X = x )\)\(\frac { 1 } { 5 }\)\(a\)\(\frac { 1 } { 10 }\)\(a\)\(\frac { 1 } { 5 }\)
    where \(a\) is a constant.
    1. Find the value of \(a\).
    2. Write down \(\mathrm { E } ( X )\).
    3. Find \(\operatorname { Var } ( X )\). The random variable \(Y = 6 - 2 X\)
    4. Find \(\operatorname { Var } ( Y )\).
    5. Calculate \(\mathrm { P } ( X \geqslant Y )\).
    Edexcel S1 Specimen Q4
    10 marks Moderate -0.8
    1. The Venn diagram in Figure 1 shows the number of students in a class who read any of 3 popular magazines \(A , B\) and \(C\).
    \begin{figure}[h]
    [diagram]
    \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} One of these students is selected at random.
    1. Show that the probability that the student reads more than one magazine is \(\frac { 1 } { 6 }\).
    2. Find the probability that the student reads \(A\) or \(B\) (or both).
    3. Write down the probability that the student reads both \(A\) and \(C\). Given that the student reads at least one of the magazines,
    4. find the probability that the student reads \(C\).
    5. Determine whether or not reading magazine \(B\) and reading magazine \(C\) are statistically independent.
    Edexcel S1 Specimen Q5
    14 marks Moderate -0.3
    1. A teacher selects a random sample of 56 students and records, to the nearest hour, the time spent watching television in a particular week.
    Hours\(1 - 10\)\(11 - 20\)\(21 - 25\)\(26 - 30\)\(31 - 40\)\(41 - 59\)
    Frequency615111383
    Mid-point5.515.52850
    1. Find the mid-points of the 21-25 hour and 31-40 hour groups. A histogram was drawn to represent these data. The 11-20 group was represented by a bar of width 4 cm and height 6 cm .
    2. Find the width and height of the 26-30 group.
    3. Estimate the mean and standard deviation of the time spent watching television by these students.
    4. Use linear interpolation to estimate the median length of time spent watching television by these students. The teacher estimated the lower quartile and the upper quartile of the time spent watching television to be 15.8 and 29.3 respectively.
    5. State, giving a reason, the skewness of these data.
    Edexcel S1 Specimen Q6
    14 marks Moderate -0.8
    1. A travel agent sells flights to different destinations from Beerow airport. The distance \(d\), measured in 100 km , of the destination from the airport and the fare \(\pounds f\) are recorded for a random sample of 6 destinations.
    Destination\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)
    \(d\)2.24.06.02.58.05.0
    \(f\)182025233228
    $$\text { [You may use } \sum d ^ { 2 } = 152.09 \quad \sum f ^ { 2 } = 3686 \quad \sum f d = 723.1 \text { ] }$$
    1. Using the axes below, complete a scatter diagram to illustrate this information.
    2. Explain why a linear regression model may be appropriate to describe the relationship between \(f\) and \(d\).
    3. Calculate \(S _ { d d }\) and \(S _ { f d }\)
    4. Calculate the equation of the regression line of \(f\) on \(d\) giving your answer in the form \(f = a + b d\).
    5. Give an interpretation of the value of \(b\). Jane is planning her holiday and wishes to fly from Beerow airport to a destination \(t \mathrm {~km}\) away. A rival travel agent charges 5 p per km.
    6. Find the range of values of \(t\) for which the first travel agent is cheaper than the rival. \includegraphics[max width=\textwidth, alt={}, center]{61983561-79f7-4883-8ae7-ab1f4955d444-20_967_1630_1722_164}
    Edexcel S1 Specimen Q7
    12 marks Moderate -0.3
    The distances travelled to work, \(D \mathrm {~km}\), by the employees at a large company are normally distributed with \(D \sim \mathrm {~N} \left( 30,8 ^ { 2 } \right)\).
    1. Find the probability that a randomly selected employee has a journey to work of more than 20 km .
    2. Find the upper quartile, \(Q _ { 3 }\), of \(D\).
    3. Write down the lower quartile, \(Q _ { 1 }\), of \(D\). An outlier is defined as any value of \(D\) such that \(D < h\) or \(D > k\) where $$h = Q _ { 1 } - 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right) \quad \text { and } \quad k = Q _ { 3 } + 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)$$
    4. Find the value of \(h\) and the value of \(k\). An employee is selected at random.
    5. Find the probability that the distance travelled to work by this employee is an outlier.
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