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Edexcel S1 Q1
10 marks Moderate -0.8
  1. Explain briefly what is meant by a discrete random variable. [1 mark] A family has 3 cats and 4 dogs. Two of the family's animals are to be chosen at random. The random variable \(X\) represents the number of dogs chosen.
  2. Copy and complete the table to show the probability distribution of \(X\):
    \(x\)012
    P\((X = x)\)
    [4 marks]
  3. Calculate
    1. E\((X)\),
    2. Var\((X)\),
    3. Var\((2X)\).
    [5 marks]
Edexcel S1 Q2
11 marks Standard +0.3
The discrete random variable \(X\) can take any value in the set \(\{1, 2, 3, 4, 5, 6, 7, 8\}\). Arthur, Beatrice and Chris each carry out trials to investigate the distribution of \(X\). Arthur finds that P\((X = 1) = 0.125\) and that E\((X) = 4.5\). Beatrice finds that P\((X = 2) =\) P\((X = 3) =\) P\((X = 4) = p\). Chris finds that the values of \(X\) greater than 4 are all equally likely, with each having probability \(q\).
  1. Calculate the values of \(p\) and \(q\). [7 marks]
  2. Give the name for the distribution of \(X\). [1 mark]
  3. Calculate the standard deviation of \(X\). [3 marks]
Edexcel S1 Q3
13 marks Moderate -0.3
The marks obtained by ten students in a Geography test and a History test were as follows:
Student\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)\(I\)\(J\)
Geography (\(x\))34574921845310776185
History (\(y\))404955407139476573
  1. Given that \(\sum y = 547\), calculate the mark obtained by student \(E\) in History. [1 mark] Given further that \(\sum x^2 = 34087\), \(\sum y^2 = 31575\) and \(\sum xy = 31342\), calculate
  2. the product moment correlation coefficient between \(x\) and \(y\), [4 marks]
  3. an equation of the regression line of \(y\) on \(x\), [4 marks]
  4. an estimate of the History mark of student \(K\), who scored 70 in Geography. [2 marks]
  5. State, with a reason, whether you would expect your answer to part (d) to be reliable. [2 marks]
Edexcel S1 Q4
13 marks Standard +0.3
The random variable \(X\) is normally distributed with mean \(\mu\) and variance \(\sigma^2\).
  1. If \(2\mu = 3\sigma\), find P\((X < 2\mu)\). [5 marks]
  2. If, instead, P\((X < 3\mu) = 0.86\),
    1. find \(\mu\) in terms of \(\sigma\), [4 marks]
    2. calculate P\((X > 0)\). [4 marks]
Edexcel S1 Q5
14 marks Moderate -0.8
The stem-and-leaf diagram shows the values taken by two variables \(A\) and \(B\).
\(A\)\(B\)
8, 7, 4, 1, 011, 1, 2, 5, 6, 8, 9
9, 8, 7, 6, 6, 5, 220, 3, 4, 6, 7, 7, 9
9, 7, 6, 4, 2, 1, 031, 4, 5, 5, 8
8, 6, 3, 2, 240, 2, 6, 6, 9, 9
6, 4, 052, 3, 5, 7
5, 3, 160, 1
Key: 3 | 1 | 2 means \(A = 13\), \(B = 12\)
  1. For each set of data, calculate estimates of the median and the quartiles. [6 marks]
  2. Calculate the 42nd percentile for \(A\). [2 marks]
  3. On graph paper, indicating your scale clearly, construct box and whisker plots for both sets of data. [4 marks]
  4. Describe the skewness of the distribution of \(A\) and of \(B\). [2 marks]
Edexcel S1 Q6
14 marks Standard +0.8
The values of the two variables \(A\) and \(B\) given in the table in Question 5 are written on cards and placed in two separate packs, which are labelled \(A\) and \(B\). One card is selected from Pack \(A\). Let \(A_i\) represent the event that the first digit on this card is \(i\).
  1. Write down the value of P\((A_2)\). [1 mark] The card taken from Pack \(A\) is now transferred into Pack \(B\), and one card is picked at random from Pack \(B\). Let \(B_i\) represent the event that the first digit on this card is \(i\).
  2. Show that P\((A_1 \cap B_1) = \frac{1}{24}\). [3 marks]
  3. Show that P\((A_6 | B_2) = \frac{4}{41}\). [5 marks]
  4. Find the value of P\((A_1 \cup B_4)\). [5 marks]
Edexcel S1 Q1
4 marks Easy -1.8
  1. Explain briefly what is meant by a random variable. [2 marks]
  2. Write down a quantity which could be modelled as
    1. a discrete random variable,
    2. a continuous random variable.
    [2 marks]
Edexcel S1 Q2
11 marks Moderate -0.8
The discrete random variable \(X\) has the probability function given by the following table:
\(x\)0123456
\(P(X = x)\)0.090.120.220.16\(p\)\(2p\)0.2
  1. Show that \(p = 0.07\) [2 marks]
  2. Find the value of \(E(X + 2)\). [4 marks]
  3. Find the value of \(\text{Var}(3X - 1)\). [5 marks]
Edexcel S1 Q3
13 marks Moderate -0.3
Twenty pairs of observations are made of two variables \(x\) and \(y\), which are believed to be related. It is found that $$\sum x = 200, \quad \sum y = 174, \quad \sum x^2 = 6201, \quad \sum y^2 = 5102, \quad \sum xy = 5200.$$ Find
  1. the product-moment correlation coefficient between \(x\) and \(y\), [3 marks]
  2. the equation of the regression line of \(y\) on \(x\). [4 marks]
Given that \(p = x + 30\) and \(q = y + 50\),
  1. find the equation of the regression line of \(q\) on \(p\), in the form \(q = mp + c\). [3 marks]
  2. Estimate the value of \(q\) when \(p = 46\), stating any assumptions you make. [3 marks]
Edexcel S1 Q4
14 marks Standard +0.8
The heights of the students at a university are assumed to follow a normal distribution. 1% of the students are over 200 cm tall and 76% are between 165 cm and 200 cm tall. Find
  1. the mean and the variance of the distribution, [9 marks]
  2. the percentage of the students who are under 158 cm tall. [3 marks]
  3. Comment briefly on the suitability of a normal distribution to model such a population. [2 marks]
Edexcel S1 Q5
16 marks Moderate -0.8
In a survey of natural habitats, the numbers of trees in sixty equal areas of land were recorded, as follows:
171292340321153422318
154510521413294369301547
356241319269312718620
22183051493550258102631
332940373844243442381123
  1. Construct a stem-and-leaf diagram to illustrate this data, using the groupings 5 - 9, 10 - 14, 15 - 19, 20 - 24, etc. [8 marks]
  2. Find the three quartiles for the distribution. [4 marks]
  3. On graph paper construct a box plot for the data, showing your scale and clearly indicating any outliers. [4 marks]
Edexcel S1 Q6
17 marks Standard +0.3
Sixteen cards have been lost from a pack, which therefore contains only 36 cards. Two cards are drawn at random from the pack. The probability that both cards are red is \(\frac{1}{3}\).
  1. Show that \(r\), the number of red cards in the pack, satisfies the equation $$r(r - 1) = 420.$$ [4 marks]
  2. Hence or otherwise find the value of \(r\). [3 marks]
  3. Find the probability that, when three cards are drawn at random from the pack,
    1. at least two are red, [6 marks]
    2. the first one is red given that at least two are red. [4 marks]
Edexcel S1 Q1
6 marks Moderate -0.8
Thirty cards, marked with the even numbers from 2 to 60 inclusive, are shuffled and one card is withdrawn at random and then replaced. The random variable \(X\) takes the value of the number on the card each time the experiment is repeated.
  1. What must be assumed about the cards if the distribution of \(X\) is modelled by a discrete uniform distribution? [1 mark]
  2. Making this modelling assumption, find the expectation and the variance of \(X\). [5 marks]
Edexcel S1 Q2
9 marks Standard +0.3
  1. Explain briefly why, for data grouped in unequal classes, the class with the highest frequency may not be the modal class. [2 marks]
In a histogram drawn to represent the annual incomes (in thousands of pounds) of 1000 families, the modal class was 15 - 20 (i.e. \(£x\), where \(15000 \leq x < 20000\)), with frequency 300. The highest frequency in a class was 400, for the class 30 - 40, and the bar representing this class was 8 cm high. The total area under the histogram was 50 cm\(^2\).
  1. Find the height and the width of the bar representing the modal class. [7 marks]
Edexcel S1 Q3
10 marks Moderate -0.8
The variable \(X\) represents the marks out of 150 scored by a group of students in an examination. The following ten values of \(X\) were obtained: 60, 66, 76, 80, 94, 106, 110, 116, 124, 140.
  1. Write down the median, \(M\), of the ten marks. [1 mark]
  2. Using the coding \(y = \frac{x - M}{2}\), and showing all your working clearly, find the mean and the standard deviation of the marks. [6 marks]
  3. Find E\((3X - 5)\). [3 marks]
Edexcel S1 Q4
11 marks Moderate -0.3
The discrete random variable \(X\) has probability function P\((X = x) = k(x + 4)\). Given that \(X\) can take any of the values \(-3, -2, -1, 0, 1, 2, 3, 4\),
  1. find the value of the constant \(k\). [3 marks]
  2. Find P\((X < 0)\). [2 marks]
  3. Show that the cumulative distribution F\((x)\) is given by $$\text{F}(x) = \lambda(x + 4)(x + 5)$$ where \(\lambda\) is a constant to be found. [6 marks]
Edexcel S1 Q5
12 marks Moderate -0.3
The events \(A\) and \(B\) are such that P\((A \cap B) = 0.24\), P\((A \cup B) = 0.88\) and P\((B) = 0.52\).
  1. Find P\((A)\). [3 marks]
  2. Determine, with reasons, whether \(A\) and \(B\) are
    1. mutually exclusive,
    2. independent.
    [4 marks]
  3. Find P\((B | A)\). [2 marks]
  4. Find P\((A' | B')\). [3 marks]
Edexcel S1 Q6
12 marks Moderate -0.3
The times taken by a group of people to complete a task are modelled by a normal distribution with mean 8 hours and standard deviation 2 hours. Use this model to calculate
  1. the probability that a person chosen at random took between 5 and 9 hours to complete the task, [4 marks]
  2. the range, symmetrical about the mean, within which 80% of the people's times lie. [5 marks]
It is found that, in fact, 80% of the people take more than 5 hours. The model is modified so that the mean is still 8 hours but the standard deviation is no longer 2 hours.
  1. Find the standard deviation of the times in the modified model. [3 marks]
Edexcel S1 Q7
15 marks Moderate -0.3
The following data was collected for seven cars, showing their engine size, \(x\) litres, and their fuel consumption, \(y\) km per litre, on a long journey.
Car\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)
\(x\)0.951.201.371.762.252.502.875
\(y\)21.317.215.519.114.711.49.0
\(\sum x = 12.905\), \(\sum x^2 = 26.8951\), \(\sum y = 108.2\), \(\sum y^2 = 1781.64\), \(\sum xy = 183.176\).
  1. Calculate the equation of the regression line of \(x\) on \(y\), expressing your answer in the form \(x = ay + b\). [6 marks]
  2. Calculate the product moment correlation coefficient between \(y\) and \(x\) and give a brief interpretation of its value. [4 marks]
  3. Use the equation of the regression line to estimate the value of \(x\) when \(y = 12\). State, with a reason, how accurate you would expect this estimate to be. [3 marks]
  4. Comment on the use of the line to find values of \(x\) as \(y\) gets very small. [2 marks]
Edexcel S1 Q1
7 marks Moderate -0.8
70% of the households in a town have a CD player and 45% have both a CD player and a personal computer (PC). 18% have neither a CD player nor a PC.
  1. Illustrate this information using a Venn diagram. [3 marks]
  2. Find the percentage of the households that do not have a PC. [2 marks]
  3. Find the probability that a household chosen at random has a CD player or a PC but not both. [2 marks]
Edexcel S1 Q2
7 marks Easy -1.3
The random variable \(X\) has the normal distribution \(N(2, 1.7^2)\).
  1. State the standard deviation of \(X\). [1 mark]
  2. Find \(P(X < 0)\). [2 marks]
  3. Find \(P(0.6 < X < 3.4)\). [4 marks]
Edexcel S1 Q3
9 marks Moderate -0.3
The discrete random variable \(X\) has probability function $$P(X = x) = \begin{cases} cx^2 & x = -3, -2, -1, 1, 2, 3 \\ 0 & \text{otherwise.} \end{cases}$$
  1. Show that \(c = \frac{1}{28}\). [3 marks]
  2. Calculate
    1. \(E(X)\),
    2. \(E(X^2)\).
    [3 marks]
  3. Calculate
    1. \(\text{Var}(X)\),
    2. \(\text{Var}(10 - 2X)\).
    [3 marks]
Edexcel S1 Q4
10 marks Moderate -0.8
The heights, \(h\) m, of eight children were measured, giving the following values of \(h\): 1.20, 1.12, 1.43, 0.98, 1.31, 1.26, 1.02, 1.41.
  1. Find the mean height of the children. [2 marks]
  2. Calculate the variance of the heights. [3 marks]
The children were also weighed. It was found that their masses, \(w\) kg, were such that $$\sum w = 324, \quad \sum w^2 = 13532, \quad \sum wh = 403.$$
  1. Calculate the product-moment correlation coefficient between \(w\) and \(h\). [4 marks]
  2. Comment briefly on the value you have obtained. [1 mark]
Edexcel S1 Q5
10 marks Standard +0.8
The ages of the residents of a retirement community are assumed to be normally distributed. 15% of the residents are under 60 years old and 5% are over 90 years old.
  1. Using this information, find the mean and the standard deviation of the ages. [7 marks]
  2. If there are 200 residents, find how many are over 80 years old. [3 marks]
Edexcel S1 Q6
15 marks Standard +0.8
Of the cars that are taken to a certain garage for an M.O.T. test, 87% pass. However, 2% of these have faults for which they should have been failed. 5% of the cars which fail are in fact roadworthy and should have passed. Using a tree diagram, or otherwise, calculate the probabilities that a car chosen at random
  1. should have passed the test, regardless of whether it actually did or not, [4 marks]
  2. failed the test, given that it should have passed. [3 marks]
The garage is told to improve its procedures. When it is inspected again a year later, it is found that the pass rate is still 87% overall and 2% of the cars passed have faults as before, but now 0.3% of the cars which should have passed are failed and \(x\)% of the cars which are failed should have passed.
  1. Find the value of \(x\). [8 marks]