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Edexcel S1 Q7
17 marks Moderate -0.8
The back-to-back stem and leaf diagram shows the journey times, to the nearest minute, of the commuter services into a big city provided by the trains of two operating companies.
Company \(A\)Company \(B\)
(3)\(4\ 3\ 1\)2\(0\ 5\ 6\ 6\ 8\ 9\)(6)
(4)\(9\ 8\ 6\ 5\)3\(1\ 3\ 4\ 7\ 9\)(5)
(4)\(8\ 8\ 6\ 2\)4\(0\ 1\ 3\ 5\ 8\)( )
(6)\(9\ 7\ 5\ 3\ 2\ 1\)5\(2\ 6\ 8\ 9\ 9\)( )
(3)\(6\ 5\ 3\)6\(3\ 4\ 7\ 7\)( )
(3)\(3\ 2\ 2\)7\(0\ 1\ 5\)( )
Key: \(4|3|6\) means 34 minutes for Company \(A\) and 36 minutes for Company \(B\).
  1. Write down the numbers needed to complete the diagram. [1 mark]
  2. Find the median and the quartiles for each company. [6 marks]
  3. On graph paper, draw box plots for the two companies. Show your scale. [6 marks]
  4. Use your plots to compare the two sets of data briefly. [2 marks]
  5. Describe the skewness of each company's distribution of times. [2 marks]
Edexcel S1 Q1
4 marks Easy -1.8
  1. Briefly explain what is meant by a sample space. [2 marks]
  2. State two properties which a function \(f(x)\) must have to be a probability function. [2 marks]
Edexcel S1 Q2
8 marks Standard +0.3
A company makes two cars, model \(A\) and model \(B\). The distance that model \(A\) travels on 10 litres of petrol is normally distributed with mean 109 km and variance 72.25 km\(^2\). The distance that model \(B\) travels on 10 litres of petrol is normally distributed with mean 108.5 km and variance 169 km\(^2\). In a trial, one of each model is filled with 10 litres of petrol and sent on a journey of 110 km. Find which model has the greater probability of completing this journey, and state the value of this probability. [8 marks]
Edexcel S1 Q3
8 marks Moderate -0.3
\(A\), \(B\) and \(C\) are three events such that \(\text{P}(A) = x\), \(\text{P}(B) = y\) and \(\text{P}(C) = x + y\). It is known that \(\text{P}(A \cup B) = 0.6\) and \(\text{P}(B \mid A) = 0.2\).
  1. Show that \(4x + 5y = 3\). [2 marks]
It is also known that \(B\) and \(C\) are mutually exclusive and that \(\text{P}(B \cup C) = 0.9\)
  1. Obtain another equation in \(x\) and \(y\) and hence find the values of \(x\) and \(y\). [4 marks]
  2. Deduce whether or not \(A\) and \(B\) are independent events. [2 marks]
Edexcel S1 Q4
12 marks Moderate -0.8
The discrete random variable \(X\) has the following probability distribution:
\(x\)012345
\(\text{P}(X = x)\)0.110.170.20.13\(p\)\(p^2\)
  1. Find the value of \(p\). [4 marks]
  2. Find
    1. \(\text{P}(0 < X \leq 2)\),
    2. \(\text{P}(X \geq 3)\).
    [3 marks]
  3. Find the mean and the variance of \(X\). [3 marks]
  4. Construct a table to represent the cumulative distribution function \(\text{F}(x)\). [2 marks]
Edexcel S1 Q5
13 marks Standard +0.3
The following marks out of 50 were given by two judges to the contestants in a talent contest:
Contestant\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)
Judge 1 (\(x\))4332402147112938
Judge 2 (\(y\))3925402236132732
Given that \(\sum x = 261\), \(\sum x^2 = 9529\) and \(\sum xy = 8373\),
  1. calculate the product-moment correlation coefficient between the two judges' marks [5 marks]
  2. Find an equation of the regression line of \(x\) on \(y\). [4 marks]
Contestant \(I\) was awarded 45 marks by Judge 2.
  1. Estimate the mark that this contestant would have received from Judge 1. [2 marks]
  2. Comment, with explanation, on the probable accuracy of your answer. [2 marks]
Edexcel S1 Q6
15 marks Moderate -0.3
1000 houses were sold in a small town in a one-year period. The selling prices were as given in the following table:
Selling PriceNumber of HousesSelling PriceNumber of Houses
Up to £50 00060Up to £150 000642
Up to £75 000227Up to £200 000805
Up to £100 000305Up to £500 000849
Up to £125 000414Up to £750 0001000
  1. Name (do not draw) a suitable type of graph for illustrating this data. [1 mark]
  2. Use interpolation to find estimates of the median and the quartiles. [6 marks]
  3. Estimate the 37th percentile. [2 marks]
Given further that the lowest price was £42 000 and the range of the prices was £690 000,
  1. draw a box plot to represent the data. Show your scale clearly. [4 marks]
In another town the median price was £149 000, and the interquartile range was £90 000.
  1. Briefly compare the prices in the two towns using this information. [2 marks]
Edexcel S1 Q7
15 marks Standard +0.3
The random variable \(X\), which can take any value from \(\{1, 2, \ldots, n\}\), is modelled by the discrete uniform distribution with mean 10.
  1. Show that \(n = 19\) and find the variance of \(X\). [4 marks]
  2. Find \(\text{P}(3 < X \leq 6)\). [2 marks]
The random variable \(Y\) is defined by \(Y = 3(X - 10)\).
  1. State the mean and the variance of \(Y\). [3 marks]
The model for the distribution of \(X\) is found to be unsatisfactory, and in a refined model the probability distribution of \(X\) is taken to be $$\text{f}(x) = \begin{cases} k(x + 1) & x = 1, 2, \ldots, 19, \\ 0 & \text{otherwise}. \end{cases}$$
  1. Show that \(k = \frac{1}{209}\). [3 marks]
  2. Find \(\text{P}(3 < X \leq 6)\) using this model. [3 marks]
Edexcel S1 Q1
8 marks Moderate -0.8
Using the coding \(y = \frac{x-90}{5}\), and showing each step in your working clearly, calculate the mean and the standard deviation of the 20 observations of a variable \(X\) given by the following table:
\(x\)7580859095100105110
Frequency12364211
[8 marks]
Edexcel S1 Q2
8 marks Standard +0.3
A darts player throws two darts, attempting to score a bull's-eye with each. The probability that he will achieve this with his first dart is \(0.25\). If he misses with his first dart, the probability that he will also miss with his second dart is \(0.7\). The probability that he will miss with at least one dart is \(0.9\).
  1. Show that the probability that he succeeds with his first dart but misses with his second is \(0.15\). [5 marks]
  2. Find the conditional probability that he misses with both darts, given that he misses with at least one. [3 marks]
Edexcel S1 Q3
8 marks Standard +0.3
The entrance to a car park is \(1.9\) m wide. It is found that this is too narrow for \(2\%\) of the vehicles which need to use the car park. The widths of these vehicles are modelled by a normal distribution with mean \(1.6\) m.
  1. Find the standard deviation of the distribution. [4 marks]
It is decided to widen the entrance so that \(99.5\%\) of vehicles will be able to use it.
  1. Find the minimum width needed to achieve this. [4 marks]
Edexcel S1 Q4
14 marks Moderate -0.8
A pack of 52 cards contains 4 cards bearing each of the integers from 1 to 13. A card is selected at random. The random variable \(X\) represents the number on the card.
  1. Find \(P(X \leq 5)\). [1 mark]
  2. Name the distribution of \(X\) and find the expectation and variance of \(X\). [4 marks]
A hand of 12 cards consists of three 2s, four 3s, two 4s, two 5s and one 6. The random variable \(Y\) represents the number on a card chosen at random from this hand.
  1. Draw up a table to show the probability distribution of \(Y\). [3 marks]
  2. Calculate \(\text{Var}(3Y - 2)\). [6 marks]
Edexcel S1 Q5
16 marks Moderate -0.8
The following data were collected by counting the number of cars that passed the gates of a college in 60 successive 5 minute intervals.
122019313235372926272017
1598111317172125272825
303237404545444742413638
353430302726292423212118
161619222628231715101213
  1. Make a stem and leaf diagram for this data, using the groups \(5-9\), \(10-14\), \(\ldots\), \(45-49\). Show the total in each group and give a key to the diagram. [7 marks]
  2. Find the three quartiles for this data. [4 marks]
  3. On graph paper, draw a box plot for the data. [4 marks]
  4. Describe the skewness of the distribution. [1 mark]
Edexcel S1 Q6
21 marks Standard +0.3
A missile was fired vertically upwards and its height above ground level, \(h\) metres, was found at various times \(t\) seconds after it was released. The results are given in the following table:
\(t\)1234567
\(h\)68126174216240252266
It is thought that this data can be fitted to the formula \(h = pt - qt^2\).
  1. Show that this equation can be written as \(\frac{h}{t} = p - qt\). [1 mark]
  2. Plot a scatter diagram of \(\frac{h}{t}\) against \(t\). [5 marks]
Given that \(\sum h = 1342\), \(\sum \frac{h}{t} = 371\) and \(\sum \frac{h^2}{t^2} = 20385\),
  1. find the equation of the regression line of \(\frac{h}{t}\) on \(t\) and hence write down the values of \(p\) and \(q\). [8 marks]
  2. Use your equation to find the value of \(h\) when \(t = 10\). Comment on the implication of your answer. [3 marks]
  3. Find the product-moment correlation coefficient between \(\frac{h}{t}\) and \(t\) and state the significance of its value. [4 marks]
Edexcel S1 Q1
4 marks Easy -1.8
Briefly describe what is meant by
  1. a statistical model, [2 marks]
  2. a refinement of a model. [2 marks]
Edexcel S1 Q2
6 marks Standard +0.3
The random variable \(X\) has the discrete uniform distribution and takes the values \(\{1, \ldots, n\}\). The standard deviation of of \(X\) is \(2\sqrt{6}\). Find
  1. the mean of \(X\), [3 marks]
  2. P\((3 \leq X < \frac{2}{5}n)\). [3 marks]
Edexcel S1 Q3
9 marks Standard +0.3
The rainfall at a weather station was recorded every day of the twentieth century. One year is selected at random from the records and the total rainfall, in cm, in January of that year is denoted by \(R\). Assuming that \(R\) can be modelled by a normal distribution with standard deviation \(12.6\), and given that P\((R > 100) = 0.0764\),
  1. find the mean of \(R\), [4 marks]
  2. calculate P\((75 < R < 80)\). [5 marks]
Edexcel S1 Q4
13 marks Standard +0.3
The length of time, in minutes, that visitors queued for a tourist attraction is given by the following table, where, for example, '\(20 -\)' means from 20 up to but not including 30 minutes.
Queuing time (mins)\(0 -\)\(10 -\)\(15 -\)\(20 -\)\(30 -\)\(40 - 60\)
Number of visitors\(15\)\(24\)\(x\)\(13\)\(10\)\(y\)
  1. State the upper class boundary of the first class. [1 mark]
A histogram is drawn to represent this data. The total area under the histogram is \(36\) cm\(^2\). The '\(10 -\)' bar has width \(1\) cm and height \(9.6\) cm. The '\(15 -\)' bar is ten times as high as the '\(40 - 60\)' bar.
  1. Find the values of \(x\) and \(y\). [7 marks]
  2. On graph paper, construct the histogram accurately. [5 marks]
Edexcel S1 Q5
13 marks Moderate -0.3
The discrete random variable \(X\) takes only the values \(4, 5, 6, 7, 8\) and \(9\). The probabilities of these values are given in the table:
\(x\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)
P\((X = x)\)\(p\)\(0.1\)\(q\)\(q\)\(0.3\)\(0.2\)
It is known that E\((X) = 6.7\). Find
  1. the values of \(p\) and \(q\), [7 marks]
  2. the value of \(a\) for which E\((2X + a) = 0\), [3 marks]
  3. Var\((X)\). [3 marks]
Edexcel S1 Q6
15 marks Standard +0.3
The marks out of 75 obtained by a group of ten students in their first and second Statistics modules were as follows:
Student\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)\(I\)\(J\)
Module 1 \((x)\)\(54\)\(33\)\(42\)\(71\)\(60\)\(27\)\(39\)\(46\)\(59\)\(64\)
Module 2 \((y)\)\(50\)\(22\)\(44\)\(58\)\(42\)\(19\)\(35\)\(46\)\(55\)\(60\)
  1. Find \(\sum x\) and \(\sum y\). [2 marks]
Given that \(\sum x^2 = 26353\) and \(\sum xy = 22991\),
  1. obtain the equation of the regression line of \(y\) on \(x\). [5 marks]
  2. Estimate the Module 2 result of a student whose mark in Module 1 was (i) 65, (ii) 5. Explain why one of these estimates is less reliable than the other. [4 marks]
The equation of the regression line of \(x\) on \(y\) is \(x = 0.921y + 9.81\).
  1. Deduce the product moment correlation coefficient between \(x\) and \(y\), and briefly interpret its value. [4 marks]
Edexcel S1 Q7
15 marks Moderate -0.3
Among the families with two children in a large city, the probability that the elder child is a boy is \(\frac{5}{12}\) and the probability that the younger child is a boy is \(\frac{9}{16}\). The probability that the younger child is a girl, given that the elder child is a girl, is \(\frac{1}{4}\). One of the families is chosen at random. Using a tree diagram, or otherwise,
  1. show that the probability that both children are boys is \(\frac{1}{8}\). [5 marks]
Find the probability that
  1. one child is a boy and the other is a girl, [3 marks]
  2. one child is a boy given that the other is a girl. [3 marks]
If three of the families are chosen at random,
  1. find the probability that exactly two of the families have two boys. [3 marks]
  2. State an assumption that you have made in answering part (d). [1 mark]
Edexcel S1 Q1
4 marks Moderate -0.8
Given that \(P(A \cup B) = 0.65\), \(P(A \cap B) = 0.15\) and \(P(A) = 0.3\), determine, with explanation, whether or not the events \(A\) and \(B\) are
  1. mutually exclusive, [1 mark]
  2. independent. [3 marks]
Edexcel S1 Q2
4 marks Easy -2.0
  1. Give one example in each case of a quantity which could be modelled as
    1. a discrete random variable,
    2. a continuous random variable.
    [2 marks]
  2. Name one discrete distribution and one continuous distribution, stating clearly which is which. [2 marks]
Edexcel S1 Q3
10 marks Moderate -0.3
A regular tetrahedron has its faces numbered 1, 2, 3 and 4. It is weighted so that when it is thrown, the probability of each face being in contact with the table is inversely proportional to the number on that face. This number is represented by the random variable \(X\).
  1. Show that \(P(X = 1) = \frac{12}{25}\) and find the probabilities of the other values of \(X\). [5 marks]
  2. Calculate the mean and the variance of \(X\). [5 marks]
Edexcel S1 Q4
10 marks Standard +0.3
The random variable \(X\) is normally distributed with mean 17. The probability that \(X\) is less than 16 is 0.3707.
  1. Calculate the standard deviation of \(X\). [4 marks]
  2. In 75 independent observations of \(X\), how many would you expect to be greater than 20? [6 marks]