Questions S1 (2020 questions)

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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]
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]