Questions — Edexcel (10514 questions)

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Edexcel S1 2011 June Q1
7 marks Moderate -0.8
On a particular day the height above sea level, \(x\) metres, and the mid-day temperature, \(y\)°C, were recorded in 8 north European towns. These data are summarised below \(S_{xx} = 3\,535\,237.5 \quad \sum y = 181 \quad \sum y^2 = 4305 \quad S_{yy} = -23\,726.25\)
  1. Find \(S_{yy}\). [2]
  2. Calculate, to 3 significant figures, the product moment correlation coefficient for these data. [2]
  3. Give an interpretation of your coefficient. [1]
A student thought that the calculations would be simpler if the height above sea level, \(h\), was measured in kilometres and used the variable \(h = \frac{x}{1000}\) instead of \(x\).
  1. Write down the value of \(S_{hh}\) [1]
  2. Write down the value of the correlation coefficient between \(h\) and \(y\). [1]
Edexcel S1 2011 June Q2
5 marks Moderate -0.8
The random variable \(X \sim \text{N}(\mu, 5^2)\) and \(\text{P}(X < 23) = 0.9192\)
  1. Find the value of \(\mu\). [4]
  2. Write down the value of \(\text{P}(\mu < X < 23)\). [1]
Edexcel S1 2011 June Q3
7 marks Easy -1.2
The discrete random variable \(Y\) has probability distribution
\(y\)1234
\(\text{P}(Y = y)\)\(a\)\(b\)0.3\(c\)
where \(a\), \(b\) and \(c\) are constants. The cumulative distribution function F(\(y\)) of \(Y\) is given in the following table
\(y\)1234
F(\(y\))0.10.5\(d\)1.0
where \(d\) is a constant.
  1. Find the value of \(a\), the value of \(b\), the value of \(c\) and the value of \(d\). [5]
  2. Find \(\text{P}(3Y + 2 \geq 8)\). [2]
Edexcel S1 2011 June Q4
7 marks Moderate -0.8
Past records show that the times, in seconds, taken to run 100 m by children at a school can be modelled by a normal distribution with a mean of 16.12 and a standard deviation of 1.60 A child from the school is selected at random.
  1. Find the probability that this child runs 100 m in less than 15 s. [3]
On sports day the school awards certificates to the fastest 30\% of the children in the 100 m race.
  1. Estimate, to 2 decimal places, the slowest time taken to run 100 m for which a child will be awarded a certificate. [4]
Edexcel S1 2011 June Q5
11 marks Moderate -0.8
A class of students had a sudoku competition. The time taken for each student to complete the sudoku was recorded to the nearest minute and the results are summarised in the table below.
TimeMid-point, \(x\)Frequency, \(f\)
2 - 852
9 - 127
13 - 15145
16 - 18178
19 - 2220.54
23 - 3026.54
(You may use \(\sum fx^2 = 8603.75\))
  1. Write down the mid-point for the 9 - 12 interval. [1]
  2. Use linear interpolation to estimate the median time taken by the students. [2]
  3. Estimate the mean and standard deviation of the times taken by the students. [5]
The teacher suggested that a normal distribution could be used to model the times taken by the students to complete the sudoku.
  1. Give a reason to support the use of a normal distribution in this case. [1]
On another occasion the teacher calculated the quartiles for the times taken by the students to complete a different sudoku and found \(Q_1 = 8.5 \quad Q_2 = 13.0 \quad Q_3 = 21.0\)
  1. Describe, giving a reason, the skewness of the times on this occasion. [2]
Edexcel S1 2011 June Q6
9 marks Moderate -0.8
Jake and Kamil are sometimes late for school. The events \(J\) and \(K\) are defined as follows \(J =\) the event that Jake is late for school \(K =\) the event that Kamil is late for school \(\text{P}(J) = 0.25\), \(\text{P}(J \cap K) = 0.15\) and \(\text{P}(J' \cap K') = 0.7\) On a randomly selected day, find the probability that
  1. at least one of Jake or Kamil are late for school, [1]
  2. Kamil is late for school. [2]
Given that Jake is late for school,
  1. find the probability that Kamil is late. [3]
The teacher suspects that Jake being late for school and Kamil being late for school are linked in some way.
  1. Determine whether or not \(J\) and \(K\) are statistically independent. [2]
  2. Comment on the teacher's suspicion in the light of your calculation in (d). [1]
Edexcel S1 2011 June Q7
12 marks Moderate -0.8
A teacher took a random sample of 8 children from a class. For each child the teacher recorded the length of their left foot, \(f\) cm, and their height, \(h\) cm. The results are given in the table below.
\(f\)2326232227242021
\(h\)135144134136140134130132
(You may use \(\sum f = 186 \quad \sum h = 1085 \quad S_{ff} = 39.5 \quad S_{hh} = 139.875 \quad \sum fh = 25291\))
  1. Calculate \(S_{fh}\) [2]
  2. Find the equation of the regression line of \(h\) on \(f\) in the form \(h = a + bf\). Give the value of \(a\) and the value of \(b\) correct to 3 significant figures. [5]
  3. Use your equation to estimate the height of a child with a left foot length of 25 cm. [2]
  4. Comment on the reliability of your estimate in (c), giving a reason for your answer. [2]
The left foot length of the teacher is 25 cm.
  1. Give a reason why the equation in (b) should not be used to estimate the teacher's height. [1]
Edexcel S1 2011 June Q8
17 marks Standard +0.3
A spinner is designed so that the score \(S\) is given by the following probability distribution.
\(s\)01245
\(\text{P}(S = s)\)\(p\)0.250.250.200.20
  1. Find the value of \(p\). [2]
  2. Find \(\text{E}(S)\). [2]
  3. Show that \(\text{E}(S^2) = 9.45\) [2]
  4. Find \(\text{Var}(S)\). [2]
Tom and Jess play a game with this spinner. The spinner is spun repeatedly and \(S\) counters are awarded on the outcome of each spin. If \(S\) is even then Tom receives the counters and if \(S\) is odd then Jess receives them. The first player to collect 10 or more counters is the winner.
  1. Find the probability that Jess wins after 2 spins. [2]
  2. Find the probability that Tom wins after exactly 3 spins. [4]
  3. Find the probability that Jess wins after exactly 3 spins. [3]
Edexcel S1 2002 November Q1
4 marks Easy -1.8
  1. Explain briefly why statistical models are used when attempting to solve real-world problems. [2]
  2. Write down the name of the distribution you would recommend as a suitable model for each of the following situations.
    1. The weight of marmalade in a jar.
    2. The number on the uppermost face of a fair die after it has been rolled.
    [2]
Edexcel S1 2002 November Q2
7 marks Moderate -0.8
There are 125 sixth-form students in a college, of whom 60 are studying only arts subjects, 40 only science subjects and the rest a mixture of both. Three students are selected at random, without replacement. Find the probability that
  1. all three students are studying only arts subjects, [4]
  2. exactly one of the three students is studying only science subjects. [3]
Edexcel S1 2002 November Q3
8 marks Moderate -0.8
The events \(A\) and \(B\) are independent such that \(P(A) = 0.25\) and \(P(B) = 0.30\). Find
  1. \(P(A \cap B)\), [2]
  2. \(P(A \cup B)\), [2]
  3. \(P(A | B')\). [4]
Edexcel S1 2002 November Q4
11 marks Moderate -0.3
Strips of metal are cut to length \(L\) cm, where \(L \sim N(\mu, 0.5^2)\).
  1. Given that 2.5\% of the cut lengths exceed 50.98 cm, show that \(\mu = 50\). [5]
  2. Find \(P(49.25 < L < 50.75)\). [4]
Those strips with length either less than 49.25 cm or greater than 50.75 cm cannot be used. Two strips of metal are selected at random.
  1. Find the probability that both strips cannot be used. [2]
Edexcel S1 2002 November Q5
12 marks Standard +0.3
An agricultural researcher collected data, in appropriate units, on the annual rainfall \(x\) and the annual yield of wheat \(y\) at 8 randomly selected places. The data were coded using \(s = x - 6\) and \(t = y - 20\) and the following summations were obtained. \(\Sigma s = 48.5\), \(\Sigma t = 65.0\), \(\Sigma s^2 = 402.11\), \(\Sigma t^2 = 701.80\), \(\Sigma st = 523.23\)
  1. Find the equation of the regression line of \(t\) on \(s\) in the form \(t = p + qs\). [7]
  2. Find the equation of the regression line of \(y\) on \(x\) in the form \(y = a + bx\), giving \(a\) and \(b\) to 3 decimal places. [3]
The value of the product moment correlation coefficient between \(s\) and \(t\) is 0.943, to 3 decimal places.
  1. Write down the value of the product moment correlation coefficient between \(x\) and \(y\). Give a justification for your answer. [2]
Edexcel S1 2002 November Q6
15 marks Moderate -0.8
The discrete random variable \(X\) has the following probability distribution.
\(x\)\(-2\)\(-1\)\(0\)\(1\)\(2\)
\(P(X = x)\)\(\alpha\)\(0.2\)\(0.1\)\(0.2\)\(\beta\)
  1. Given that \(E(X) = -0.2\), find the value of \(\alpha\) and the value of \(\beta\). [6]
  2. Write down \(F(0.8)\). [1]
  3. Evaluate \(\text{Var}(X)\). [4]
Find the value of
  1. \(E(3X - 2)\), [2]
  2. \(\text{Var}(2X + 6)\). [2]
Edexcel S1 2002 November Q7
18 marks Moderate -0.8
The following stem and leaf diagram shows the aptitude scores \(x\) obtained by all the applicants for a particular job.
Aptitude score\(3|1\) means 31
31 2 9(3)
42 4 6 8 9(5)
51 3 3 5 6 7 9(7)
60 1 3 3 3 5 6 8 8 9(10)
71 2 2 2 4 5 5 5 6 8 8 8 8 9(14)
80 1 2 3 5 8 8 9(8)
90 1 2(3)
  1. Write down the modal aptitude score. [1]
  2. Find the three quartiles for these data. [3]
Outliers can be defined to be outside the limits \(Q_1 - 1.0(Q_3 - Q_1)\) and \(Q_3 + 1.0(Q_3 - Q_1)\).
  1. On a graph paper, draw a box plot to represent these data. [7]
For these data, \(\Sigma x = 3363\) and \(\Sigma x^2 = 238305\).
  1. Calculate, to 2 decimal places, the mean and the standard deviation for these data. [3]
  2. Use two different methods to show that these data are negatively skewed. [4]
Edexcel S1 Specimen Q1
4 marks Easy -1.8
  1. Explain what you understand by a statistical model. [2]
  2. Write down a random variable which could be modelled by
    1. a discrete uniform distribution,
    2. a normal distribution.
    [2]
Edexcel S1 Specimen Q2
9 marks Standard +0.8
A group of students believes that the time taken to travel to college, \(T\) minutes, can be assumed to be normally distributed. Within the college 5\% of students take at least 55 minutes to travel to college and 0.1\% take less than 10 minutes. Find the mean and standard deviation of \(T\). [9]
Edexcel S1 Specimen Q3
14 marks Moderate -0.8
The discrete random variable \(X\) has probability function $$P(X = x) = \begin{cases} kx, & x = 1, 2, 3, 4, 5, \\ 0, & \text{otherwise.} \end{cases}$$
  1. Show that \(k = \frac{1}{15}\). [3]
Find the value of
  1. E\((2X + 3)\), [5]
  2. Var\((2X - 4)\). [6]
Edexcel S1 Specimen Q4
14 marks Moderate -0.3
A drilling machine can run at various speeds, but in general the higher the speed the sooner the drill needs to be replaced. Over several months, 15 pairs of observations relating to speed, \(s\) revolutions per minute, and life of drill, \(h\) hours, are collected. For convenience the data are coded so that \(x = s - 20\) and \(y = h - 100\) and the following summations obtained. \(\Sigma x = 143; \Sigma y = 391; \Sigma x^2 = 2413; \Sigma y^2 = 22441; \Sigma xy = 484\).
  1. Find the equation of the regression line of \(h\) on \(s\). [10]
  2. Interpret the slope of your regression line. [2]
Estimate the life of a drill revolving at 30 revolutions per minute. [2]
Edexcel S1 Specimen Q5
16 marks Easy -1.2
  1. Explain briefly the advantages and disadvantages of using the quartiles to summarise a set of data. [4]
  2. Describe the main features and uses of a box plot. [3]
The distances, in kilometres, travelled to school by the teachers in two schools, \(A\) and \(B\), in the same town were recorded. The data for School \(A\) are summarised in Diagram 1. \includegraphics{figure_1} For School \(B\), the least distance travelled was 3 km and the longest distance travelled was 55 km. The three quartiles were 17, 24 and 31 respectively. 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.
  1. Draw a box plot for School \(B\). [5]
  2. Compare and contrast the two box plots. [4]
Edexcel S1 Specimen Q6
18 marks Moderate -0.8
For any married couple who are members of a tennis club, the probability that the husband has a degree is \(\frac{3}{5}\) and the probability that the wife has a degree is \(\frac{1}{2}\). The probability that the husband has a degree, given that the wife has a degree, is \(\frac{11}{12}\). A married couple is chosen at random.
  1. Show that the probability that both of them have degrees is \(\frac{11}{24}\). [2]
  2. Draw a Venn diagram to represent these data. [5]
Find the probability that
  1. only one of them has a degree, [2]
  2. neither of them has a degree. [3]
Two married couples are chosen at random.
  1. Find the probability that only one of the two husbands and only one of the two wives have degrees. [6]
Edexcel S2 2016 January Q1
5 marks Easy -1.2
The manager of a clothing shop wishes to investigate how satisfied customers are with the quality of service they receive. A database of the shop's customers is used as a sampling frame for this investigation.
  1. Identify one potential problem with this sampling frame. [1]
Customers are asked to complete a survey about the quality of service they receive. Past information shows that 35\% of customers complete the survey. A random sample of 20 customers is taken.
  1. Write down a suitable distribution to model the number of customers in this sample that complete the survey. [2]
  2. Find the probability that more than half of the customers in the sample complete the survey. [2]
Edexcel S2 2016 January Q2
10 marks Moderate -0.3
The continuous random variable \(X\) is uniformly distributed over the interval \([a, b]\) Given that \(\mathrm{P}(3 < X < 5) = \frac{1}{8}\) and \(\mathrm{E}(X) = 4\)
  1. find the value of \(a\) and the value of \(b\) [3]
  2. find the value of the constant, \(c\), such that \(\mathrm{E}(cX - 2) = 0\) [2]
  3. find the exact value of \(\mathrm{E}(X^2)\) [3]
  4. find \(\mathrm{P}(2X - b > a)\) [2]
Edexcel S2 2016 January Q3
11 marks Moderate -0.3
Left-handed people make up 10\% of a population. A random sample of 60 people is taken from this population. The discrete random variable \(Y\) represents the number of left-handed people in the sample.
    1. Write down an expression for the exact value of \(\mathrm{P}(Y \leq 1)\)
    2. Evaluate your expression, giving your answer to 3 significant figures. [3]
  2. Using a Poisson approximation, estimate \(\mathrm{P}(Y \leq 1)\) [2]
  3. Using a normal approximation, estimate \(\mathrm{P}(Y \leq 1)\) [5]
  4. Give a reason why the Poisson approximation is a more suitable estimate of \(\mathrm{P}(Y \leq 1)\) [1]
Edexcel S2 2016 January Q4
12 marks Standard +0.3
A continuous random variable \(X\) has cumulative distribution function $$\mathrm{F}(x) = \begin{cases} 0 & x < 0 \\ \frac{1}{4}x & 0 \leq x \leq 1 \\ \frac{1}{20}x^4 + \frac{1}{5} & 1 < x \leq d \\ 1 & x > d \end{cases}$$
  1. Show that \(d = 2\) [2]
  2. Find \(\mathrm{P}(X < 1.5)\) [2]
  3. Write down the value of the lower quartile of \(X\) [1]
  4. Find the median of \(X\) [3]
  5. Find, to 3 significant figures, the value of \(k\) such that \(\mathrm{P}(X > 1.9) = \mathrm{P}(X < k)\) [4]