Questions — Edexcel S1 (606 questions)

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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 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]