Questions — Edexcel S1 (606 questions)

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Edexcel S1 2002 January Q2
7 marks Moderate -0.8
A meteorologist measured the number of hours of sunshine, to the nearest hour, each day for 100 days. The results are summarised in the table below.
Hours of sunshineDays
116
2-432
5-628
712
89
9-112
121
  1. On graph paper, draw a histogram to represent these data. [5]
  2. Calculate an estimate of the number of days that had between 6 and 9 hours of sunshine. [2]
Edexcel S1 2002 January Q3
7 marks Moderate -0.8
A discrete random variable \(X\) has the probability function shown in the table below.
\(x\)012
P(\(X = x\))\(\frac{1}{3}\)\(a\)\(\frac{2}{3} - a\)
  1. Given that E(\(X\)) = \(\frac{2}{3}\), find \(a\). [3]
  2. Find the exact value of Var (\(X\)). [3]
  3. Find the exact value of P(\(X \leq 15\)). [1]
Edexcel S1 2002 January Q4
10 marks Moderate -0.8
A contractor bids for two building projects. He estimates that the probability of winning the first project is 0.5, the probability of winning the second is 0.3 and the probability of winning both projects is 0.2.
  1. Find the probability that he does not win either project. [3]
  2. Find the probability that he wins exactly one project. [2]
  3. Given that he does not win the first project, find the probability that he wins the second. [2]
  4. By calculation, determine whether or not winning the first contract and winning the second contract are independent events. [3]
Edexcel S1 2002 January Q5
11 marks Standard +0.3
The duration of the pregnancy of a certain breed of cow is normally distributed with mean \(\mu\) days and standard deviation \(\sigma\) days. Only 2.5\% of all pregnancies are shorter than 235 days and 15\% are longer than 286 days.
  1. Show that \(\mu - 235 = 1.96\sigma\). [2]
  2. Obtain a second equation in \(\mu\) and \(\sigma\). [3]
  3. Find the value of \(\mu\) and the value of \(\sigma\). [4]
  4. Find the values between which the middle 68.3\% of pregnancies lie. [2]
Edexcel S1 2002 January Q6
17 marks Easy -1.2
Hospital records show the number of babies born in a year. The number of babies delivered by 15 male doctors is summarised by the stem and leaf diagram below. Babies \quad (4|5 means 45) \quad Totals 0 \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad (0) 1|9 \quad \quad \quad \quad \quad \quad \quad \quad \quad (1) 2|1 6 7 7 \quad \quad \quad \quad \quad \quad (4) 3|2 2 3 4 8 \quad \quad \quad \quad \quad (5) 4|5 \quad \quad \quad \quad \quad \quad \quad \quad \quad (1) 5|1 \quad \quad \quad \quad \quad \quad \quad \quad \quad (1) 6|0 \quad \quad \quad \quad \quad \quad \quad \quad \quad (1) 7 \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad (0) 8|6 7 \quad \quad \quad \quad \quad \quad \quad \quad (2)
  1. Find the median and inter-quartile range of these data. [3]
  2. Given that there are no outliers, draw a box plot on graph paper to represent these data. Start your scale at the origin. [4]
  3. Calculate the mean and standard deviation of these data. [5]
The records also contain the number of babies delivered by 10 female doctors. 34 \quad 30 \quad 20 \quad 15 \quad 6 32 \quad 26 \quad 19 \quad 11 \quad 4 The quartiles are 11, 19.5 and 30.
  1. Using the same scale as in part (b) and on the same graph paper draw a box plot for the data for the 10 female doctors. [3]
  2. Compare and contrast the box plots for the data for male and female doctors. [2]
Edexcel S1 2002 January Q7
19 marks Moderate -0.3
A number of people were asked to guess the calorific content of 10 foods. The mean \(s\) of the guesses for each food and the true calorific content \(t\) are given in the table below.
Food\(t\)\(s\)
Packet of biscuits170420
1 potato90160
1 apple80110
Crisp breads1070
Chocolate bar260360
1 slice white bread75135
1 slice brown bread60115
Portion of beef curry270350
Portion of rice pudding165390
Half a pint of milk160200
[You may assume that \(\Sigma t = 1340\), \(\Sigma s = 2310\), \(\Sigma ts = 396775\), \(\Sigma t^2 = 246050\), \(\Sigma s^2 = 694650\).]
  1. Draw a scatter diagram, indicating clearly which is the explanatory (independent) and which is the response (dependent) variable. [3]
  2. Calculate, to 3 significant figures, the product moment correlation coefficient for the above data. [7]
  3. State, with a reason, whether or not the value of the product moment correlation coefficient changes if all the guesses are 50 calories higher than the values in the table. [2]
The mean of the guesses for the portion of rice pudding and for the packet of biscuits are outside the linear relation of the other eight foods.
  1. Find the equation of the regression line of \(s\) on \(t\) excluding the values for rice pudding and biscuits. [3]
[You may now assume that \(S_{tt} = 72587\), \(S_{st} = 63671.875\), \(\bar{t} = 125.625\), \(\bar{s} = 187.5\).]
  1. Draw the regression line on your scatter diagram. [2]
  2. State, with a reason, what the effect would be on the regression line of including the values for a portion of rice pudding and a packet of biscuits. [2]
Edexcel S1 2010 January Q1
5 marks Easy -1.3
A jar contains 2 red, 1 blue and 1 green bead. Two beads are drawn at random from the jar without replacement.
  1. In the space below, draw a tree diagram to illustrate all the possible outcomes and associated probabilities. State your probabilities clearly. [3]
  2. Find the probability that a blue bead and a green bead are drawn from the jar. [2]
Edexcel S1 2010 January Q2
9 marks Easy -1.2
The 19 employees of a company take an aptitude test. The scores out of 40 are illustrated in the stem and leaf diagram below. \(2|6\) means a score of 26 \begin{align} 0 & | 7 & (1)
1 & | 88 & (2)
2 & | 4468 & (4)
3 & | 2333459 & (7)
4 & | 00000 & (5) \end{align} Find
  1. the median score, [1]
  2. the interquartile range. [3]
The company director decides that any employees whose scores are so low that they are outliers will undergo retraining. An outlier is an observation whose value is less than the lower quartile minus 1.0 times the interquartile range.
  1. Explain why there is only one employee who will undergo retraining. [2]
  2. On the graph paper on page 5, draw a box plot to illustrate the employees' scores. [3]
Edexcel S1 2010 January Q3
11 marks Moderate -0.8
The birth weights, in kg, of 1500 babies are summarised in the table below.
Weight (kg)Midpoint, \(x\)kgFrequency, \(f\)
\(0.0 - 1.0\)\(0.50\)\(1\)
\(1.0 - 2.0\)\(1.50\)\(6\)
\(2.0 - 2.5\)\(2.25\)\(60\)
\(2.5 - 3.0\)\(280\)
\(3.0 - 3.5\)\(3.25\)\(820\)
\(3.5 - 4.0\)\(3.75\)\(320\)
\(4.0 - 5.0\)\(4.50\)\(10\)
\(5.0 - 6.0\)\(3\)
[You may use \(\sum fx = 4841\) and \(\sum fx^2 = 15889.5\)]
  1. Write down the missing midpoints in the table above. [2]
  2. Calculate an estimate of the mean birth weight. [2]
  3. Calculate an estimate of the standard deviation of the birth weight. [3]
  4. Use interpolation to estimate the median birth weight. [2]
  5. Describe the skewness of the distribution. Give a reason for your answer. [2]
Edexcel S1 2010 January Q4
9 marks Moderate -0.3
There are 180 students at a college following a general course in computing. Students on this course can choose to take up to three extra options. 112 take systems support, 70 take developing software, 81 take networking, 35 take developing software and systems support, 28 take networking and developing software, 40 take systems support and networking, 4 take all three extra options.
  1. In the space below, draw a Venn diagram to represent this information. [5]
A student from the course is chosen at random. Find the probability that this student takes
  1. none of the three extra options, [1]
  2. networking only. [1]
Students who want to become technicians take systems support and networking. Given that a randomly chosen student wants to become a technician,
  1. find the probability that this student takes all three extra options. [2]
Edexcel S1 2010 January Q5
10 marks Moderate -0.8
The probability function of a discrete random variable \(X\) is given by $$p(x) = kx^2 \quad x = 1, 2, 3$$ where \(k\) is a positive constant.
  1. Show that \(k = \frac{1}{14}\) [2]
Find
  1. P\((X \geq 2)\) [2]
  2. E\((X)\) [2]
  3. Var\((1-X)\) [4]
Edexcel S1 2010 January Q6
18 marks Moderate -0.8
The blood pressures, \(p\) mmHg, and the ages, \(t\) years, of 7 hospital patients are shown in the table below.
PatientABCDEFG
\(t\)42744835562660
\(p\)981301208818280135
[\(\sum t = 341\), \(\sum p = 833\), \(\sum t^2 = 18181\), \(\sum p^2 = 106397\), \(\sum tp = 42948\)]
  1. Find \(S_{tt}\), \(S_{pp}\) and \(S_t\) for these data. [4]
  2. Calculate the product moment correlation coefficient for these data. [3]
  3. Interpret the correlation coefficient. [1]
  4. On the graph paper on page 17, draw the scatter diagram of blood pressure against age for these 7 patients. [2]
  5. Find the equation of the regression line of \(p\) on \(t\). [4]
  6. Plot your regression line on your scatter diagram. [2]
  7. Use your regression line to estimate the blood pressure of a 40 year old patient. [2]
Edexcel S1 2010 January Q7
13 marks Standard +0.3
The heights of a population of women are normally distributed with mean \(\mu\) cm and standard deviation \(\sigma\) cm. It is known that 30% of the women are taller than 172 cm and 5% are shorter than 154 cm.
  1. Sketch a diagram to show the distribution of heights represented by this information. [3]
  2. Show that \(\mu = 154 + 1.6449\sigma\). [3]
  3. Obtain a second equation and hence find the value of \(\mu\) and the value of \(\sigma\). [4]
A woman is chosen at random from the population.
  1. Find the probability that she is taller than 160 cm. [3]
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]