Questions — Edexcel S1 (606 questions)

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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]
Edexcel S1 Q5
13 marks Moderate -0.8
The students in a large Sixth Form can choose to do exactly one of Community Service, Games or Private Study on Wednesday afternoons. The probabilities that a randomly chosen student does Games and Private Study are \(\frac{3}{8}\) and \(\frac{1}{5}\) respectively. It may be assumed that the number of students is large enough for these probabilities to be treated as constant.
  1. Find the probability that a randomly chosen student does Community Service. [2 marks]
  2. If two students are chosen at random, find the probability that they both do the same activity. [3 marks]
  3. If three students are chosen at random, find the probability that exactly one of them does Games. [3 marks]
Two-fifths of the students are girls, and a quarter of these girls do Private Study.
  1. Find the probability that a randomly chosen student who does Private Study is a boy. [5 marks]
Edexcel S1 Q6
13 marks Standard +0.3
Two variables \(x\) and \(y\) are such that, for a sample of ten pairs of values, $$\sum x = 104.5, \quad \sum y = 113.6, \quad \sum x^2 = 1954.1, \quad \sum y^2 = 2100.6.$$ The regression line of \(x\) on \(y\) has gradient 0.8. Find
  1. \(\sum xy\), [4 marks]
  2. the equation of the regression line of \(y\) on \(x\), [5 marks]
  3. the product moment correlation coefficient between \(y\) and \(x\). [3 marks]
  4. Describe the kind of correlation indicated by your answer to (c). [1 mark]
Edexcel S1 Q7
21 marks Standard +0.3
The following table gives the weights, in grams, of 60 items delivered to a company in a day.
Weight (g)0 - 1010 - 2020 - 3030 - 4040 - 5050 - 6060 - 80
No. of items2111812962
  1. Use interpolation to calculate estimated values of
    1. the median weight,
    2. the interquartile range,
    3. the thirty-third percentile.
    [7 marks]
Outliers are defined to be outside the range from \(2.5Q_1 - 1.5Q_2\) to \(2.5Q_2 - 1.5Q_1\).
  1. Given that the lightest item weighed 3 g and the two heaviest weighed 65 g and 79 g, draw on graph paper an accurate box-and-whisker plot of the data. Indicate any outliers clearly. [5 marks]
  2. Describe the skewness of the distribution. [1 mark]
The mean weight was 32.0 g and the standard deviation of the weights was 14.9 g.
  1. State, with a reason, whether you would choose to summarise the data by using the mean and standard deviation or the median and interquartile range. [2 marks]
On another day, items were delivered whose weights ranged from 14 g to 58 g; the median was 32 g, the lower quartile was 24 g and the interquartile range was 26 g.
  1. Draw a further box plot for these data on the same diagram. Briefly compare the two sets of data using your plots. [6 marks]
Edexcel S1 Q1
8 marks Moderate -0.8
An athlete believes that her times for running 200 metres in races are normally distributed with a mean of 22.8 seconds.
  1. Given that her time is over 23.3 seconds in 20\% of her races, calculate the variance of her times. [5]
  2. The record over this distance for women at her club is 21.82 seconds. According to her model, what is the chance that she will beat this record in her next race? [3]
Edexcel S1 Q2
10 marks Moderate -0.3
The events \(A\) and \(B\) are such that $$\text{P}(A) = \frac{5}{16}, \text{P}(B) = \frac{1}{2} \text{ and P}(A|B) = \frac{1}{4}$$ Find
  1. P\((A \cap B)\). [2]
  2. P\((B'|A)\). [3]
  3. P\((A' \cup B)\). [2]
  4. Determine, with a reason, whether or not the events \(A\) and \(B\) are independent. [3]
Edexcel S1 Q3
10 marks Moderate -0.3
A group of 60 children were each asked to choose an integer value between 1 and 9 inclusive. Their choices are summarised in the table below.
Value chosen123456789
Number of children345101213742
  1. Calculate the mean and standard deviation of the values chosen. [6]
It is suggested that the value chosen could be modelled by a discrete uniform distribution.
  1. Write down the mean that this model would predict. [2]
Given also that the standard deviation according to this model would be 2.58,
  1. explain why this model is not suitable and suggest why this is the case. [2]
Edexcel S1 Q4
13 marks Moderate -0.3
A six-sided die is biased such that there is an equal chance of scoring each of the numbers from 1 to 5 but a score of 6 is three times more likely than each of the other numbers.
  1. Write down the probability distribution for the random variable, \(X\), the score on a single throw of the die. [4]
  2. Show that E\((X) = \frac{33}{8}\). [3]
  3. Find E\((4X - 1)\). [2]
  4. Find Var\((X)\). [4]
Edexcel S1 Q5
17 marks Moderate -0.3
The number of patients attending a hospital trauma clinic each day was recorded over several months, giving the data in the table below.
Number of patients10 - 1920 - 2930 - 3435 - 3940 - 4445 - 4950 - 69
Frequency218243027145
These data are represented by a histogram. Given that the bar representing the 20 - 29 group is 2 cm wide and 7.2 cm high,
  1. calculate the dimensions of the bars representing the groups
    1. 30 - 34
    2. 50 - 69
    [6]
  2. Use linear interpolation to estimate the median and quartiles of these data. [6]
The lowest and highest numbers of patients recorded were 14 and 67 respectively.
  1. Represent these data with a boxplot drawn on graph paper and describe the skewness of the distribution. [5]
Edexcel S1 Q6
17 marks Moderate -0.3
Penshop have stores selling stationary in each of 6 towns. The population, \(P\), in tens of thousands and the monthly turnover, \(T\), in thousands of pounds for each of the shops are as recorded below.
TownAbbertonBemberClasterDellerEdgetonFigland
\(P\) (0.000's)3.27.65.29.08.14.8
\(T\) (£ 000's)11.112.413.319.317.911.8
  1. Represent these data on a scatter diagram with \(T\) on the vertical axis. [4]
    1. Which town's shop might appear to be underachieving given the populations of the towns?
    2. Suggest two other factors that might affect each shop's turnover. [3]
You may assume that $$\Sigma P = 37.9, \quad \Sigma T = 85.8, \quad \Sigma P^2 = 264.69, \quad \Sigma T^2 = 1286, \quad \Sigma PT = 574.25.$$
  1. Find the equation of the regression line of \(T\) on \(P\). [7]
  2. Estimate the monthly turnover that might be expected if a shop were opened in Gratton, a town with a population of 68 000. [2]
  3. Why might the management of Penshop be reluctant to use the regression line to estimate the monthly turnover they could expect if a shop were opened in Haggin, a town with a population of 172 000? [1]
Edexcel S1 Q1
6 marks Moderate -0.8
  1. Draw two separate scatter diagrams, each with eight points, to illustrate the relationship between \(x\) and \(y\) in the cases where they have a product moment correlation coefficient equal to
    1. exactly \(+1\),
    2. about \(-0.4\). [4 marks]
  2. Explain briefly how the conclusion you would draw from a product moment correlation coefficient of \(+0.3\) would vary according to the number of pairs of data used in its calculation. [2 marks]
Edexcel S1 Q2
6 marks Moderate -0.8
A histogram was drawn to show the distribution of age in completed years of the participants on an outward-bound course. There were 32 people aged 30-34 years on the course. The height of the rectangle representing this group was 19.2 cm and it was 1 cm in width. Given that there were 28 people aged 35-39 years,
  1. find the height of the rectangle representing this group. [3 marks]
Given that the height of the rectangle representing people aged 40-59 years was 2.7 cm,
  1. find the number of people on the course in this age group. [3 marks]
Edexcel S1 Q3
9 marks Moderate -0.3
The events \(A\) and \(B\) are such that $$\text{P}(A) = \frac{7}{12}, \quad \text{P}(A \cap B) = \frac{1}{4} \quad \text{and} \quad \text{P}(A|B) = \frac{2}{3}.$$ Find
  1. P\((B)\), [3 marks]
  2. P\((A \cup B)\), [3 marks]
  3. P\((B|A')\). [3 marks]
Edexcel S1 Q4
12 marks Standard +0.3
The owner of a mobile burger-bar believes that hot weather reduces his sales. To investigate the effect on his business he collected data on his daily sales, \(£P\), and the maximum temperature, \(T\)°C, on each of 20 days. He then coded the data, using \(x = T - 20\) and \(y = P - 300\), and calculated the summary statistics given below. $$\Sigma x = 57, \quad \Sigma y = 2222, \quad \Sigma x^2 = 401, \quad \Sigma y^2 = 305576, \quad \Sigma xy = 3871.$$
  1. Find an equation of the regression line of \(P\) on \(T\). [9 marks]
The owner of the bar doesn't believe it is profitable for him to run the bar if he takes less than £460 in a day.
  1. According to your regression line at what maximum daily temperature, to the nearest degree Celsius, does it become unprofitable for him to run the bar? [3 marks]
Edexcel S1 Q5
13 marks Moderate -0.8
The discrete random variable \(X\) has the probability function shown below. $$P(X = x) = \begin{cases} kx, & x = 2, 3, 4, 5, 6, \\ 0, & \text{otherwise}. \end{cases}$$
  1. Find the value of \(k\). [2 marks]
  2. Show that E\((X) = \frac{9}{2}\). [3 marks]
Find
  1. P\([X > \text{E}(X)]\), [2 marks]
  2. E\((2X - 5)\), [2 marks]
  3. Var\((X)\). [4 marks]
Edexcel S1 Q6
14 marks Standard +0.3
A geologist is analysing the size of quartz crystals in a sample of granite. She estimates that the longest diameter of 75% of the crystals is greater than 2 mm, but only 10% of the crystals have a longest diameter of more than 6 mm. The geologist believes that the distribution of the longest diameters of the quartz crystals can be modelled by a normal distribution.
  1. Find the mean and variance of this normal distribution. [9 marks]
The geologist also estimated that only 2% of the longest diameters were smaller than 1 mm.
  1. Calculate the corresponding percentage that would be predicted by a normal distribution with the parameters you calculated in part \((a)\). [3 marks]
  2. Hence, comment on the suitability of the normal distribution as a model in this situation. [2 marks]
Edexcel S1 Q7
15 marks Moderate -0.8
Jane and Tahira play together in a basketball team. The list below shows the number of points that Jane scored in each of 30 games.
39192830182123153424
29174312242541192640
45232132372418152436
  1. Construct a stem and leaf diagram for these data. [3 marks]
  2. Find the median and quartiles for these data. [4 marks]
  3. Represent these data with a boxplot. [3 marks]
Tahira played in the same 30 games and her lowest and highest points total in a game were 19 and 41 respectively. The quartiles for Tahira were 27, 31 and 35 respectively.
  1. Using the same scale draw a boxplot for Tahira's points totals. [2 marks]
  2. Compare and contrast the number of points scored per game by Jane and Tahira. [3 marks]