Questions S1 (2020 questions)

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Edexcel S1 Q2
6 marks Standard +0.3
The random variable \(X\) has the discrete uniform distribution and takes the values \(\{1, \ldots, n\}\). The standard deviation of of \(X\) is \(2\sqrt{6}\). Find
  1. the mean of \(X\), [3 marks]
  2. P\((3 \leq X < \frac{2}{5}n)\). [3 marks]
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]
OCR S1 2010 January Q1
9 marks Moderate -0.8
Andy makes repeated attempts to thread a needle. The number of attempts up to and including his first success is denoted by \(X\).
  1. State two conditions necessary for \(X\) to have a geometric distribution. [2]
  2. Assuming that \(X\) has the distribution Geo(0.3), find
    1. P\((X = 5)\), [2]
    2. P\((X > 5)\). [3]
  3. Suggest a reason why one of the conditions you have given in part (i) might not be satisfied in this context. [2]
OCR S1 2010 January Q2
13 marks Moderate -0.8
40 people were asked to guess the length of a certain road. Each person gave their guess, \(l\) km, correct to the nearest kilometre. The results are summarised below.
\(l\)10-1213-1516-2021-30
Frequency113206
    1. Use appropriate formulae to calculate estimates of the mean and standard deviation of \(l\). [6]
    2. Explain why your answers are only estimates. [1]
  2. A histogram is to be drawn to illustrate the data. Calculate the frequency density of the block for the 16-20 class. [2]
  3. Explain which class contains the median value of \(l\). [2]
  4. Later, the person whose guess was between 10 km and 12 km changed his guess to between 13 km and 15 km. Without calculation state whether the following will increase, decrease or remain the same:
    1. the mean of \(l\), [1]
    2. the standard deviation of \(l\). [1]
OCR S1 2010 January Q3
7 marks Moderate -0.8
The heights, \(h\) m, and weights, \(m\) kg, of five men were measured. The results are plotted on the diagram. \includegraphics{figure_3} The results are summarised as follows. \(n = 5\) \(\Sigma h = 9.02\) \(\Sigma m = 377.7\) \(\Sigma h^2 = 16.382\) \(\Sigma m^2 = 28558.67\) \(\Sigma hm = 681.612\)
  1. Use the summarised data to calculate the value of the product moment correlation coefficient, \(r\). [3]
  2. Comment on your value of \(r\) in relation to the diagram. [2]
  3. It was decided to re-calculate the value of \(r\) after converting the heights to feet and the masses to pounds. State what effect, if any, this will have on the value of \(r\). [1]
  4. One of the men had height 1.63 m and mass 78.4 kg. The data for this man were removed and the value of \(r\) was re-calculated using the original data for the remaining four men. State in general terms what effect, if any, this will have on the value of \(r\). [1]
OCR S1 2010 January Q4
10 marks Moderate -0.3
A certain four-sided die is biased. The score, \(X\), on each throw is a random variable with probability distribution as shown in the table. Throws of the die are independent.
\(x\)0123
P\((X = x)\)\(\frac{1}{2}\)\(\frac{1}{4}\)\(\frac{1}{8}\)\(\frac{1}{8}\)
  1. Calculate E\((X)\) and Var\((X)\). [5]
The die is thrown 10 times.
  1. Find the probability that there are not more than 4 throws on which the score is 1. [2]
  2. Find the probability that there are exactly 4 throws on which the score is 2. [3]
OCR S1 2010 January Q5
6 marks Moderate -0.8
A washing-up bowl contains 6 spoons, 5 forks and 3 knives. Three of these 14 items are removed at random, without replacement. Find the probability that
  1. all three items are of different kinds, [3]
  2. all three items are of the same kind. [3]
OCR S1 2010 January Q6
7 marks Standard +0.3
  1. A student calculated the values of the product moment correlation coefficient, \(r\), and Spearman's rank correlation coefficient, \(r_s\), for two sets of bivariate data, \(A\) and \(B\). His results are given below. $$A: \quad r = 0.9 \text{ and } r_s = 1$$ $$B: \quad r = 1 \quad \text{and } r_s = 0.9$$ With the aid of a diagram where appropriate, explain why the student's results for \(A\) could both be correct but his results for \(B\) cannot both be correct. [3]
  2. An old research paper has been partially destroyed. The surviving part of the paper contains the following incomplete information about some bivariate data from an experiment. \includegraphics{figure_6} The mean of \(x\) is 4.5. The equation of the regression line of \(y\) on \(x\) is \(y = 2.4x + 3.7\). The equation of the regression line of \(x\) on \(y\) is \(x = 0.40y\) + [missing constant] Calculate the missing constant at the end of the equation of the second regression line. [4]
OCR S1 2010 January Q7
6 marks Moderate -0.8
The table shows the numbers of male and female members of a vintage car club who own either a Jaguar or a Bentley. No member owns both makes of car.
MaleFemale
Jaguar2515
Bentley128
One member is chosen at random from these 60 members.
  1. Given that this member is male, find the probability that he owns a Jaguar. [2]
Now two members are chosen at random from the 60 members. They are chosen one at a time, without replacement.
  1. Given that the first one of these members is female, find the probability that both own Jaguars. [4]
OCR S1 2010 January Q8
7 marks Moderate -0.8
The five letters of the word NEVER are arranged in random order in a straight line.
  1. How many different orders of the letters are possible? [2]
  2. In how many of the possible orders are the two Es next to each other? [2]
  3. Find the probability that the first two letters in the order include exactly one letter E. [3]
OCR S1 2010 January Q9
7 marks Standard +0.8
\(R\) and \(S\) are independent random variables each having the distribution Geo\((p)\).
  1. Find P\((R = 1\) and \(S = 1)\) in terms of \(p\). [1]
  2. Show that P\((R = 3\) and \(S = 3) = p^2q^4\), where \(q = 1 - p\). [1]
  3. Use the formula for the sum to infinity of a geometric series to show that $$\text{P}(R = S) = \frac{p}{2-p}.$$ [5]
OCR S1 2013 January Q1
7 marks Moderate -0.8
When a four-sided spinner is spun, the number on which it lands is denoted by \(X\), where \(X\) is a random variable taking values 2, 4, 6 and 8. The spinner is biased so that P(\(X = x\)) = \(kx\), where \(k\) is a constant.
  1. Show that P(\(X = 6\)) = \(\frac{3}{10}\). [2]
  2. Find E(\(X\)) and Var(\(X\)). [5]
OCR S1 2013 January Q2
6 marks Moderate -0.8
  1. Kathryn is allowed three attempts at a high jump. If she succeeds on any attempt, she does not jump again. The probability that she succeeds on her first attempt is \(\frac{1}{4}\). If she fails on her first attempt, the probability that she succeeds on her second attempt is \(\frac{1}{3}\). If she fails on her first two attempts, the probability that she succeeds on her third attempt is \(\frac{1}{2}\). Find the probability that she succeeds. [3]
  2. Khaled is allowed two attempts to pass an examination. If he succeeds on his first attempt, he does not make a second attempt. The probability that he passes at the first attempt is 0.4 and the probability that he passes on either the first or second attempt is 0.58. Find the probability that he passes on the second attempt, given that he failed on the first attempt. [3]
OCR S1 2013 January Q3
12 marks Moderate -0.3
The Gross Domestic Product per Capita (GDP), \(x\) dollars, and the Infant Mortality Rate per thousand (IMR), \(y\), of 6 African countries were recorded and summarised as follows. \(n = 6\) \quad \(\sum x = 7000\) \quad \(\sum x^2 = 8700000\) \quad \(\sum y = 456\) \quad \(\sum y^2 = 36262\) \quad \(\sum xy = 509900\)
  1. Calculate the equation of the regression line of \(y\) on \(x\) for these 6 countries. [4]
The original data were plotted on a scatter diagram and the regression line of \(y\) on \(x\) was drawn, as shown below. \includegraphics{figure_3}
  1. The GDP for another country, Tanzania, is 1300 dollars. Use the regression line in the diagram to estimate the IMR of Tanzania. [1]
  2. The GDP for Nigeria is 2400 dollars. Give two reasons why the regression line is unlikely to give a reliable estimate for the IMR for Nigeria. [2]
  3. The actual value of the IMR for Tanzania is 96. The data for Tanzania (\(x = 1300, y = 96\)) is now included with the original 6 countries. Calculate the value of the product moment correlation coefficient, \(r\), for all 7 countries. [4]
  4. The IMR is now redefined as the infant mortality rate per hundred instead of per thousand, and the value of \(r\) is recalculated for all 7 countries. Without calculation state what effect, if any, this would have on the value of \(r\) found in part (iv). [1]