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Edexcel S1 2004 November Q3
12 marks Standard +0.3
3. The random variable \(X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\). It is known that $$\mathrm { P } ( X \leq 66 ) = 0.0359 \text { and } \mathrm { P } ( X \geq 81 ) = 0.1151 .$$
  1. In the space below, give a clearly labelled sketch to represent these probabilities on a Normal curve.
    1. Show that the value of \(\sigma\) is 5 .
    2. Find the value of \(\mu\).
  3. Find \(\mathrm { P } ( 69 \leq X \leq 83 )\).
Edexcel S1 2004 November Q4
14 marks Easy -1.3
4. The discrete random variable \(X\) has probability function $$\mathrm { P } ( X = x ) = \begin{array} { l l } 0.2 , & x = - 3 , - 2 \\ \alpha , & x = - 1,0 \\ 0.1 , & x = 1,2 . \end{array}$$ Find
  1. \(\alpha\),
  2. \(\mathrm { P } ( - 1 \leq X < 2 )\),
  3. \(\mathrm { F } ( 0.6 )\),
  4. the value of \(a\) such that \(\mathrm { E } ( a X + 3 ) = 1.2\),
  5. \(\operatorname { Var } ( X )\),
  6. \(\operatorname { Var } ( 3 X - 2 )\).
Edexcel S1 2004 November Q5
7 marks Easy -1.3
5. The events \(A\) and \(B\) are such that \(\mathrm { P } ( A ) = \frac { 1 } { 2 } , \mathrm { P } ( B ) = \frac { 1 } { 3 }\) and \(\mathrm { P } ( A \cap B ) = \frac { 1 } { 4 }\).
  1. Using the space below, represent these probabilities in a Venn diagram. Hence, or otherwise, find
  2. \(\mathrm { P } ( A \cup B )\),
  3. \(\mathrm { P } \left( \begin{array} { l l } A & B ^ { \prime } \end{array} \right)\)
Edexcel S1 2004 November Q6
18 marks Easy -1.2
6. Students in Mr Brawn's exercise class have to do press-ups and sit-ups. The number of press-ups \(x\) and the number of sit-ups \(y\) done by a random sample of 8 students are summarised below. $$\begin{array} { l l } \Sigma x = 272 , & \Sigma x ^ { 2 } = 10164 , \quad \Sigma x y = 11222 , \\ \Sigma y = 320 , & \Sigma y ^ { 2 } = 13464 . \end{array}$$
  1. Evaluate \(S _ { x x } , S _ { y y }\) and \(S _ { x y }\).
  2. Calculate, to 3 decimal places, the product moment correlation coefficient between \(x\) and \(y\).
  3. Give an interpretation of your coefficient.
  4. Calculate the mean and the standard deviation of the number of press-ups done by these students. Mr Brawn assumes that the number of press-ups that can be done by any student can be modelled by a normal distribution with mean \(\mu\) and standard deviation \(\sigma\). Assuming that \(\mu\) and \(\sigma\) take the same values as those calculated in part (d),
  5. find the value of \(a\) such that \(\mathrm { P } ( \mu - a < X < \mu + a ) = 0.95\).
  6. Comment on Mr Brawn's assumption of normality.
Edexcel S1 2004 November Q7
6 marks Easy -1.8
7. A college organised a 'fun run'. The times, to the nearest minute, of a random sample of 100 students who took part are summarised in the table below.
TimeNumber of students
\(40 - 44\)10
\(45 - 47\)15
4823
\(49 - 51\)21
\(52 - 55\)16
\(56 - 60\)15
  1. Give a reason to support the use of a histogram to represent these data.
  2. Write down the upper class boundary and the lower class boundary of the class 40-44.
  3. On graph paper, draw a histogram to represent these data. END
Edexcel S1 Specimen Q1
4 marks Standard +0.3
  1. (a) Explain what you understand by a statistical model.
    (2)
    (b) Write down a random variable which could be modelled by
    1. a discrete uniform distribution,
    2. a normal distribution.
    3. 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 \(T\).
Edexcel S1 Specimen Q3
14 marks Moderate -0.8
3. The discrete random variable \(X\) has probability function \(\mathrm { P } ( X = x ) = \begin{cases} k x , & x = 1,2,3,4,5 , \\ 0 , & \text { otherwise } . \end{cases}\)
  1. Show that \(k = \frac { 1 } { 15 }\). Find the value of
  2. \(\mathrm { E } ( 2 X + 3 )\),
  3. \(\operatorname { Var } ( 2 X - 4 )\).
    (6)
Edexcel S1 Specimen Q4
14 marks Standard +0.3
4. 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. \(\Sigma x = 143 ; \Sigma y = 391 ; \Sigma x ^ { 2 } = 2413 ; \Sigma y ^ { 2 } = 22441 ; \Sigma x y = 484\).
  1. Find the equation of the regression line of \(h\) on \(s\).
  2. Interpret the slope of your regression line. Estimate the life of a drill revolving at 30 revolutions per minute.
    (2)
Edexcel S1 Specimen Q5
16 marks Easy -1.2
5. (a) Explain briefly the advantages and disadvantages of using the quartiles to summarise a set of data.
(b) Describe the main features and uses of a box plot. 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. \section*{Diagram 1}
\includegraphics[max width=\textwidth, alt={}]{516911a4-d55e-4008-bad5-7c97bea94f9f-4_540_1244_772_390}
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.
(c) Draw a box plot for School B.
(d) Compare and contrast the two box plots.
(4)
Edexcel S1 Specimen Q6
18 marks Moderate -0.8
6. 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. Draw a Venn diagram to represent these data. Find the probability that
  3. only one of them has a degree,
  4. neither of them has a degree. Two married couples are chosen at random.
  5. Find the probability that only one of the two husbands and only one of the two wives have degrees.
Edexcel S2 2014 January Q1
8 marks Moderate -0.3
  1. The probability of a leaf cutting successfully taking root is 0.05
Find the probability that, in a batch of 10 randomly selected leaf cuttings, the number taking root will be
    1. exactly 1
    2. more than 2 A second random sample of 160 leaf cuttings is selected.
  2. Using a suitable approximation, estimate the probability of at least 10 leaf cuttings taking root.
Edexcel S2 2014 January Q2
10 marks Moderate -0.3
2. Bill owns a restaurant. Over the next four weeks Bill decides to carry out a sample survey to obtain the customers’ opinions.
  1. Suggest a suitable sampling frame for the sample survey.
  2. Identify the sampling units.
  3. Give one advantage and one disadvantage of taking a census rather than a sample survey. Bill believes that only \(30 \%\) of customers would like a greater choice on the menu. He takes a random sample of 50 customers and finds that 20 of them would like a greater choice on the menu.
  4. Test, at the \(5 \%\) significance level, whether or not the percentage of customers who would like a greater choice on the menu is more than Bill believes. State your hypotheses clearly.
Edexcel S2 2014 January Q3
11 marks Standard +0.3
  1. The continuous random variable \(X\) has cumulative distribution function given by
$$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < 0 \\ \frac { 1 } { 6 } x ( x + 1 ) & 0 \leqslant x \leqslant 2 \\ 1 & x > 2 \end{array} \right.$$
  1. Find the value of \(a\) such that \(\mathrm { P } ( X > a ) = 0.4\) Give your answer to 3 significant figures.
  2. Use calculus to find (i) \(\mathrm { E } ( X )\) (ii) \(\operatorname { Var } ( X )\).
Edexcel S2 2014 January Q4
7 marks Standard +0.3
  1. The number of telephone calls per hour received by a business is a random variable with distribution \(\operatorname { Po } ( \lambda )\).
Charlotte records the number of calls, \(C\), received in 4 hours. A test of the null hypothesis \(\mathrm { H } _ { 0 } : \lambda = 1.5\) is carried out. \(\mathrm { H } _ { 0 }\) is rejected if \(C > 10\)
  1. Write down the alternative hypothesis.
  2. Find the significance level of the test. Given that \(\mathrm { P } ( C > 10 ) < 0.1\)
  3. find the largest possible value of \(\lambda\) that can be found by using the tables.
Edexcel S2 2014 January Q5
12 marks Standard +0.8
5. A school photocopier breaks down randomly at a rate of 15 times per year.
  1. Find the probability that there will be exactly 3 breakdowns in the next month.
  2. Show that the probability that there will be at least 2 breakdowns in the next month is 0.355 to 3 decimal places.
  3. Find the probability of at least 2 breakdowns in each of the next 4 months. The teachers would like a new photocopier. The head teacher agrees to monitor the situation for the next 12 months. The head teacher decides he will buy a new photocopier if there is more than 1 month when the photocopier has at least 2 breakdowns.
  4. Find the probability that the head teacher will buy a new photocopier.
Edexcel S2 2014 January Q6
15 marks Standard +0.3
  1. The continuous random variable \(X\) has probability density function given by
$$f ( x ) = \left\{ \begin{array} { c c } k ( x + 1 ) ^ { 2 } & - 1 \leqslant x \leqslant 1 \\ k ( 6 - 2 x ) & 1 < x \leqslant 3 \\ 0 & \text { otherwise } \end{array} \right.$$ where \(k\) is a positive constant.
  1. Sketch the graph of \(\mathrm { f } ( x )\).
  2. Show that the value of \(k\) is \(\frac { 3 } { 20 }\)
  3. Define fully the cumulative distribution function \(\mathrm { F } ( x )\).
  4. Find the median of \(X\), giving your answer to 3 significant figures.
Edexcel S2 2014 January Q7
12 marks Challenging +1.2
  1. The random variable \(Y \sim \mathrm {~B} ( n , p )\).
Using a normal approximation the probability that \(Y\) is at least 65 is 0.2266 and the probability that \(Y\) is more than 52 is 0.8944 Find the value of \(n\) and the value of \(p\).
Edexcel S2 2015 January Q1
16 marks Standard +0.8
  1. The number of cars caught speeding per day, by a particular camera, has a Poisson distribution with mean 0.8
    1. Find the probability that in a given 4 day period exactly 3 cars will be caught speeding by this camera.
    A car has been caught speeding by this camera.
  2. Find the probability that the period of time that elapses before the next car is caught speeding by this camera is less than 48 hours. Given that 4 cars were caught speeding by this camera in a two day period,
  3. find the probability that 1 was caught on the first day and 3 were caught on the second day. Each car that is caught speeding by this camera is fined \(\pounds 60\)
  4. Using a suitable approximation, find the probability that, in 90 days, the total amount of fines issued will be more than \(\pounds 5000\)
Edexcel S2 2015 January Q2
11 marks Moderate -0.8
2. A continuous random variable \(X\) has cumulative distribution function $$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < 1 \\ \frac { 1 } { 5 } ( x - 1 ) & 1 \leqslant x \leqslant 6 \\ 1 & x > 6 \end{array} \right.$$
  1. Find \(\mathrm { P } ( X > 4 )\)
  2. Write down the value of \(\mathrm { P } ( X \neq 4 )\)
  3. Find the probability density function of \(X\), specifying it for all values of \(X\)
  4. Write down the value of \(\mathrm { E } ( X )\)
  5. Find \(\operatorname { Var } ( X )\)
  6. Hence or otherwise find \(\mathrm { E } \left( 3 X ^ { 2 } + 1 \right)\)
Edexcel S2 2015 January Q3
11 marks Moderate -0.8
3. Explain what you understand by
  1. a statistic,
  2. a sampling distribution. A factory stores screws in packets. A small packet contains 100 screws and a large packet contains 200 screws. The factory keeps small and large packets in the ratio 4:3 respectively.
  3. Find the mean and the variance of the number of screws in the packets stored at the factory. A random sample of 3 packets is taken from the factory and \(Y _ { 1 } , Y _ { 2 }\) and \(Y _ { 3 }\) denote the number of screws in each of these packets.
  4. List all the possible samples.
  5. Find the sampling distribution of \(\bar { Y }\)
Edexcel S2 2015 January Q4
7 marks Standard +0.3
4. Accidents occur randomly at a crossroads at a rate of 0.5 per month. A researcher records the number of accidents, \(X\), which occur at the crossroads in a year.
  1. Find \(\mathrm { P } ( 5 \leqslant X < 7 )\) A new system is introduced at the crossroads. In the first 18 months, 4 accidents occur at the crossroads.
  2. Test, at the \(5 \%\) level of significance, whether or not there is reason to believe that the new system has led to a reduction in the mean number of accidents per month. State your hypotheses clearly.
Edexcel S2 2015 January Q5
9 marks Standard +0.8
5. The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by $$f ( x ) = \left\{ \begin{array} { c c } k \left( x ^ { 2 } + a \right) & - 1 < x \leqslant 2 \\ 3 k & 2 < x \leqslant 3 \\ 0 & \text { otherwise } \end{array} \right.$$ where \(k\) and \(a\) are constants.
Given that \(\mathrm { E } ( X ) = \frac { 17 } { 12 }\)
  1. find the value of \(k\) and the value of \(a\)
  2. Write down the mode of \(X\)
Edexcel S2 2015 January Q6
13 marks Standard +0.8
6. The Headteacher of a school claims that \(30 \%\) of parents do not support a new curriculum. In a survey of 20 randomly selected parents, the number, \(X\), who do not support the new curriculum is recorded. Assuming that the Headteacher's claim is correct, find
  1. the probability that \(X = 5\)
  2. the mean and the standard deviation of \(X\) The Director of Studies believes that the proportion of parents who do not support the new curriculum is greater than \(30 \%\). Given that in the survey of 20 parents 8 do not support the new curriculum,
  3. test, at the \(5 \%\) level of significance, the Director of Studies' belief. State your hypotheses clearly. The teachers believe that the sample in the original survey was biased and claim that only \(25 \%\) of the parents are in support of the new curriculum. A second random sample, of size \(2 n\), is taken and exactly half of this sample supports the new curriculum. A test is carried out at a 10\% level of significance of the teachers' belief using this sample of size \(2 n\) Using the hypotheses \(\mathrm { H } _ { 0 } : p = 0.25\) and \(\mathrm { H } _ { 1 } : p > 0.25\)
  4. find the minimum value of \(n\) for which the outcome of the test is that the teachers' belief is rejected.
Edexcel S2 2015 January Q7
8 marks Standard +0.8
7. A multiple choice examination paper has \(n\) questions where \(n > 30\) Each question has 5 answers of which only 1 is correct. A pass on the paper is obtained by answering 30 or more questions correctly. The probability of obtaining a pass by randomly guessing the answer to each question should not exceed 0.0228 Use a normal approximation to work out the greatest number of questions that could be used.
Edexcel S2 2016 January Q1
5 marks Moderate -0.8
  1. 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.
    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.
  2. Write down a suitable distribution to model the number of customers in this sample that complete the survey.
  3. Find the probability that more than half of the customers in the sample complete the survey.