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Edexcel S2 2024 January Q6
9 marks Moderate -0.3
  1. A bag contains a large number of counters with an odd number or an even number written on each.
Odd and even numbered counters occur in the ratio \(4 : 1\) In a game a player takes a random sample of 4 counters from the bag.
The player scores
5 points for each counter taken that has an even number written on it
2 points for each counter taken that has an odd number written on it
The random variable \(X\) represents the total score, in points, from the 4 counters.
  1. Find the sampling distribution of \(X\) A random sample of \(n\) sets of 4 counters is taken. The random variable \(Y\) represents the number of these \(n\) sets that have a total score of exactly 14
  2. Calculate the minimum value of \(n\) such that \(\mathrm { P } ( Y \geqslant 1 ) > 0.95\)
Edexcel S2 2024 January Q7
8 marks Challenging +1.2
  1. A continuous random variable \(X\) has cumulative distribution function \(\mathrm { F } ( x )\) given by
$$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < 1 \\ k \left( a x + b x ^ { 3 } - x ^ { 4 } - 4 \right) & 1 \leqslant x \leqslant 2 \\ 1 & x > 2 \end{array} \right.$$ where \(a\), \(b\) and \(k\) are non-zero constants.
Given that the mode of \(X\) is 1.5
  1. show that \(b = 3\)
  2. Hence show that \(a = 2\)
  3. Show that the median of \(X\) lies between 1.4 and 1.5
Edexcel S2 2014 June Q1
7 marks Easy -1.2
  1. (a) State the conditions under which the Poisson distribution may be used as an approximation to the binomial distribution.
A farmer supplies a bakery with eggs. The manager of the bakery claims that the proportion of eggs having a double yolk is 0.009 The farmer claims that the proportion of his eggs having a double yolk is more than 0.009
(b) State suitable hypotheses for testing these claims. In a batch of 500 eggs the baker records 9 eggs with a double yolk.
(c) Using a suitable approximation, test at the \(5 \%\) level of significance whether or not this supports the farmer's claim.
Edexcel S2 2014 June Q2
16 marks Moderate -0.3
2. The amount of flour used by a factory in a week is \(Y\) thousand kg where \(Y\) has probability density function $$\mathrm { f } ( y ) = \left\{ \begin{array} { c c } k \left( 4 - y ^ { 2 } \right) & 0 \leqslant y \leqslant 2 \\ 0 & \text { otherwise } \end{array} \right.$$
  1. Show that the value of \(k\) is \(\frac { 3 } { 16 }\) Use algebraic integration to find
  2. the mean number of kilograms of flour used by the factory in a week,
  3. the standard deviation of the number of kilograms of flour used by the factory in a week,
  4. the probability that more than 1500 kg of flour will be used by the factory next week.
Edexcel S2 2014 June Q3
7 marks Moderate -0.8
  1. The continuous random variable \(T\) is uniformly distributed on the interval \([ \alpha , \beta ]\) where \(\beta > \alpha\)
Given that \(\mathrm { E } ( T ) = 2\) and \(\operatorname { Var } ( T ) = \frac { 16 } { 3 }\), find
  1. the value of \(\alpha\) and the value of \(\beta\),
  2. \(\mathrm { P } ( T < 3.4 )\)
Edexcel S2 2014 June Q4
12 marks Standard +0.3
4. Pieces of ribbon are cut to length \(L \mathrm {~cm}\) where \(L \sim \mathrm {~N} \left( \mu , 0.5 ^ { 2 } \right)\)
  1. Given that \(30 \%\) of the pieces of ribbon have length more than 100 cm , find the value of \(\mu\) to the nearest 0.1 cm . John selects 12 pieces of ribbon at random.
  2. Find the probability that fewer than 3 of these pieces of ribbon have length more than 100 cm . Aditi selects 400 pieces of ribbon at random.
  3. Using a suitable approximation, find the probability that more than 127 of these pieces of ribbon will have length more than 100 cm .
Edexcel S2 2014 June Q5
7 marks Standard +0.3
5. A company claims that \(35 \%\) of its peas germinate. In order to test this claim Ann decides to plant 15 of these peas and record the number which germinate.
    1. State suitable hypotheses for a two-tailed test of this claim.
    2. Using a \(5 \%\) level of significance, find an appropriate critical region for this test. The probability in each of the tails should be as close to \(2.5 \%\) as possible.
  2. Ann found that 8 of the 15 peas germinated. State whether or not the company's claim is supported. Give a reason for your answer.
  3. State the actual significance level of this test.
Edexcel S2 2014 June Q6
10 marks Standard +0.3
6. A continuous random variable \(X\) has cumulative distribution function \(\mathrm { F } ( x )\) given by $$F ( x ) = \left\{ \begin{array} { l c } 0 & x < 0 \\ \frac { x ^ { 2 } } { 20 } ( 9 - 2 x ) & 0 \leqslant x \leqslant 2 \\ 1 & x > 2 \end{array} \right.$$
  1. Verify that the median of \(X\) lies between 1.23 and 1.24
  2. Specify fully the probability density function \(\mathrm { f } ( x )\).
  3. Find the mode of \(X\).
  4. Describe the skewness of this distribution. Justify your answer.
Edexcel S2 2014 June Q7
16 marks Standard +0.8
7. Flaws occur at random in a particular type of material at a mean rate of 2 per 50 m .
  1. Find the probability that in a randomly chosen 50 m length of this material there will be exactly 5 flaws. This material is sold in rolls of length 200 m . Susie buys 4 rolls of this material.
  2. Find the probability that only one of these rolls will have fewer than 7 flaws. A piece of this material of length \(x \mathrm {~m}\) is produced. Using a normal approximation, the probability that this piece of material contains fewer than 26 flaws is 0.5398
  3. Find the value of \(x\).
Edexcel S2 2015 June Q1
10 marks Moderate -0.3
  1. A continuous random variable \(X\) has cumulative distribution function
$$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 & x < 2 \\ \frac { 1 } { 20 } \left( x ^ { 2 } - 4 \right) & 2 \leqslant x \leqslant 4 \\ \frac { 1 } { 5 } ( 2 x - 5 ) & 4 < x \leqslant 5 \\ 1 & x > 5 \end{array} \right.$$
  1. Calculate \(\mathrm { P } ( X > 4 )\)
  2. Find the probability density function of \(X\), specifying it for all values of \(x\).
  3. Find the value of \(a\) such that \(\mathrm { P } ( 3 < X < a ) = 0.642\)
  4. Find the probability density function of \(X\), specifying it for all values of \(x\).
Edexcel S2 2015 June Q2
15 marks Standard +0.3
2. A company produces chocolate chip biscuits. The number of chocolate chips per biscuit has a Poisson distribution with mean 8
  1. Find the probability that one of these biscuits, selected at random, does not contain 8 chocolate chips. A small packet contains 4 of these biscuits, selected at random.
  2. Find the probability that each biscuit in the packet contains at least 8 chocolate chips. A large packet contains 9 of these biscuits, selected at random.
  3. Use a suitable approximation to find the probability that there are more than 75 chocolate chips in the packet. A shop sells packets of biscuits, randomly, at a rate of 1.5 packets per hour. Following an advertising campaign, 11 packets are sold in 4 hours.
  4. Test, at the \(5 \%\) level of significance, whether or not there is evidence that the rate of sales of packets of biscuits has increased. State your hypotheses clearly.
Edexcel S2 2015 June Q3
11 marks Standard +0.8
  1. A piece of spaghetti has length \(2 c\), where \(c\) is a positive constant. It is cut into two pieces at a random point. The continuous random variable \(X\) represents the length of the longer piece and is uniformly distributed over the interval \([ c , 2 c ]\).
    1. Sketch the graph of the probability density function of \(X\)
    2. Use integration to prove that \(\operatorname { Var } ( X ) = \frac { c ^ { 2 } } { 12 }\)
    3. Find the probability that the longer piece is more than twice the length of the shorter piece.
Edexcel S2 2015 June Q4
5 marks Challenging +1.2
  1. A single observation \(x\) is to be taken from a Poisson distribution with parameter \(\lambda\) This observation is to be used to test, at a \(5 \%\) level of significance,
$$\mathrm { H } _ { 0 } : \lambda = k \quad \mathrm { H } _ { 1 } : \lambda \neq k$$ where \(k\) is a positive integer.
Given that the critical region for this test is \(( X = 0 ) \cup ( X \geqslant 9 )\)
  1. find the value of \(k\), justifying your answer.
  2. Find the actual significance level of this test.
Edexcel S2 2015 June Q5
9 marks Standard +0.3
5. A bag contains a large number of counters with \(35 \%\) of the counters having a value of 6 and \(65 \%\) of the counters having a value of 9 A random sample of size 2 is taken from the bag and the value of each counter is recorded as \(X _ { 1 }\) and \(X _ { 2 }\) respectively. The statistic \(Y\) is calculated using the formula $$Y = \frac { 2 X _ { 1 } + X _ { 2 } } { 3 }$$
  1. List all the possible values of \(Y\).
  2. Find the sampling distribution of \(Y\).
  3. Find \(\mathrm { E } ( Y )\).
Edexcel S2 2015 June Q6
15 marks Moderate -0.3
6. Past information at a computer shop shows that \(40 \%\) of customers buy insurance when they purchase a product. In a random sample of 30 customers, \(X\) buy insurance.
  1. Write down a suitable model for the distribution of \(X\).
  2. State an assumption that has been made for the model in part (a) to be suitable. The probability that fewer than \(r\) customers buy insurance is less than 0.05
  3. Find the largest possible value of \(r\). A second random sample, of 100 customers, is taken.
    The probability that at least \(t\) of these customers buy insurance is 0.938 , correct to 3 decimal places.
  4. Using a suitable approximation, find the value of \(t\). The shop now offers an extended warranty on all products. Following this, a random sample of 25 customers is taken and 6 of them buy insurance.
  5. Test, at the \(10 \%\) level of significance, whether or not there is evidence that the proportion of customers who buy insurance has decreased. State your hypotheses clearly.
Edexcel S2 2015 June Q7
10 marks Standard +0.3
  1. A random variable \(X\) has probability density function
$$f ( x ) = \begin{cases} \frac { 2 x } { 15 } & 0 \leqslant x \leqslant k \\ \frac { 1 } { 5 } ( 5 - x ) & k < x \leqslant 5 \\ 0 & \text { otherwise } \end{cases}$$
  1. Showing your working clearly, find the value of \(k\).
  2. Write down the mode of \(X\).
  3. Find \(\mathrm { P } \left( \left. X \leqslant \frac { k } { 2 } \right\rvert \, X \leqslant k \right)\) 郬
Edexcel S2 2016 June Q1
11 marks Standard +0.3
  1. During a typical day, a school website receives visits randomly at a rate of 9 per hour.
The probability that the school website receives fewer than \(v\) visits in a randomly selected one hour period is less than 0.75
  1. Find the largest possible value of \(v\)
  2. Find the probability that in a randomly selected one hour period, the school website receives at least 4 but at most 11 visits.
  3. Find the probability that in a randomly selected 10 minute period, the school website receives more than 1 visit.
  4. Using a suitable approximation, find the probability that in a randomly selected 8 hour period the school website receives more than 80 visits.
Edexcel S2 2016 June Q2
10 marks Standard +0.3
2. The random variable \(X \sim \mathrm {~B} ( 10 , p )\)
    1. Write down an expression for \(\mathrm { P } ( X = 3 )\) in terms of \(p\)
    2. Find the value of \(p\) such that \(\mathrm { P } ( X = 3 )\) is 16 times the value of \(\mathrm { P } ( X = 7 )\) The random variable \(Y \sim \operatorname { Po } ( \lambda )\)
  2. Find the value of \(\lambda\) such that \(\mathrm { P } ( Y = 3 )\) is 5 times the value of \(\mathrm { P } ( Y = 5 )\) The random variable \(W \sim \mathrm {~B} ( n , 0.4 )\)
  3. Find the value of \(n\) and the value of \(\alpha\) such that \(W\) can be approximated by the normal distribution, \(\mathrm { N } ( 32 , \alpha )\)
Edexcel S2 2016 June Q3
6 marks Moderate -0.3
3. A single observation \(x\) is to be taken from \(X \sim \mathrm {~B} ( 12 , p )\) This observation is used to test \(\mathrm { H } _ { 0 } : p = 0.45\) against \(\mathrm { H } _ { 1 } : p > 0.45\)
  1. Using a \(5 \%\) level of significance, find the critical region for this test.
  2. State the actual significance level of this test. The value of the observation is found to be 9
  3. State the conclusion that can be made based on this observation.
  4. State whether or not this conclusion would change if the same test was carried out at the
    1. 10\% level of significance,
    2. \(1 \%\) level of significance.
Edexcel S2 2016 June Q4
13 marks Moderate -0.3
  1. The waiting times, in minutes, between flight take-offs at an airport are modelled by the continuous random variable \(X\) with probability density function
$$f ( x ) = \begin{cases} \frac { 1 } { 5 } & 2 \leqslant x \leqslant 7 \\ 0 & \text { otherwise } \end{cases}$$
  1. Write down the name of this distribution. A randomly selected flight takes off at 9am
  2. Find the probability that the next flight takes off before 9.05 am
  3. Find the probability that at least 1 of the next 5 flights has a waiting time of more than 6 minutes.
  4. Find the cumulative distribution function of \(X\), for all \(x\)
  5. Sketch the cumulative distribution function of \(X\) for \(2 \leqslant x \leqslant 7\) On foggy days, an extra 2 minutes is added to each waiting time.
  6. Find the mean and variance of the waiting times between flight take-offs on foggy days.
Edexcel S2 2016 June Q5
9 marks Challenging +1.2
5. A bag contains a large number of coins. It contains only \(1 \mathrm { p } , 5 \mathrm { p }\) and 10 p coins. The fraction of 1 p coins in the bag is \(q\), the fraction of 5 p coins in the bag is \(r\) and the fraction of 10p coins in the bag is \(s\). Two coins are selected at random from the bag and the coin with the highest value is recorded. Let \(M\) represent the value of the highest coin. The sampling distribution of \(M\) is given below
\(m\)1510
\(\mathrm { P } ( M = m )\)\(\frac { 1 } { 25 }\)\(\frac { 13 } { 80 }\)\(\frac { 319 } { 400 }\)
  1. List all the possible samples of two coins which may be selected.
  2. Find the value of \(q\), the value of \(r\) and the value of \(s\)
Edexcel S2 2016 June Q6
11 marks Standard +0.3
6. A continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} a x - b x ^ { 2 } & 0 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$ Given that the mode is 1
  1. show that \(a = 2 b\)
  2. Find the value of \(a\) and the value of \(b\)
  3. Calculate F(1.5)
  4. State whether the upper quartile of \(X\) is greater than 1.5, equal to 1.5, or less than 1.5 Give a reason for your answer.
Edexcel S2 2016 June Q7
15 marks Standard +0.3
7. Last year \(4 \%\) of cars tested in a large chain of garages failed an emissions test. A random sample of \(n\) of these cars is taken. The number of cars that fail the test is represented by \(X\) Given that the standard deviation of \(X\) is 1.44
    1. find the value of \(n\)
    2. find \(\mathrm { E } ( X )\) A random sample of 20 of the cars tested is taken.
  2. Find the probability that all of these cars passed the emissions test. Given that at least 1 of these cars failed the emissions test,
  3. find the probability that exactly 3 of these cars failed the emissions test. A car mechanic claims that more than \(4 \%\) of the cars tested at the garage chain this year are failing the emissions test. A random sample of 125 of these cars is taken and 10 of these cars fail the emissions test.
  4. Using a suitable approximation, test whether or not there is evidence to support the mechanic's claim. Use a \(5 \%\) level of significance and state your hypotheses clearly.
Edexcel S2 2017 June Q1
11 marks Standard +0.3
  1. At a particular junction on a train line, signal failures are known to occur randomly at a rate of 1 every 4 days.
    1. Find the probability that there are no signal failures on a randomly selected day.
    2. Find the probability that there is at least 1 signal failure on each of the next 3 days.
    3. Find the probability that in a randomly selected 7 -day week, there are exactly 5 days with no signal failures.
    Repair works are carried out on the line. After these repair works, the number, \(f\), of signal failures in a 32-day period is recorded. A test is carried out, at the \(5 \%\) level of significance, to determine whether or not there has been a decrease in the rate of signal failures following the repair works.
  2. State the hypotheses for this test.
  3. Find the largest value of \(f\) for which the null hypothesis should be rejected.
Edexcel S2 2017 June Q2
12 marks Standard +0.3
2. Crispy-crisps produces packets of crisps. During a promotion, a prize is placed in \(25 \%\) of the packets. No more than 1 prize is placed in any packet. A box contains 6 packets of crisps.
    1. Write down a suitable distribution to model the number of prizes found in a box.
    2. Write down one assumption required for the model.
  2. Find the probability that in 2 randomly selected boxes, only 1 box contains exactly 1 prize.
  3. Find the probability that a randomly selected box contains at least 2 prizes. Neha buys 80 boxes of crisps.
  4. Using a normal approximation, find the probability that no more than 30 of the boxes contain at least 2 prizes.