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Edexcel S2 2010 June Q6
15 marks Standard +0.3
  1. A company claims that a quarter of the bolts sent to them are faulty. To test this claim the number of faulty bolts in a random sample of 50 is recorded.
    1. Give two reasons why a binomial distribution may be a suitable model for the number of faulty bolts in the sample.
    2. Using a 5\% significance level, find the critical region for a two-tailed test of the hypothesis that the probability of a bolt being faulty is \(\frac { 1 } { 4 }\). The probability of rejection in either tail should be as close as possible to 0.025
    3. Find the actual significance level of this test.
    In the sample of 50 the actual number of faulty bolts was 8 .
  2. Comment on the company's claim in the light of this value. Justify your answer. The machine making the bolts was reset and another sample of 50 bolts was taken. Only 5 were found to be faulty.
  3. Test at the \(1 \%\) level of significance whether or not the probability of a faulty bolt has decreased. State your hypotheses clearly.
Edexcel S2 2011 June Q1
3 marks Easy -1.8
  1. A factory produces components. Each component has a unique identity number and it is assumed that \(2 \%\) of the components are faulty. On a particular day, a quality control manager wishes to take a random sample of 50 components.
    1. Identify a sampling frame.
    The statistic \(F\) represents the number of faulty components in the random sample of size 50.
  2. Specify the sampling distribution of \(F\).
Edexcel S2 2011 June Q2
10 marks Standard +0.3
2. A traffic officer monitors the rate at which vehicles pass a fixed point on a motorway. When the rate exceeds 36 vehicles per minute he must switch on some speed restrictions to improve traffic flow.
  1. Suggest a suitable model to describe the number of vehicles passing the fixed point in a 15 s interval. The traffic officer records 12 vehicles passing the fixed point in a 15 s interval.
  2. Stating your hypotheses clearly, and using a \(5 \%\) level of significance, test whether or not the traffic officer has sufficient evidence to switch on the speed restrictions.
  3. Using a \(5 \%\) level of significance, determine the smallest number of vehicles the traffic officer must observe in a 10 s interval in order to have sufficient evidence to switch on the speed restrictions.
Edexcel S2 2011 June Q3
10 marks Challenging +1.2
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6e4af10e-ee8d-493f-bd72-34b231003d97-05_455_1026_242_484} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the probability density function \(\mathrm { f } ( x )\) of the random variable \(X\).
For \(0 \leqslant x \leqslant 3 , \mathrm { f } ( x )\) is represented by a curve \(O B\) with equation \(\mathrm { f } ( x ) = k x ^ { 2 }\), where \(k\) is a constant. For \(3 \leqslant x \leqslant a\), where \(a\) is a constant, \(\mathrm { f } ( x )\) is represented by a straight line passing through \(B\) and the point ( \(a , 0\) ). For all other values of \(x , \mathrm { f } ( x ) = 0\).
Given that the mode of \(X =\) the median of \(X\), find
  1. the mode,
  2. the value of \(k\),
  3. the value of \(a\). Without calculating \(\mathrm { E } ( X )\) and with reference to the skewness of the distribution
  4. state, giving your reason, whether \(\mathrm { E } ( X ) < 3 , \mathrm { E } ( X ) = 3\) or \(\mathrm { E } ( X ) > 3\).
Edexcel S2 2011 June Q4
8 marks Standard +0.3
  1. In a game, players select sticks at random from a box containing a large number of sticks of different lengths. The length, in cm , of a randomly chosen stick has a continuous uniform distribution over the interval [7,10].
A stick is selected at random from the box.
  1. Find the probability that the stick is shorter than 9.5 cm . To win a bag of sweets, a player must select 3 sticks and wins if the length of the longest stick is more than 9.5 cm .
  2. Find the probability of winning a bag of sweets. To win a soft toy, a player must select 6 sticks and wins the toy if more than four of the sticks are shorter than 7.6 cm .
  3. Find the probability of winning a soft toy.
Edexcel S2 2011 June Q5
13 marks Standard +0.3
5. Defects occur at random in planks of wood with a constant rate of 0.5 per 10 cm length. Jim buys a plank of length 100 cm .
  1. Find the probability that Jim's plank contains at most 3 defects. Shivani buys 6 planks each of length 100 cm .
  2. Find the probability that fewer than 2 of Shivani's planks contain at most 3 defects.
  3. Using a suitable approximation, estimate the probability that the total number of defects on Shivani's 6 planks is less than 18.
Edexcel S2 2011 June Q6
14 marks Standard +0.3
  1. A shopkeeper knows, from past records, that \(15 \%\) of customers buy an item from the display next to the till. After a refurbishment of the shop, he takes a random sample of 30 customers and finds that only 1 customer has bought an item from the display next to the till.
    1. Stating your hypotheses clearly, and using a \(5 \%\) level of significance, test whether or not there has been a change in the proportion of customers buying an item from the display next to the till.
    During the refurbishment a new sandwich display was installed. Before the refurbishment \(20 \%\) of customers bought sandwiches. The shopkeeper claims that the proportion of customers buying sandwiches has now increased. He selects a random sample of 120 customers and finds that 31 of them have bought sandwiches.
  2. Using a suitable approximation and stating your hypotheses clearly, test the shopkeeper's claim. Use a \(10 \%\) level of significance.
Edexcel S2 2011 June Q7
17 marks Standard +0.3
  1. The continuous random variable \(X\) has probability density function given by
$$f ( x ) = \left\{ \begin{array} { c c } \frac { 3 } { 32 } ( x - 1 ) ( 5 - x ) & 1 \leqslant x \leqslant 5 \\ 0 & \text { otherwise } \end{array} \right.$$
  1. Sketch \(\mathrm { f } ( x )\) showing clearly the points where it meets the \(x\)-axis.
  2. Write down the value of the mean, \(\mu\), of \(X\).
  3. Show that \(\mathrm { E } \left( X ^ { 2 } \right) = 9.8\)
  4. Find the standard deviation, \(\sigma\), of \(X\). The cumulative distribution function of \(X\) is given by $$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < 1 \\ \frac { 1 } { 32 } \left( a - 15 x + 9 x ^ { 2 } - x ^ { 3 } \right) & 1 \leqslant x \leqslant 5 \\ 1 & x > 5 \end{array} \right.$$ where \(a\) is a constant.
  5. Find the value of \(a\).
  6. Show that the lower quartile of \(X , q _ { 1 }\), lies between 2.29 and 2.31
  7. Hence find the upper quartile of \(X\), giving your answer to 1 decimal place.
  8. Find, to 2 decimal places, the value of \(k\) so that $$\mathrm { P } ( \mu - k \sigma < X < \mu + k \sigma ) = 0.5$$
Edexcel S2 2012 June Q1
7 marks Moderate -0.3
  1. A manufacturer produces sweets of length \(L \mathrm {~mm}\) where \(L\) has a continuous uniform distribution with range [15, 30].
    1. Find the probability that a randomly selected sweet has a length greater than 24 mm .
    These sweets are randomly packed in bags of 20 sweets.
  2. Find the probability that a randomly selected bag will contain at least 8 sweets with length greater than 24 mm .
  3. Find the probability that 2 randomly selected bags will both contain at least 8 sweets with length greater than 24 mm .
Edexcel S2 2012 June Q2
5 marks Standard +0.3
2. A test statistic has a distribution \(\mathrm { B } ( 25 , p )\). Given that $$\mathrm { H } _ { 0 } : p = 0.5 \quad \mathrm { H } _ { 1 } : p \neq 0.5$$
  1. find the critical region for the test statistic such that the probability in each tail is as close as possible to \(2.5 \%\).
  2. State the probability of incorrectly rejecting \(\mathrm { H } _ { 0 }\) using this critical region.
Edexcel S2 2012 June Q3
9 marks Moderate -0.8
3. (a) Write down two conditions needed to approximate the binomial distribution by the Poisson distribution. A machine which manufactures bolts is known to produce \(3 \%\) defective bolts. The machine breaks down and a new machine is installed. A random sample of 200 bolts is taken from those produced by the new machine and 12 bolts were defective.
(b) Using a suitable approximation, test at the \(5 \%\) level of significance whether or not the proportion of defective bolts is higher with the new machine than with the old machine. State your hypotheses clearly.
Edexcel S2 2012 June Q4
14 marks Standard +0.3
4. The number of houses sold by an estate agent follows a Poisson distribution, with a mean of 2 per week.
  1. Find the probability that in the next 4 weeks the estate agent sells,
    1. exactly 3 houses,
    2. more than 5 houses. The estate agent monitors sales in periods of 4 weeks.
  2. Find the probability that in the next twelve of these 4 week periods there are exactly nine periods in which more than 5 houses are sold. The estate agent will receive a bonus if he sells more than 25 houses in the next 10 weeks.
  3. Use a suitable approximation to estimate the probability that the estate agent receives a bonus.
Edexcel S2 2012 June Q5
12 marks Standard +0.3
  1. The queueing time, \(X\) minutes, of a customer at a till of a supermarket has probability density function
$$f ( x ) = \left\{ \begin{array} { c c } \frac { 3 } { 32 } x ( k - x ) & 0 \leqslant x \leqslant k \\ 0 & \text { otherwise } \end{array} \right.$$
  1. Show that the value of \(k\) is 4
  2. Write down the value of \(\mathrm { E } ( X )\).
  3. Calculate \(\operatorname { Var } ( X )\).
  4. Find the probability that a randomly chosen customer's queueing time will differ from the mean by at least half a minute.
Edexcel S2 2012 June Q6
6 marks Moderate -0.3
6. A bag contains a large number of balls. 65\% are numbered 1 35\% are numbered 2 A random sample of 3 balls is taken from the bag.
Find the sampling distribution for the range of the numbers on the 3 selected balls.
Edexcel S2 2012 June Q7
14 marks Standard +0.3
7. The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by $$\mathrm { f } ( x ) = \left\{ \begin{array} { c c } \frac { x ^ { 2 } } { 45 } & 0 \leqslant x \leqslant 3 \\ \frac { 1 } { 5 } & 3 < x < 4 \\ \frac { 1 } { 3 } - \frac { x } { 30 } & 4 \leqslant x \leqslant 10 \\ 0 & \text { otherwise } \end{array} . \right.$$
  1. Sketch \(\mathrm { f } ( x )\) for \(0 \leqslant x \leqslant 10\)
  2. Find the cumulative distribution function \(\mathrm { F } ( x )\) for all values of \(x\).
  3. Find \(\mathrm { P } ( X \leqslant 8 )\).
Edexcel S2 2012 June Q8
8 marks Standard +0.3
  1. In a large restaurant an average of 3 out of every 5 customers ask for water with their meal.
A random sample of 10 customers is selected.
  1. Find the probability that
    1. exactly 6 ask for water with their meal,
    2. less than 9 ask for water with their meal. A second random sample of 50 customers is selected.
  2. Find the smallest value of \(n\) such that $$\mathrm { P } ( X < n ) \geqslant 0.9$$ where the random variable \(X\) represents the number of these customers who ask for water.
Edexcel S2 2013 June Q1
5 marks Moderate -0.8
  1. A bag contains a large number of counters. A third of the counters have a number 5 on them and the remainder have a number 1 .
A random sample of 3 counters is selected.
  1. List all possible samples.
  2. Find the sampling distribution for the range.
Edexcel S2 2013 June Q2
7 marks Standard +0.3
2. The continuous random variable \(Y\) has cumulative distribution function $$\mathrm { F } ( y ) = \left\{ \begin{array} { c c } 0 & y < 0 \\ \frac { 1 } { 4 } \left( y ^ { 3 } - 4 y ^ { 2 } + k y \right) & 0 \leqslant y \leqslant 2 \\ 1 & y > 2 \end{array} \right.$$ where \(k\) is a constant.
  1. Find the value of \(k\).
  2. Find the probability density function of \(Y\), specifying it for all values of \(y\).
  3. Find \(\mathrm { P } ( Y > 1 )\).
Edexcel S2 2013 June Q3
8 marks Standard +0.3
3. The random variable \(X\) has a continuous uniform distribution on \([ a , b ]\) where \(a\) and \(b\) are positive numbers. Given that \(\mathrm { E } ( X ) = 23\) and \(\operatorname { Var } ( X ) = 75\)
  1. find the value of \(a\) and the value of \(b\). Given that \(\mathrm { P } ( X > c ) = 0.32\)
  2. find \(\mathrm { P } ( 23 < X < c )\).
Edexcel S2 2013 June Q4
11 marks Standard +0.3
4. The random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by $$f ( x ) = \left\{ \begin{array} { c c } k \left( 3 + 2 x - x ^ { 2 } \right) & 0 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{array} \right.$$ where \(k\) is a constant.
  1. Show that \(k = \frac { 1 } { 9 }\)
  2. Find the mode of \(X\).
  3. Use algebraic integration to find \(\mathrm { E } ( X )\). By comparing your answers to parts (b) and (c),
  4. describe the skewness of \(X\), giving a reason for your answer.
Edexcel S2 2013 June Q5
13 marks Moderate -0.3
  1. In a village shop the customers must join a queue to pay. The number of customers joining the queue in a 10 minute interval is modelled by a Poisson distribution with mean 3
Find the probability that
  1. exactly 4 customers join the queue in the next 10 minutes,
  2. more than 10 customers join the queue in the next 20 minutes. When a customer reaches the front of the queue the customer pays the assistant. The time each customer takes paying the assistant, \(T\) minutes, has a continuous uniform distribution over the interval \([ 0,5 ]\). The random variable \(T\) is independent of the number of people joining the queue.
  3. Find \(\mathrm { P } ( T > 3.5 )\) In a random sample of 5 customers, the random variable \(C\) represents the number of customers who took more than 3.5 minutes paying the assistant.
  4. Find \(\mathrm { P } ( C \geqslant 3 )\) Bethan has just reached the front of the queue and starts paying the assistant.
  5. Find the probability that in the next 4 minutes Bethan finishes paying the assistant and no other customers join the queue.
Edexcel S2 2013 June Q6
14 marks Standard +0.8
6. Frugal bakery claims that their packs of 10 muffins contain on average 80 raisins per pack. A Poisson distribution is used to describe the number of raisins per muffin. A muffin is selected at random to test whether or not the mean number of raisins per muffin has changed.
  1. Find the critical region for a two-tailed test using a \(10 \%\) level of significance. The probability of rejection in each tail should be less than 0.05
  2. Find the actual significance level of this test. The bakery has a special promotion claiming that their muffins now contain even more raisins. A random sample of 10 muffins is selected and is found to contain a total of 95 raisins.
  3. Use a suitable approximation to test the bakery's claim. You should state your hypotheses clearly and use a \(5 \%\) level of significance.
Edexcel S2 2013 June Q7
17 marks Moderate -0.3
7. As part of a selection procedure for a company, applicants have to answer all 20 questions of a multiple choice test. If an applicant chooses answers at random the probability of choosing a correct answer is 0.2 and the number of correct answers is represented by the random variable \(X\).
  1. Suggest a suitable distribution for \(X\).
    (2) Each applicant gains 4 points for each correct answer but loses 1 point for each incorrect answer. The random variable \(S\) represents the final score, in points, for an applicant who chooses answers to this test at random.
  2. Show that \(S = 5 X - 20\)
  3. Find \(\mathrm { E } ( S )\) and \(\operatorname { Var } ( S )\). An applicant who achieves a score of at least 20 points is invited to take part in the final stage of the selection process.
  4. Find \(\mathrm { P } ( S \geqslant 20 )\) (4) Cameron is taking the final stage of the selection process which is a multiple choice test consisting of 100 questions. He has been preparing for this test and believes that his chance of answering each question correctly is 0.4
  5. Using a suitable approximation, estimate the probability that Cameron answers more than half of the questions correctly.
Edexcel S2 2013 June Q1
11 marks Standard +0.3
  1. A bag contains a large number of \(1 \mathrm { p } , 2 \mathrm { p }\) and 5 p coins. \(50 \%\) are 1 p coins \(20 \%\) are \(2 p\) coins
    30\% are 5p coins
    A random sample of 3 coins is chosen from the bag.
    1. List all the possible samples of size 3 with median 5p.
    2. Find the probability that the median value of the sample is 5 p .
    3. Find the sampling distribution of the median of samples of size 3
Edexcel S2 2013 June Q2
10 marks Moderate -0.3
  1. The number of defects per metre in a roll of cloth has a Poisson distribution with mean 0.25
Find the probability that
  1. a randomly chosen metre of cloth has 1 defect,
  2. the total number of defects in a randomly chosen 6 metre length of cloth is more than 2 A tailor buys 300 metres of cloth.
  3. Using a suitable approximation find the probability that the tailor's cloth will contain less than 90 defects.