Questions — Edexcel S2 (494 questions)

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Edexcel S2 2012 June Q3
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
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
  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. 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
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
  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
  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
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
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
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
  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
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
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
  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
  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.
Edexcel S2 2013 June Q3
  1. An online shop sells a computer game at an average rate of 1 per day.
    1. Find the probability that the shop sells more than 10 games in a 7 day period.
    Once every 7 days the shop has games delivered before it opens.
  2. Find the least number of games the shop should have in stock immediately after a delivery so that the probability of running out of the game before the next delivery is less than 0.05 In an attempt to increase sales of the computer game, the price is reduced for six months. A random sample of 28 days is taken from these six months. In the sample of 28 days, 36 computer games are sold.
  3. Using a suitable approximation and a \(5 \%\) level of significance, test whether or not the average rate of sales per day has increased during these six months. State your hypotheses clearly.
Edexcel S2 2013 June Q4
  1. A continuous random variable \(X\) is uniformly distributed over the interval [ \(b , 4 b\) ] where \(b\) is a constant.
    1. Write down \(\mathrm { E } ( X )\).
    2. Use integration to show that \(\operatorname { Var } ( X ) = \frac { 3 b ^ { 2 } } { 4 }\).
    3. Find \(\operatorname { Var } ( 3 - 2 X )\).
    Given that \(b = 1\) find
  2. the cumulative distribution function of \(X , \mathrm {~F} ( x )\), for all values of \(x\),
  3. the median of \(X\).
Edexcel S2 2013 June Q5
  1. The continuous random variable \(X\) has a cumulative distribution function
$$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 & x < 1
\frac { x ^ { 3 } } { 10 } + \frac { 3 x ^ { 2 } } { 10 } + a x + b & 1 \leqslant x \leqslant 2
1 & x > 2 \end{array} \right.$$ where \(a\) and \(b\) are constants.
  1. Find the value of \(a\) and the value of \(b\).
  2. Show that \(\mathrm { f } ( x ) = \frac { 3 } { 10 } \left( x ^ { 2 } + 2 x - 2 \right) , \quad 1 \leqslant x \leqslant 2\)
  3. Use integration to find \(\mathrm { E } ( X )\).
  4. Show that the lower quartile of \(X\) lies between 1.425 and 1.435
Edexcel S2 2013 June Q6
6. In a manufacturing process \(25 \%\) of articles are thought to be defective. Articles are produced in batches of 20
  1. A batch is selected at random. Using a \(5 \%\) significance level, find the critical region for a two tailed test that the probability of an article chosen at random being defective is 0.25
    You should state the probability in each tail which should be as close as possible to 0.025 The manufacturer changes the production process to try to reduce the number of defective articles. She then chooses a batch at random and discovers there are 3 defective articles.
  2. Test at the \(5 \%\) level of significance whether or not there is evidence that the changes to the process have reduced the percentage of defective articles. State your hypotheses clearly.
Edexcel S2 2013 June Q7
  1. A telesales operator is selling a magazine. Each day he chooses a number of people to telephone. The probability that each person he telephones buys the magazine is 0.1
    1. Suggest a suitable distribution to model the number of people who buy the magazine from the telesales operator each day.
    2. On Monday, the telesales operator telephones 10 people. Find the probability that he sells at least 4 magazines.
    3. Calculate the least number of people he needs to telephone on Tuesday, so that the probability of selling at least 1 magazine, on that day, is greater than 0.95
    A call centre also sells the magazine. The probability that a telephone call made by the call centre sells a magazine is 0.05 The call centre telephones 100 people every hour.
  2. Using a suitable approximation, find the probability that more than 10 people telephoned by the call centre buy a magazine in a randomly chosen hour.
Edexcel S2 2014 June Q1
  1. Before Roger will use a tennis ball he checks it using a "bounce" test. The probability that a ball from Roger's usual supplier fails the bounce test is 0.2 . A new supplier claims that the probability of one of their balls failing the bounce test is less than 0.2 . Roger checks a random sample of 40 balls from the new supplier and finds that 3 balls fail the bounce test.
Stating your hypotheses clearly, use a \(5 \%\) level of significance to test the new supplier's claim.
Edexcel S2 2014 June Q2
2. A bag contains a large number of counters. Each counter has a single digit number on it and the mean of all the numbers in the bag is the unknown parameter \(\mu\). The number 2 is on \(40 \%\) of the counters and the number 5 is on \(25 \%\) of the counters. All the remaining counters have numbers greater than 5 on them. A random sample of 10 counters is taken from the bag.
  1. State whether or not each of the following is a statistic
    1. \(S =\) the sum of the numbers on the counters in the sample,
    2. \(D =\) the difference between the highest number in the sample and \(\mu\),
    3. \(F =\) the number of counters in the sample with a number 5 on them. The random variable \(T\) represents the number of counters in a random sample of 10 with the number 2 on them.
  2. Specify the sampling distribution of \(T\). The counters are selected one by one.
  3. Find the probability that the third counter selected is the first counter with the number 2 on it.
Edexcel S2 2014 June Q3
3. Accidents occur randomly at a road junction at a rate of 18 every year. The random variable \(X\) represents the number of accidents at this road junction in the next 6 months.
  1. Write down the distribution of \(X\).
  2. Find \(\mathrm { P } ( X > 7 )\).
  3. Show that the probability of at least one accident in a randomly selected month is 0.777 (correct to 3 decimal places).
  4. Find the probability that there is at least one accident in exactly 4 of the next 6 months.
Edexcel S2 2014 June Q4
4. The random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by $$\mathrm { f } ( x ) = \left\{ \begin{array} { c c } 3 k & 0 \leqslant x < 1
k x ( 4 - x ) & 1 \leqslant x \leqslant 4
0 & \text { otherwise } \end{array} \right.$$ where \(k\) is a constant.
  1. Sketch f (x).
  2. Write down the mode of \(X\). Given that \(\mathrm { E } ( X ) = \frac { 29 } { 16 }\)
  3. describe, giving a reason, the skewness of the distribution.
  4. Use integration to find the value of \(k\).
  5. Write down the lower quartile of \(X\). Given also that \(\mathrm { P } ( 2 < X < 3 ) = \frac { 11 } { 36 }\)
  6. find the exact value of \(\mathrm { P } ( X > 3 )\).
Edexcel S2 2014 June Q5
  1. Sammy manufactures wallpaper. She knows that defects occur randomly in the manufacturing process at a rate of 1 every 8 metres. Once a week the machinery is cleaned and reset. Sammy then takes a random sample of 40 metres of wallpaper from the next batch produced to test if there has been any change in the rate of defects.
    1. Stating your hypotheses clearly and using a \(10 \%\) level of significance, find the critical region for this test. You should choose your critical region so that the probability of rejection is less than 0.05 in each tail.
    2. State the actual significance level of this test.
    Thomas claims that his new machine would reduce the rate of defects and invites Sammy to test it. Sammy takes a random sample of 200 metres of wallpaper produced on Thomas' machine and finds 19 defects.
  2. Using a suitable approximation, test Thomas' claim. You should use a \(5 \%\) level of significance and state your hypotheses clearly.