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Edexcel S2 2008 January Q4
7 marks Standard +0.3
  1. The continuous random variable \(Y\) has cumulative distribution function \(\mathrm { F } ( y )\) given by
$$\mathrm { F } ( y ) = \left\{ \begin{array} { c l } 0 & y < 1 \\ k \left( y ^ { 4 } + y ^ { 2 } - 2 \right) & 1 \leqslant y \leqslant 2 \\ 1 & y > 2 \end{array} \right.$$
  1. Show that \(k = \frac { 1 } { 18 }\).
  2. Find \(\mathrm { P } ( Y > 1.5 )\).
  3. Specify fully the probability density function f(y).
Edexcel S2 2008 January Q5
7 marks Moderate -0.3
  1. Dhriti grows tomatoes. Over a period of time, she has found that there is a probability 0.3 of a ripe tomato having a diameter greater than 4 cm . She decides to try a new fertiliser. In a random sample of 40 ripe tomatoes, 18 have a diameter greater than 4 cm . Dhriti claims that the new fertiliser has increased the probability of a ripe tomato being greater than 4 cm in diameter.
Test Dhriti's claim at the 5\% level of significance. State your hypotheses clearly.
Edexcel S2 2008 January Q6
12 marks Standard +0.3
6. The probability that a sunflower plant grows over 1.5 metres high is 0.25 . A random sample of 40 sunflower plants is taken and each sunflower plant is measured and its height recorded.
  1. Find the probability that the number of sunflower plants over 1.5 m high is between 8 and 13 (inclusive) using
    1. a Poisson approximation,
    2. a Normal approximation.
  2. Write down which of the approximations used in part (a) is the most accurate estimate of the probability. You must give a reason for your answer.
Edexcel S2 2008 January Q7
14 marks Standard +0.3
  1. (a) Explain what you understand by
    1. a hypothesis test,
    2. a critical region.
    During term time, incoming calls to a school are thought to occur at a rate of 0.45 per minute. To test this, the number of calls during a random 20 minute interval, is recorded.
    (b) Find the critical region for a two-tailed test of the hypothesis that the number of incoming calls occurs at a rate of 0.45 per 1 minute interval. The probability in each tail should be as close to \(2.5 \%\) as possible.
    (c) Write down the actual significance level of the above test. In the school holidays, 1 call occurs in a 10 minute interval.
    (d) Test, at the \(5 \%\) level of significance, whether or not there is evidence that the rate of incoming calls is less during the school holidays than in term time.
Edexcel S2 2008 January Q8
13 marks Moderate -0.3
  1. The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by
$$f ( x ) = \left\{ \begin{array} { c c } 2 ( x - 2 ) & 2 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{array} \right.$$
  1. Sketch \(\mathrm { f } ( x )\) for all values of \(x\).
  2. Write down the mode of \(X\). Find
  3. \(\mathrm { E } ( X )\),
  4. the median of \(X\).
  5. Comment on the skewness of this distribution. Give a reason for your answer.
Edexcel S2 2009 January Q1
11 marks Standard +0.3
  1. A botanist is studying the distribution of daisies in a field. The field is divided into a number of equal sized squares. The mean number of daisies per square is assumed to be 3. The daisies are distributed randomly throughout the field.
Find the probability that, in a randomly chosen square there will be
  1. more than 2 daisies,
  2. either 5 or 6 daisies. The botanist decides to count the number of daisies, \(x\), in each of 80 randomly selected squares within the field. The results are summarised below $$\sum x = 295 \quad \sum x ^ { 2 } = 1386$$
  3. Calculate the mean and the variance of the number of daisies per square for the 80 squares. Give your answers to 2 decimal places.
  4. Explain how the answers from part (c) support the choice of a Poisson distribution as a model.
  5. Using your mean from part (c), estimate the probability that exactly 4 daisies will be found in a randomly selected square.
Edexcel S2 2009 January Q2
9 marks Easy -1.2
  1. The continuous random variable \(X\) is uniformly distributed over the interval \([ - 2,7 ]\).
    1. Write down fully the probability density function \(\mathrm { f } ( x )\) of \(X\).
    2. Sketch the probability density function \(\mathrm { f } ( x )\) of \(X\).
    Find
  2. \(\mathrm { E } \left( X ^ { 2 } \right)\),
  3. \(\mathrm { P } ( - 0.2 < X < 0.6 )\).
Edexcel S2 2009 January Q3
7 marks Standard +0.3
3. A single observation \(x\) is to be taken from a Binomial distribution \(\mathrm { B } ( 20 , p )\). This observation is used to test \(\mathrm { H } _ { 0 } : p = 0.3\) against \(\mathrm { H } _ { 1 } : p \neq 0.3\)
  1. Using a \(5 \%\) level of significance, find the critical region for this test. The probability of rejecting either tail should be as close as possible to \(2.5 \%\).
  2. State the actual significance level of this test. The actual value of \(x\) obtained is 3 .
  3. State a conclusion that can be drawn based on this value giving a reason for your answer.
Edexcel S2 2009 January Q4
12 marks Moderate -0.8
4. The length of a telephone call made to a company is denoted by the continuous random variable \(T\). It is modelled by the probability density function $$\mathrm { f } ( t ) = \left\{ \begin{array} { c l } k t & 0 \leqslant t \leqslant 10 \\ 0 & \text { otherwise } \end{array} \right.$$
  1. Show that the value of \(k\) is \(\frac { 1 } { 50 }\).
  2. Find \(\mathrm { P } ( T > 6 )\).
  3. Calculate an exact value for \(\mathrm { E } ( T )\) and for \(\operatorname { Var } ( T )\).
  4. Write down the mode of the distribution of \(T\). It is suggested that the probability density function, \(\mathrm { f } ( t )\), is not a good model for \(T\).
  5. Sketch the graph of a more suitable probability density function for \(T\).
Edexcel S2 2009 January Q5
9 marks Standard +0.3
  1. A factory produces components of which \(1 \%\) are defective. The components are packed in boxes of 10 . A box is selected at random.
    1. Find the probability that the box contains exactly one defective component.
    2. Find the probability that there are at least 2 defective components in the box.
    3. Using a suitable approximation, find the probability that a batch of 250 components contains between 1 and 4 (inclusive) defective components.
    4. A web server is visited on weekdays, at a rate of 7 visits per minute. In a random one minute on a Saturday the web server is visited 10 times.
      1. Test, at the \(10 \%\) level of significance, whether or not there is evidence that the rate of visits is greater on a Saturday than on weekdays. State your hypotheses clearly.
      2. State the minimum number of visits required to obtain a significant result.
    6. State an assumption that has been made about the visits to the server.
    In a random two minute period on a Saturday the web server is visited 20 times.
  2. Using a suitable approximation, test at the \(10 \%\) level of significance, whether or not the rate of visits is greater on a Saturday.
Edexcel S2 2009 January Q7
13 marks Moderate -0.3
  1. A random variable \(X\) has probability density function given by
$$f ( x ) = \left\{ \begin{array} { c l } - \frac { 2 } { 9 } x + \frac { 8 } { 9 } & 1 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{array} \right.$$
  1. Show that the cumulative distribution function \(\mathrm { F } ( x )\) can be written in the form \(a x ^ { 2 } + b x + c\), for \(1 \leqslant x \leqslant 4\) where \(a , b\) and \(c\) are constants.
  2. Define fully the cumulative distribution function \(\mathrm { F } ( x )\).
  3. Show that the upper quartile of \(X\) is 2.5 and find the lower quartile. Given that the median of \(X\) is 1.88
  4. describe the skewness of the distribution. Give a reason for your answer.
Edexcel S2 2010 January Q1
8 marks Easy -1.2
  1. A manufacturer supplies DVD players to retailers in batches of 20 . It has \(5 \%\) of the players returned because they are faulty.
    1. Write down a suitable model for the distribution of the number of faulty DVD players in a batch.
    Find the probability that a batch contains
  2. no faulty DVD players,
  3. more than 4 faulty DVD players.
  4. Find the mean and variance of the number of faulty DVD players in a batch.
Edexcel S2 2010 January Q2
10 marks Moderate -0.8
2. A continuous random variable \(X\) has cumulative distribution function $$\mathrm { F } ( x ) = \begin{cases} 0 , & x < - 2 \\ \frac { x + 2 } { 6 } , & - 2 \leqslant x \leqslant 4 \\ 1 , & x > 4 \end{cases}$$
  1. Find \(\mathrm { P } ( X < 0 )\).
  2. Find the probability density function \(\mathrm { f } ( x )\) of \(X\).
  3. Write down the name of the distribution of \(X\).
  4. Find the mean and the variance of \(X\).
  5. Write down the value of \(\mathrm { P } ( X = 1 )\).
Edexcel S2 2010 January Q3
10 marks Moderate -0.3
  1. A robot is programmed to build cars on a production line. The robot breaks down at random at a rate of once every 20 hours.
    1. Find the probability that it will work continuously for 5 hours without a breakdown.
    Find the probability that, in an 8 hour period,
  2. the robot will break down at least once,
  3. there are exactly 2 breakdowns. In a particular 8 hour period, the robot broke down twice.
  4. Write down the probability that the robot will break down in the following 8 hour period. Give a reason for your answer.
Edexcel S2 2010 January Q4
17 marks Standard +0.3
  1. The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by
$$f ( x ) = \begin{cases} k \left( x ^ { 2 } - 2 x + 2 \right) & 0 < x \leqslant 3 \\ 3 k & 3 < x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = \frac { 1 } { 9 }\)
  2. Find the cumulative distribution function \(\mathrm { F } ( x )\).
  3. Find the mean of \(X\).
  4. Show that the median of \(X\) lies between \(x = 2.6\) and \(x = 2.7\)
Edexcel S2 2010 January Q5
9 marks Standard +0.3
  1. A café serves breakfast every morning. Customers arrive for breakfast at random at a rate of 1 every 6 minutes.
Find the probability that
  1. fewer than 9 customers arrive for breakfast on a Monday morning between 10 am and 11 am. The café serves breakfast every day between 8 am and 12 noon.
  2. Using a suitable approximation, estimate the probability that more than 50 customers arrive for breakfast next Tuesday.
Edexcel S2 2010 January Q6
10 marks Standard +0.3
6. (a) Define the critical region of a test statistic. A discrete random variable \(X\) has a Binomial distribution \(\mathrm { B } ( 30 , p )\). A single observation is used to test \(\mathrm { H } _ { 0 } : p = 0.3\) against \(\mathrm { H } _ { 1 } : p \neq 0.3\)
(b) Using a \(1 \%\) level of significance find the critical region of this test. You should state the probability of rejection in each tail which should be as close as possible to 0.005
(c) Write down the actual significance level of the test. The value of the observation was found to be 15 .
(d) Comment on this finding in light of your critical region.
Edexcel S2 2010 January Q7
11 marks Moderate -0.5
  1. A bag contains a large number of coins. It contains only \(1 p\) and \(2 p\) coins in the ratio \(1 : 3\)
    1. Find the mean \(\mu\) and the variance \(\sigma ^ { 2 }\) of the values of this population of coins.
    A random sample of size 3 is taken from the bag.
  2. List all the possible samples.
  3. Find the sampling distribution of the mean value of the samples.
Edexcel S2 2011 January Q1
10 marks Moderate -0.3
  1. A disease occurs in \(3 \%\) of a population.
    1. State any assumptions that are required to model the number of people with the disease in a random sample of size \(n\) as a binomial distribution.
    2. Using this model, find the probability of exactly 2 people having the disease in a random sample of 10 people.
    3. Find the mean and variance of the number of people with the disease in a random sample of 100 people.
    A doctor tests a random sample of 100 patients for the disease. He decides to offer all patients a vaccination to protect them from the disease if more than 5 of the sample have the disease.
  2. Using a suitable approximation, find the probability that the doctor will offer all patients a vaccination.
Edexcel S2 2011 January Q2
6 marks Moderate -0.8
2. A student takes a multiple choice test. The test is made up of 10 questions each with 5 possible answers. The student gets 4 questions correct. Her teacher claims she was guessing the answers. Using a one tailed test, at the \(5 \%\) level of significance, test whether or not there is evidence to reject the teacher's claim. State your hypotheses clearly.
Edexcel S2 2011 January Q3
11 marks Standard +0.3
3. The continuous random variable \(X\) is uniformly distributed over the interval \([ - 1,3 ]\). Find
  1. \(\mathrm { E } ( X )\)
  2. \(\operatorname { Var } ( X )\)
  3. \(\mathrm { E } \left( X ^ { 2 } \right)\)
  4. \(\mathrm { P } ( X < 1.4 )\) A total of 40 observations of \(X\) are made.
  5. Find the probability that at least 10 of these observations are negative.
Edexcel S2 2011 January Q4
6 marks Moderate -0.3
  1. Richard regularly travels to work on a ferry. Over a long period of time, Richard has found that the ferry is late on average 2 times every week. The company buys a new ferry to improve the service. In the 4-week period after the new ferry is launched, Richard finds the ferry is late 3 times and claims the service has improved. Assuming that the number of times the ferry is late has a Poisson distribution, test Richard's claim at the \(5 \%\) level of significance. State your hypotheses clearly.
    (6)
  2. A continuous random variable \(X\) has the probability density function \(\mathrm { f } ( x )\) shown in Figure 1.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{58e5aa9e-f177-48ad-8bb8-54c0e2c21e6d-07_591_689_358_630} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure}
  1. Show that \(\mathrm { f } ( x ) = 4 - 8 x\) for \(0 \leqslant x \leqslant 0.5\) and specify \(\mathrm { f } ( x )\) for all real values of \(x\).
  2. Find the cumulative distribution function \(\mathrm { F } ( x )\).
  3. Find the median of \(X\).
  4. Write down the mode of \(X\).
  5. State, with a reason, the skewness of \(X\).
Edexcel S2 2011 January Q6
16 marks Standard +0.3
6. Cars arrive at a motorway toll booth at an average rate of 150 per hour.
  1. Suggest a suitable distribution to model the number of cars arriving at the toll booth, \(X\), per minute.
  2. State clearly any assumptions you have made by suggesting this model. Using your model,
  3. find the probability that in any given minute
    1. no cars arrive,
    2. more than 3 cars arrive.
  4. In any given 4 minute period, find \(m\) such that \(\mathrm { P } ( X > m ) = 0.0487\)
  5. Using a suitable approximation find the probability that fewer than 15 cars arrive in any given 10 minute period.
    January 2011
Edexcel S2 2011 January Q7
13 marks Standard +0.3
7. The queuing time in minutes, \(X\), of a customer at a post office is modelled by the probability density function $$f ( x ) = \begin{cases} k x \left( 81 - x ^ { 2 } \right) & 0 \leqslant x \leqslant 9 \\ 0 & \text { otherwise } \end{cases}$$
  1. Show that \(k = \frac { 4 } { 6561 }\). Using integration, find
  2. the mean queuing time of a customer,
  3. the probability that a customer will queue for more than 5 minutes. Three independent customers shop at the post office.
  4. Find the probability that at least 2 of the customers queue for more than 5 minutes.
Edexcel S2 2012 January Q1
8 marks Moderate -0.8
  1. The time in minutes that Elaine takes to checkout at her local supermarket follows a continuous uniform distribution defined over the interval [3,9].
Find
  1. Elaine's expected checkout time,
  2. the variance of the time taken to checkout at the supermarket,
  3. the probability that Elaine will take more than 7 minutes to checkout. Given that Elaine has already spent 4 minutes at the checkout,
  4. find the probability that she will take a total of less than 6 minutes to checkout.