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Edexcel S2 2004 January Q7
18 marks Standard +0.3
7. The continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} k x ( 5 - x ) , & 0 \leq x \leq 4 \\ 0 , & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = \frac { 3 } { 56 }\).
  2. Find the cumulative distribution function \(\mathrm { F } ( x )\) for all values of \(x\).
  3. Evaluate \(\mathrm { E } ( X )\).
  4. Find the modal value of \(X\).
  5. Verify that the median value of \(X\) lies between 2.3 and 2.5.
  6. Comment on the skewness of \(X\). Justify your answer. \section*{END}
Edexcel S2 2005 January Q1
4 marks Easy -1.2
  1. The random variables \(R , S\) and \(T\) are distributed as follows
$$R \sim \mathrm {~B} ( 15,0.3 ) , \quad S \sim \mathrm { Po } ( 7.5 ) , \quad T \sim \mathrm {~N} \left( 8,2 ^ { 2 } \right) .$$ Find
  1. \(\mathrm { P } ( R = 5 )\),
  2. \(\mathrm { P } ( S = 5 )\),
  3. \(\mathrm { P } ( T = 5 )\).
Edexcel S2 2005 January Q2
7 marks Easy -1.8
2. (a) Explain what you understand by (i) a population and (ii) a sampling frame. The population and the sampling frame may not be the same.
(b) Explain why this might be the case.
(c) Give an example, justifying your choices, to illustrate when you might use
  1. a census,
  2. a sample.
Edexcel S2 2005 January Q3
8 marks Standard +0.8
3. A rod of length \(2 l\) was broken into 2 parts. The point at which the rod broke is equally likely to be anywhere along the rod. The length of the shorter piece of rod is represented by the random variable \(X\).
  1. Write down the name of the probability density function of \(X\), and specify it fully.
  2. Find \(\mathrm { P } \left( X < \frac { 1 } { 3 } l \right)\).
  3. Write down the value of \(\mathrm { E } ( X )\). Two identical rods of length \(2 l\) are broken.
  4. Find the probability that both of the shorter pieces are of length less than \(\frac { 1 } { 3 } l\).
Edexcel S2 2005 January Q4
10 marks Moderate -0.3
4. In an experiment, there are 250 trials and each trial results in a success or a failure.
  1. Write down two other conditions needed to make this into a binomial experiment. It is claimed that \(10 \%\) of students can tell the difference between two brands of baked beans. In a random sample of 250 students, 40 of them were able to distinguish the difference between the two brands.
  2. Using a normal approximation, test at the \(1 \%\) level of significance whether or not the claim is justified. Use a one-tailed test.
  3. Comment on the acceptability of the assumptions you needed to carry out the test.
Edexcel S2 2005 January Q5
13 marks Moderate -0.3
5. From company records, a manager knows that the probability that a defective article is produced by a particular production line is 0.032 . A random sample of 10 articles is selected from the production line.
  1. Find the probability that exactly 2 of them are defective. On another occasion, a random sample of 100 articles is taken.
  2. Using a suitable approximation, find the probability that fewer than 4 of them are defective. At a later date, a random sample of 1000 is taken.
  3. Using a suitable approximation, find the probability that more than 42 are defective.
    (6)
Edexcel S2 2005 January Q6
16 marks Standard +0.3
6. Over a long period of time, accidents happened on a stretch of road at random at a rate of 3 per month. Find the probability that
  1. in a randomly chosen month, more than 4 accidents occurred,
  2. in a three-month period, more than 4 accidents occurred. At a later date, a speed restriction was introduced on this stretch of road. During a randomly chosen month only one accident occurred.
  3. Test, at the \(5 \%\) level of significance, whether or not there is evidence to support the claim that this speed restriction reduced the mean number of road accidents occurring per month. The speed restriction was kept on this road. Over a two-year period, 55 accidents occurred.
  4. Test, at the \(5 \%\) level of significance, whether or not there is now evidence that this speed restriction reduced the mean number of road accidents occurring per month.
Edexcel S2 2005 January Q7
17 marks Standard +0.3
7. The random variable \(X\) has probability density function $$\mathrm { f } ( x ) = \begin{cases} k \left( - x ^ { 2 } + 5 x - 4 \right) , & 1 \leq x \leq 4 \\ 0 , & \text { otherwise } \end{cases}$$
  1. Show that \(k = \frac { 2 } { 9 }\). Find
  2. \(\mathrm { E } ( X )\),
  3. the mode of \(X\).
  4. the cumulative distribution function \(\mathrm { F } ( x )\) for all \(x\).
  5. Evaluate \(\mathrm { P } ( X \leq 2.5 )\),
  6. Deduce the value of the median and comment on the shape of the distribution.
Edexcel S2 2006 January Q1
7 marks Easy -1.3
  1. A fair coin is tossed 4 times.
Find the probability that
  1. an equal number of head and tails occur
  2. all the outcomes are the same,
  3. the first tail occurs on the third throw.
Edexcel S2 2006 January Q2
8 marks Moderate -0.3
2. Accidents on a particular stretch of motorway occur at an average rate of 1.5 per week.
  1. Write down a suitable model to represent the number of accidents per week on this stretch of motorway. Find the probability that
  2. there will be 2 accidents in the same week,
  3. there is at least one accident per week for 3 consecutive weeks,
  4. there are more than 4 accidents in a 2 week period.
Edexcel S2 2006 January Q3
8 marks Easy -1.2
3. The random variable \(X\) is uniformly distributed over the interval \([ - 1,5 ]\).
  1. Sketch the probability density function \(\mathrm { f } ( x )\) of \(X\). Find
  2. \(\mathrm { E } ( X )\),
  3. \(\operatorname { Var } ( \mathrm { X } )\),
  4. \(\mathrm { P } ( - 0.3 < X < 3.3 )\).
Edexcel S2 2006 January Q4
4 marks Standard +0.3
4. The random variable \(X \sim \mathrm {~B} ( 150,0.02 )\). Use a suitable approximation to estimate \(\mathrm { P } ( X > 7 )\).
Edexcel S2 2006 January Q5
15 marks Standard +0.3
5. A continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) where $$f ( x ) = \begin{cases} k x ( x - 2 ) , & 2 \leq x \leq 3 \\ 0 , & \text { otherwise } \end{cases}$$ where \(k\) is a positive constant.
  1. Show that \(k = \frac { 3 } { 4 }\). Find
  2. \(\mathrm { E } ( X )\),
  3. the cumulative distribution function \(\mathrm { F } ( x )\).
  4. Show that the median value of \(X\) lies between 2.70 and 2.75.
Edexcel S2 2006 January Q6
13 marks Moderate -0.8
6. A bag contains a large number of coins. Half of them are 1 p coins, one third are 2 p coins and the remainder are 5p coins.
  1. Find the mean and variance of the value of the coins. A random sample of 2 coins is chosen from the bag.
  2. List all the possible samples that can be drawn.
  3. Find the sampling distribution of the mean value of these samples.
Edexcel S2 2006 January Q7
19 marks Standard +0.3
7. A teacher thinks that \(20 \%\) of the pupils in a school read the Deano comic regularly. He chooses 20 pupils at random and finds 9 of them read the Deano.
    1. Test, at the \(5 \%\) level of significance, whether or not there is evidence that the percentage of pupils that read the Deano is different from 20\%. State your hypotheses clearly.
    2. State all the possible numbers of pupils that read the Deano from a sample of size 20 that will make the test in part (a)(i) significant at the \(5 \%\) level.
      (9) The teacher takes another 4 random samples of size 20 and they contain 1, 3, 1 and 4 pupils that read the Deano.
  2. By combining all 5 samples and using a suitable approximation test, at the \(5 \%\) level of significance, whether or not this provides evidence that the percentage of pupils in the school that read the Deano is different from 20\%.
  3. Comment on your results for the tests in part (a) and part (b).
Edexcel S2 2007 January Q1
4 marks Easy -1.8
  1. (a) Define a statistic.
A random sample \(X _ { 1 } , X _ { 2 } , \ldots , X _ { \mathrm { n } }\) is taken from a population with unknown mean \(\mu\).
(b) For each of the following state whether or not it is a statistic.
  1. \(\frac { X _ { 1 } + X _ { 4 } } { 2 }\),
  2. \(\frac { \sum X ^ { 2 } } { n } - \mu ^ { 2 }\).
Edexcel S2 2007 January Q2
5 marks Moderate -0.8
2. The random variable \(J\) has a Poisson distribution with mean 4.
  1. Find \(\mathrm { P } ( J \geqslant 10 )\). The random variable \(K\) has a binomial distribution with parameters \(n = 25 , p = 0.27\).
  2. Find \(\mathrm { P } ( K \leqslant 1 )\).
Edexcel S2 2007 January Q3
15 marks Standard +0.3
3. For a particular type of plant \(45 \%\) have white flowers and the remainder have coloured flowers. Gardenmania sells plants in batches of 12. A batch is selected at random. Calculate the probability that this batch contains
  1. exactly 5 plants with white flowers,
  2. more plants with white flowers than coloured ones. Gardenmania takes a random sample of 10 batches of plants.
  3. Find the probability that exactly 3 of these batches contain more plants with white flowers than coloured ones. Due to an increasing demand for these plants by large companies, Gardenmania decides to sell them in batches of 50 .
  4. Use a suitable approximation to calculate the probability that a batch of 50 plants contains more than 25 plants with white flowers.
Edexcel S2 2007 January Q4
12 marks Moderate -0.8
4. (a) State the condition under which the normal distribution may be used as an approximation to the Poisson distribution.
(b) Explain why a continuity correction must be incorporated when using the normal distribution as an approximation to the Poisson distribution. A company has yachts that can only be hired for a week at a time. All hiring starts on a Saturday.
During the winter the mean number of yachts hired per week is 5 .
(c) Calculate the probability that fewer than 3 yachts are hired on a particular Saturday in winter. During the summer the mean number of yachts hired per week increases to 25 . The company has only 30 yachts for hire.
(d) Using a suitable approximation find the probability that the demand for yachts cannot be met on a particular Saturday in the summer. In the summer there are 16 Saturdays on which a yacht can be hired.
(e) Estimate the number of Saturdays in the summer that the company will not be able to meet the demand for yachts.
Edexcel S2 2007 January Q5
12 marks Moderate -0.3
5. The continuous random variable \(X\) is uniformly distributed over the interval \(\alpha < x < \beta\).
  1. Write down the probability density function of \(X\), for all \(x\).
  2. Given that \(\mathrm { E } ( X ) = 2\) and \(\mathrm { P } ( X < 3 ) = \frac { 5 } { 8 }\) find the value of \(\alpha\) and the value of \(\beta\). A gardener has wire cutters and a piece of wire 150 cm long which has a ring attached at one end. The gardener cuts the wire, at a randomly chosen point, into 2 pieces. The length, in cm, of the piece of wire with the ring on it is represented by the random variable \(X\). Find
  3. \(\mathrm { E } ( X )\),
  4. the standard deviation of \(X\),
  5. the probability that the shorter piece of wire is at most 30 cm long.
Edexcel S2 2007 January Q6
13 marks Standard +0.3
6. Past records from a large supermarket show that \(20 \%\) of people who buy chocolate bars buy the family size bar. On one particular day a random sample of 30 people was taken from those that had bought chocolate bars and 2 of them were found to have bought a family size bar.
  1. Test at the \(5 \%\) significance level, whether or not the proportion \(p\), of people who bought a family size bar of chocolate that day had decreased. State your hypotheses clearly. The manager of the supermarket thinks that the probability of a person buying a gigantic chocolate bar is only 0.02 . To test whether this hypothesis is true the manager decides to take a random sample of 200 people who bought chocolate bars.
  2. Find the critical region that would enable the manager to test whether or not there is evidence that the probability is different from 0.02 . The probability of each tail should be as close to \(2.5 \%\) as possible.
  3. Write down the significance level of this test.
Edexcel S2 2007 January Q7
14 marks Standard +0.3
7. The continuous random variable \(X\) has cumulative distribution function $$\mathrm { F } ( x ) = \begin{cases} 0 , & x < 0 \\ 2 x ^ { 2 } - x ^ { 3 } , & 0 \leqslant x \leqslant 1 \\ 1 , & x > 1 \end{cases}$$
  1. Find \(\mathrm { P } ( X > 0.3 )\).
  2. Verify that the median value of \(X\) lies between \(x = 0.59\) and \(x = 0.60\).
  3. Find the probability density function \(\mathrm { f } ( x )\).
  4. Evaluate \(\mathrm { E } ( X )\).
  5. Find the mode of \(X\).
  6. Comment on the skewness of \(X\). Justify your answer.
Edexcel S2 2008 January Q1
4 marks Easy -2.0
  1. (a) Explain what you understand by a census.
Each cooker produced at GT Engineering is stamped with a unique serial number. GT Engineering produces cookers in batches of 2000. Before selling them, they test a random sample of 5 to see what electric current overload they will take before breaking down.
(b) Give one reason, other than to save time and cost, why a sample is taken rather than a census.
(c) Suggest a suitable sampling frame from which to obtain this sample.
(d) Identify the sampling units.
Edexcel S2 2008 January Q2
7 marks Standard +0.3
2. The probability of a bolt being faulty is 0.3 . Find the probability that in a random sample of 20 bolts there are
  1. exactly 2 faulty bolts,
  2. more than 3 faulty bolts. These bolts are sold in bags of 20. John buys 10 bags.
  3. Find the probability that exactly 6 of these bags contain more than 3 faulty bolts.
Edexcel S2 2008 January Q3
11 marks Moderate -0.8
3. (a) State two conditions under which a Poisson distribution is a suitable model to use in statistical work. The number of cars passing an observation point in a 10 minute interval is modelled by a Poisson distribution with mean 1.
(b) Find the probability that in a randomly chosen 60 minute period there will be
  1. exactly 4 cars passing the observation point,
  2. at least 5 cars passing the observation point. The number of other vehicles, other than cars, passing the observation point in a 60 minute interval is modelled by a Poisson distribution with mean 12.
    (c) Find the probability that exactly 1 vehicle, of any type, passes the observation point in a 10 minute period.