Questions S2 (1690 questions)

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Edexcel S2 Q6
14 marks Standard +0.3
6. Patients suffering from 'flu are treated with a drug. The number of days, \(t\), that it then takes for them to recover is modelled by the continuous random variable \(T\) with the probability density function $$\begin{array} { l l } \mathrm { f } ( t ) = \frac { 3 t ^ { 2 } ( 4 - t ) } { 64 } & 0 \leq t \leq 4 \\ \mathrm { f } ( t ) = 0 & \text { otherwise. } \end{array}$$
  1. Find the mean and standard deviation of \(T\).
  2. Find the probability that a patient takes more than 3 days to recover.
  3. Two patients are selected at random. Find the probability that they both recover within three days.
  4. Comment on the suitability of the model.
Edexcel S2 Q7
17 marks Standard +0.8
7. In a certain field, daisies are randomly distributed, at an average density of 0.8 daisies per \(\mathrm { cm } ^ { 2 }\). One particular patch, of area \(1 \mathrm {~cm} ^ { 2 }\), is selected at random. Assuming that the number of daisies per \(\mathrm { cm } ^ { 2 }\) has a Poisson distribution,
  1. find the probability that the chosen patch contains
    1. no daisies,
    2. one daisy. Ten such patches are chosen. Using your answers to part (a),
  2. find the probability that the total number of daisies is less than two.
  3. By considering the distribution of daisies over patches of \(10 \mathrm {~cm} ^ { 2 }\), use the Poisson distribution to find the probability that a particular area of \(10 \mathrm {~cm} ^ { 2 }\) contains no more than one daisy.
  4. Compare your answers to parts (b) and (c).
  5. Use a suitable approximation to find the probability that a patch of area \(1 \mathrm {~m} ^ { 2 }\) contains more than 8100 daisies.
Edexcel S2 Q1
4 marks Easy -1.8
Explain what is meant by
  1. a population,
  2. a sampling unit. Suggest suitable sampling frames for surveys of
  3. families who have holidays in Greece,
  4. mothers with children under two years old.
Edexcel S2 Q2
6 marks Easy -1.3
2. A continuous random variable \(X\) has the probability density function $$\begin{array} { l l } \mathrm { f } ( x ) = k & 5 \leq x \leq 15 , \\ \mathrm { f } ( x ) = 0 & \text { otherwise. } \end{array}$$
  1. Find \(k\) and specify the cumulative density function \(\mathrm { F } ( x )\).
  2. Write down the value of \(\mathrm { P } ( X < 8 )\).
Edexcel S2 Q3
8 marks Moderate -0.3
3. A coin is tossed 20 times, giving 16 heads.
  1. Test at the \(1 \%\) significance level whether the coin is fair, stating your hypotheses clearly.
  2. Find the critical region for the same test at the \(0.1 \%\) significance level.
Edexcel S2 Q4
9 marks Standard +0.8
4. Alison and Gemma play table tennis. Alison starts by serving for the first five points. The probability that she wins a point when serving is \(p\).
  1. Show that the probability that Alison is ahead at the end of her five serves is given by $$p ^ { 3 } \left( 6 p ^ { 2 } - 15 p + 10 \right) .$$
  2. Evaluate this probability when \(p = 0.6\).
Edexcel S2 Q5
15 marks Moderate -0.3
5. In a certain school, \(32 \%\) of Year 9 pupils are left-handed. A random sample of 10 Year 9 pupils is chosen.
  1. Find the probability that none are left-handed.
  2. Find the probability that at least two are left-handed.
  3. Use a suitable approximation to find the probability of getting more than 5 but less than 15 left-handed pupils in a group of 35 randomly selected Year 9 pupils.
    Explain what adjustment is necessary when using this approximation. \section*{STATISTICS 2 (A) TEST PAPER 3 Page 2}
Edexcel S2 Q6
15 marks Standard +0.3
A sample of radioactive material decays randomly, with an approximate mean of 1.5 counts per minute.
  1. Name a distribution that would be suitable for modelling the number of counts per minute. Give any parameters required for the model.
  2. Find the probability of at least 4 counts in a randomly chosen minute.
  3. Find the probability of 3 counts or fewer in a random interval lasting 5 minutes. More careful measurements, over 50 one-minute intervals, give the following data for \(x\), the number of counts per minute: $$\sum x = 84 , \quad \sum x ^ { 2 } = 226$$
  4. Decide whether these data support your answer to part (a).
  5. Use the improved data to find probability of exactly two counts in a given one-minute interval.
Edexcel S2 Q7
18 marks Standard +0.3
7. Each day on the way to work, a commuter encounters a similar traffic jam. The length of time, in 10-minute units, spent waiting in the traffic jam is modelled by the random variable \(T\) with the cumulative distribution function: $$\begin{array} { l l } \mathrm { F } ( t ) = 0 & t < 0 , \\ \mathrm {~F} ( t ) = \frac { t ^ { 2 } \left( 3 t ^ { 2 } - 16 t + 24 \right) } { 16 } & 0 \leq t \leq 2 , \\ \mathrm {~F} ( t ) = 1 & t > 2 . \end{array}$$
  1. Show that 0.77 is approximately the median value of \(T\).
  2. Given that he has already waited for 12 minutes, find the probability that he will have to wait another 3 minutes.
  3. Find, and sketch, the probability density function of \(T\).
  4. Hence find the modal value of \(T\).
  5. Comment on the validity of this model.
Edexcel S2 Q1
4 marks Easy -1.8
A random sample is to be taken from the A-level results obtained by the final-year students in a Sixth Form College. Suggest
  1. suitable sampling units,
  2. a suitable sampling frame.
  3. Would it be advisable simply to use the results of all those doing A-level Maths? Explain your answer.
Edexcel S2 Q2
5 marks Moderate -0.8
2. The random variable \(X\), which can take any value in the interval \(1 \leq X \leq n\), is modelled by the continuous uniform distribution with mean 12.
  1. Show that \(n = 23\) and find the variance of \(X\).
  2. Find \(\mathrm { P } ( 10 < X < 14 )\).
Edexcel S2 Q3
10 marks Moderate -0.8
3. The Driving Theory Test includes 30 questions which require one answer to be selected from four options.
  1. Phil ticks answers at random. Find how many of the 30 he should expect to get right.
  2. If he gets 15 correct, decide whether this is evidence that he has actually done some revision. Use a \(5 \%\) significance level. Another candidate, Sarah, has revised and has a 0.9 probability of getting each question right.
  3. Determine the expected number of answers that Sarah will get right.
  4. Find the probability that Sarah gets more than 25 correct answers out of 30.
Edexcel S2 Q4
10 marks Moderate -0.3
4. A continuous random variable \(X\) has probability density function $$\begin{array} { l l } \mathrm { f } ( x ) = 0 & x < 1 , \\ \mathrm { f } ( x ) = k x & 1 \leq x \leq 4 , \\ \mathrm { f } ( x ) = 0 & x > 4 . \end{array}$$
  1. Sketch a graph of \(\mathrm { f } ( x )\), and hence find the value of \(k\).
  2. Calculate the mean and the variance of \(X\). \section*{STATISTICS 2 (A)TEST PAPER 4 Page 2}
Edexcel S2 Q5
14 marks Standard +0.3
In World War II, the number of V2 missiles that landed on each square mile of London was, on average, \(3 \cdot 5\). Assuming that the hits were randomly distributed throughout London,
  1. suggest a suitable model for the number of hits on each square mile, giving a suitable value for any parameters.
  2. calculate the probability that a particular square mile received
    1. no hits,
    2. more than 7 hits.
  3. State, with a reason, whether the model is likely to be accurate. In contrast, the number of bombs weighing more than 1 ton landing on each square mile was 45 .
  4. Use a suitable approximation to find the probability that a randomly selected square mile received more than 60 such bombs. Explain what adjustment must be made when using this approximation.
Edexcel S2 Q6
16 marks Standard +0.3
6. In a fruit packing plant, apples are packed on to trays of 10 , and then checked for blemishes. The chance of any particular apple having a blemish is \(5 \%\). If a tray is selected at random, find
  1. the probability that at least two of the apples in it are blemished,
  2. the probability that exactly two are blemished. Trays are now packed in boxes of 50 trays each. In one such box, find
  3. the probability that at most one tray contains at least two blemished apples,
  4. the expected number of trays containing at least two blemished apples.
  5. Use a suitable approximation to find the probability that in a random selection of 20 trays there are more than 10 blemished apples.
Edexcel S2 Q7
16 marks Standard +0.8
7. The time, in hours, taken to run the London marathon is modelled by a continuous random variable \(T\) with the probability density function $$f ( t ) = \begin{cases} c ( t - 2 ) & 2 \leq t < 4 \\ \frac { 2 c ( 7 - t ) } { 3 } & 4 \leq t \leq 7 \\ 0 & \text { otherwise } \end{cases}$$
  1. Sketch the function \(\mathrm { f } ( t )\), and show that \(c = \frac { 1 } { 5 }\).
  2. Calculate the median value of \(T\).
  3. Make two critical comments about the model.
Edexcel S2 Q1
5 marks Easy -1.3
  1. Explain the difference between a discrete and a continuous variable. A random number generator on a calculator generates numbers, \(X\), to 3 decimal places, in the range 0 to 1 , e.g. 0.386 . The variable \(X\) may be modelled by a continuous uniform distribution, having the probability density function \(\mathrm { f } ( x )\), where $$\begin{array} { l l } \mathrm { f } ( x ) = 1 & 0 < x < 1 \\ \mathrm { f } ( x ) = 0 & \text { otherwise } \end{array}$$
  2. Explain why this model is not totally accurate.
  3. Sketch the cumulative distribution function of \(X\).
Edexcel S2 Q2
6 marks Easy -1.8
2. A video rental shop needs to find out whether or not videos have been rewound when they are returned; it will do this by taking a sample of returned videos
  1. State one advantage and one disadvantage of taking a sample.
  2. Suggest a suitable sampling frame.
  3. Describe the sampling units.
  4. Criticise the sampling method of looking at just one particular shelf of videos.
Edexcel S2 Q3
7 marks Standard +0.3
3. The random variable \(X\) is modelled by a binomial distribution \(\mathrm { B } ( n , p )\), with \(n = 20\) and \(p\) unknown. It is suspected that \(p = 0 \cdot 4\).
  1. Find the critical region for the test of \(\mathrm { H } _ { 0 } : p = 0.4\) against \(\mathrm { H } _ { 1 } : p \neq 0.4\), at the \(5 \%\) significance level.
  2. Find the critical region if, instead, the alternative hypothesis is \(\mathrm { H } _ { 1 } : p < 0.4\).
Edexcel S2 Q4
11 marks Moderate -0.8
4. A random variable \(X\) has the distribution \(\mathrm { B } ( 80,0.375 )\).
  1. Write down the mean and variance of \(X\).
  2. Use the Normal approximation to the binomial distribution to estimate \(\mathrm { P } ( X > 40 )\).
Edexcel S2 Q5
13 marks Standard +0.3
5. A traffic analyst is interested in the number of heavy lorries passing a certain junction. He counts the numbers of lorries in 100 five-minute intervals, and gets the following results:
Number of lorries in
five-minute interval, \(X\)
01234567
Number of intervals7132519151074
Q. 5 continued on next page ... \section*{STATISTICS 2 (A) TEST PAPER 9 Page 2} continued ...
  1. Show that the mean of \(X\) is 3 , and find the variance of \(X\).
  2. Give two reasons for thinking that \(X\) can be modelled by a Poisson distribution. (2 marks) After a new landfill site has been established nearby, a member of an environmental group notices that 18 lorries pass the junction in a period of 15 minutes. The group claims that this is evidence that the mean number of lorries per five-minute interval has increased.
  3. Test whether the group's claim is valid. Work at the \(5 \%\) significance level, and state your hypotheses clearly.
Edexcel S2 Q6
16 marks Standard +0.3
6. In a particular parliamentary constituency, the percentage of Conservative voters at the last election was \(35 \%\), and the percentage who voted for the Monster Raving Loony party was \(2 \%\).
  1. Find the probability that a random sample of 10 electors includes at least two Conservative voters. Use suitable approximations to find
  2. the probability that a random sample of 500 electors will include at least 200 who voted either Conservative or Monster Raving Loony,
  3. the probability that a random sample of 200 electors will have at least 5 Monster Raving Loony voters in it.
  4. One of (b) or (c) requires an adjustment to be made before a calculation is done. Explain what this adjustment is, and why it is necessary.
Edexcel S2 Q7
17 marks Standard +0.3
7. The fraction of sky covered by cloud is modelled by the random variable \(X\) with probability density function $$\begin{array} { l l } \mathrm { f } ( x ) = 0 & x < 0 \\ \mathrm { f } ( x ) = k x ^ { 2 } ( 1 - x ) & 0 \leq x \leq 1 , \\ \mathrm { f } ( x ) = 0 & x > 1 . \end{array}$$
  1. Find \(k\) and sketch the graph of \(\mathrm { f } ( x )\).
  2. Find the mean and the variance of \(X\).
  3. Find the cumulative distribution function \(\mathrm { F } ( x )\).
  4. Given that flying is prohibited when \(85 \%\) of the sky is covered by cloud, show that cloud conditions allow flying nearly \(90 \%\) of the time.
Edexcel S2 Q1
7 marks Moderate -0.3
  1. The continuous random variable \(X\) has the following cumulative distribution function:
$$F ( x ) = \begin{cases} 0 , & x < 2 \\ k \left( 19 x - x ^ { 2 } - 34 \right) , & 2 \leq x \leq 5 \\ 1 , & x > 5 \end{cases}$$
  1. Show that \(k = \frac { 1 } { 36 }\).
  2. Find \(\mathrm { P } ( X > 4 )\).
  3. Find and specify fully the probability density function \(\mathrm { f } ( x )\) of \(X\).
Edexcel S2 Q2
9 marks Easy -1.2
2. Suggest, with reasons, suitable distributions for modelling each of the following:
  1. the number of times the letter J occurs on each page of a magazine,
  2. the length of string left over after cutting as many 3 metre long pieces as possible from partly used balls of string,
  3. the number of heads obtained when spinning a coin 15 times.