Questions — Edexcel S2 (562 questions)

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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
  1. 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 .
  2. 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. (a) 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 $$\begin{array} { l l } \mathrm { f } ( x ) = 1 & 0 < x < 1 \\ \mathrm { f } ( x ) = 0 & \text { otherwise } \end{array}$$ (b) Explain why this model is not totally accurate.
(c) 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}
  1. 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.
  2. 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.
Edexcel S2 Q3
9 marks Standard +0.3
3. A primary school teacher finds that exactly half of his year 6 class have mobile phones. He decides to investigate whether the proportion of pupils with mobile phones is different from 0.5 in the year 5 class at his school. There are 25 pupils in the year 5 class.
  1. State the hypotheses that he should use.
  2. Find the largest critical region for this test such that the probability in each "tail" is less than \(2.5 \%\).
  3. Determine the significance level of this test. He finds that eight of the year 5 pupils have mobile phones and concludes that there is not sufficient evidence of the proportion being different from 0.5
  4. Stating the new hypotheses clearly, find if the number of year 5 pupils with mobile phones would have been significant if he had tested whether or not the proportion was less than 0.5 and used the largest critical region with a probability of less than \(5 \%\).
    (3 marks)
Edexcel S2 Q4
10 marks Standard +0.3
4. A hardware store is open on six days each week. On average the store sells 8 of a particular make of electric drill each week. Find the probability that the store sells
  1. no more than 4 of the drills in a week,
  2. more than 2 of the drills in one day. The store receives one delivery of drills at the same time each week.
  3. Find the number of drills that need to be in stock after a delivery for there to be at most a 5\% chance of the store not having sufficient drills to meet demand before the next delivery.
    (3 marks)
    [0pt]
Edexcel S2 Q5
10 marks Moderate -0.3
5. In a party game, a bottle is spun and whoever it points to when it stops has to play next. The acute angle, in degrees, that the bottle makes with the side of the room is modelled by a rectangular distribution over the interval [0,90]. Find the probability that on one spin this angle is
  1. between \(25 ^ { \circ }\) and \(38 ^ { \circ }\),
  2. \(45 ^ { \circ }\) to the nearest degree. The bottle is spun ten times.
  3. Find the probability that the acute angle it makes with the side of the room is less than \(10 ^ { \circ }\) more than twice.
Edexcel S2 Q6
12 marks Standard +0.3
6. A teacher is monitoring attendance at lessons in her department. She believes that the number of students absent from each lesson follows a Poisson distribution and wished to test the null hypothesis that the mean is 2.5 against the alternative hypothesis that it is greater than 2.5 She visits one lesson and decides on a critical region of 6 or more students absent.
  1. Find the significance level of this test.
  2. State any assumptions made in carrying out this test and comment on their validity. The teacher decides to undertake a wider study by looking at a sample of all the lessons that have taken place in the department during the previous four weeks.
  3. Suggest a suitable sampling frame. She finds that there have been 96 pupils absent from the 30 lessons in her sample.
  4. Using a suitable approximation, test at the \(5 \%\) level of significance the null hypothesis that the mean is 2.5 students absent per lesson against the alternative hypothesis that it is greater than 2.5. You may assume that the number of absences follows a Poisson distribution.
    (6 marks)
Edexcel S2 Q7
18 marks Standard +0.3
7. In a competition at a funfair, participants have to stay on a log being rotated in a pool of water for as long as possible. The length of time, in tens of seconds, that the competitors stay on the log is modelled by the random variable \(T\) with the following probability density function: $$\mathrm { f } ( t ) = \begin{cases} k ( t - 3 ) ^ { 2 } , & 0 \leq t \leq 3 \\ 0 , & \text { otherwise } \end{cases}$$
  1. Show that \(k = \frac { 1 } { 9 }\).
  2. Sketch f \(( t )\) for all values of \(t\).
  3. Show that the mean time that competitors stay on the \(\log\) is 7.5 seconds. When the competition is next run the organisers decide to make it easier at first by spinning the log more slowly and then increasing the speed of rotation. The length of time, in tens of seconds, that the competitors now stay on the log is modelled by the random variable \(S\) with the following probability density function: $$f ( s ) = \begin{cases} \frac { 1 } { 12 } \left( 8 - s ^ { 3 } \right) , & 0 \leq s \leq 2 \\ 0 , & \text { otherwise } \end{cases}$$
  4. Find the change in the mean time that competitors stay on the log.
Edexcel S2 Q1
7 marks Easy -1.2
  1. (a) The random variable \(X\) follows a Poisson distribution with a mean of 1.4
Find \(\mathrm { P } ( X \leq 3 )\).
(b) The random variable \(Y\) follows a binomial distribution such that \(Y \sim \mathrm {~B} ( 20,0.6 )\). Find \(\mathrm { P } ( Y \leq 12 )\).
(4 marks)
Edexcel S2 Q2
9 marks Moderate -0.8
2. A driving instructor keeps records of all the learners she has taught. In order to analyse her success rate she wishes to take a random sample of 120 of these learners.
  1. Suggest a suitable sampling frame and identify the sampling units. She believes that only 1 in 20 of the people she teaches fail to pass their test in their first two attempts. She decides to use her sample to test whether or not the proportion is different from this.
  2. Using a suitable approximation and stating clearly the hypotheses she should use, find the largest critical region for this test such that the probability in each "tail" is less than \(2.5 \%\).
  3. State the significance level of this test.
Edexcel S2 Q3
9 marks Standard +0.3
3. In an old computer game a white square representing a ball appears at random at the top of the playing area, which is 24 cm wide, and moves down the screen. The continuous random variable \(X\) represents the distance, in centimetres, of the dot from the left-hand edge of the screen when it appears. The distribution of \(X\) is rectangular over the interval [4,28].
  1. Find the mean and variance of \(X\).
  2. Find \(\mathrm { P } ( | X - 16 | < 3 )\). During a single game, a player receives 12 "balls".
  3. Find the probability that the ball appears within 3 cm of the middle of the top edge of the playing area more than four times in a single game.
    (3 marks)
Edexcel S2 Q4
14 marks Moderate -0.8
4. A music website is visited by an average of 30 different people per hour on a weekday evening. The site's designer believes that the number of visitors to the site per hour can be modelled by a Poisson distribution.
  1. State the conditions necessary for a Poisson distribution to be applicable and comment on their validity in this case. Assuming that the number of visitors does follow a Poisson distribution, find the probability that there will be
  2. less than two visitors in a 10 -minute interval,
  3. at least ten visitors in a 15-minute interval.
  4. Using a suitable approximation, find the probability of the site being visited by more than 100 people between 6 pm and 9 pm on a Thursday evening.
    (5 marks)
Edexcel S2 Q5
17 marks Standard +0.3
5. Four coins are flipped together and the random variable \(H\) represents the number of heads obtained. Assuming that the coins are fair,
  1. suggest with reasons a suitable distribution for modelling \(H\) and give the value of any parameters needed,
  2. show that the probability of obtaining more heads than tails is \(\frac { 5 } { 16 }\). The four coins are flipped 5 times and more heads are obtained than tails 4 times.
  3. Stating your hypotheses clearly, test at the \(5 \%\) level of significance whether or not there is evidence of the probability of getting more heads than tails being more than \(\frac { 5 } { 16 }\). Given that the four coins are all biased such that the chance of each one showing a head is 50\% more than the chance of it showing a tail,
  4. find the probability of obtaining more heads than tails when the four coins are flipped together.
Edexcel S2 Q6
19 marks Standard +0.3
6. The continuous random variable \(X\) has the following probability density function: $$f ( x ) = \begin{cases} \frac { 1 } { 16 } x , & 2 \leq x \leq 6 \\ 0 , & \text { otherwise } \end{cases}$$
  1. Sketch \(\mathrm { f } ( x )\) for all values of \(x\).
  2. Find \(\mathrm { E } ( X )\).
  3. Show that \(\operatorname { Var } ( X ) = \frac { 11 } { 9 }\).
  4. Define fully the cumulative distribution function \(\mathrm { F } ( x )\) of \(X\).
  5. Show that the interquartile range of \(X\) is \(2 ( \sqrt { } 7 - \sqrt { 3 } )\). END