Questions S2 (1597 questions)

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Edexcel S2 Q2
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
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
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
  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
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
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
  1. (a) Explain briefly why it is often useful to take a sample from a population.
    (b) Suggest a suitable sampling frame for a local council to use to survey attitudes towards a proposed new shopping centre.
  2. A certain type of lettuce seed has a \(12 \%\) failure rate for germination. In a new sample of 25 genetically modified seeds, only 1 did not germinate.
    Clearly stating your hypotheses, test, at the \(1 \%\) significance level, whether the GM seeds are better.
  3. A random variable \(X\) has a Poisson distribution with a mean, \(\lambda\), which is assumed to equal 5 .
    (a) Find \(\mathrm { P } ( X = 0 )\).
    (b) In 100 measurements, the value 0 occurs three times. Find the highest significance level at which you should reject the original hypothesis in favour of \(\lambda < 5\).
  4. The waiting time, in minutes, at a dentist is modelled by the continuous random variable \(T\) with probability density function
$$\begin{array} { l l } \mathrm { f } ( t ) = k ( 10 - t ) & 0 \leq t \leq 10
\mathrm { f } ( t ) = 0 & \text { otherwise. } \end{array}$$ (a) Sketch the graph of \(\mathrm { f } ( t )\) and find the value of \(k\).
(b) Find the mean value of \(T\).
(c) Find the 95th percentile of \(T\).
(d) State whether you consider this function to be a sensible model for \(T\) and suggest how it could be modified to provide a better model.
Edexcel S2 Q5
5. A textbook contains, on average, 1.2 misprints per page. Assuming that the misprints are randomly distributed throughout the book,
  1. specify a suitable model for \(X\), the random variable representing the number of misprints on a given page.
  2. Find the probability that a particular page has more than 2 misprints.
  3. Find the probability that Chapter 1, with 8 pages, has no misprints at all. Chapter 2 is longer, at 20 pages.
  4. Use a suitable approximation to find the probability that Chapter 2 has less than ten misprints altogether. Explain what adjustment is necessary when making this approximation. \section*{STATISTICS 2 (A) TEST PAPER 5 Page 2}
Edexcel S2 Q6
  1. On a production line, bags are filled with cement and weighed as they emerge. It is found that \(20 \%\) of the bags are underweight. In a random sample consisting of \(n\) bags, the variance of the number of underweight bags is found to be \(2 \cdot 4\).
    1. Show that \(n = 15\).
    2. Use cumulative binomial probability tables to find the probability that, in a further random sample of 15 bags, the number that are underweight is
      1. less than 3 ,
      2. at least 5 .
    Ten samples of 15 bags each are tested. Find the probability that
  2. all these batches contain less than 5 underweight bags,
  3. the fourth batch tested is the first to contain less than 5 underweight bags.
Edexcel S2 Q7
7. A continuous random variable \(X\) has a probability density function given by $$\begin{array} { l l } \mathrm { f } ( x ) = \frac { x ^ { 2 } } { 312 } & 4 \leq x \leq 10
\mathrm { f } ( x ) = 0 & \text { otherwise. } \end{array}$$
  1. Find \(\mathrm { E } ( X )\).
  2. Find the variance of \(X\).
  3. Find the cumulative distribution function \(\mathrm { F } ( x )\), for all values of \(x\).
  4. Hence find the median value of \(X\).
  5. Write down the modal value of \(X\). It is sometimes suggested that, for most distributions, $$2 \times ( \text { median } - \text { mean } ) \approx \text { mode } - \text { median } .$$
  6. Show that this result is not satisfied in this case, and suggest a reason why.
Edexcel S2 Q1
\begin{enumerate} \item An insurance company is investigating how often its customers crash their cars.
  1. Suggest an appropriate sampling frame.
  2. Describe the sampling units.
  3. State the advantage of a sample survey over a census in this case. \item A searchlight is rotating in a horizontal circle. It is assumed that that, at any moment, the centre of its beam is equally likely to be pointing in any direction. The random variable \(X\) represents this direction, expressed as a bearing in the range \(000 ^ { \circ }\) to \(360 ^ { \circ }\).
Edexcel S2 Q4
4. A certain Sixth Former is late for school once a week, on average. In a particular week of 5 days, find the probability that
  1. he is not late at all,
  2. he is late more than twice. In a half term of seven weeks, lateness on more than ten occasions results in loss of privileges the following half term.
  3. Use the Normal approximation to estimate the probability that he loses his privileges. \section*{STATISTICS 2 (A)TEST PAPER 6 Page 2}
Edexcel S2 Q5
  1. A certain type of steel is produced in a foundry. It has flaws (small bubbles) randomly distributed, and these can be detected by X-ray analysis. On average, there are 0.1 bubbles per \(\mathrm { cm } ^ { 3 }\), and the number of bubbles per \(\mathrm { cm } ^ { 3 }\) has a Poisson distribution.
    In an ingot of \(40 \mathrm {~cm} ^ { 3 }\), find
    1. the probability that there are less than two bubbles,
    2. the probability that there are more than 3 but less than 10 bubbles.
    A new machine is being considered. Its manufacturer claims that it produces fewer bubbles per \(\mathrm { cm } ^ { 3 }\). In a sample ingot of \(60 \mathrm {~cm} ^ { 3 }\), there is just one bubble.
  2. Carry out a hypothesis test at the \(1 \%\) significance level to decide whether the new machine is better. State your hypotheses and conclusion carefully.
Edexcel S2 Q6
6. A random variable \(X\) has a probability density function given by $$\begin{array} { l l } \mathrm { f } ( x ) = \frac { 4 x ^ { 2 } ( 3 - x ) } { 27 } & 0 \leq x \leq 3
\mathrm { f } ( x ) = 0 & \text { otherwise. } \end{array}$$
  1. Find the mode of \(X\).
  2. Find the mean of \(X\).
  3. Specify completely the cumulative distribution function of \(X\).
  4. Deduce that the median, \(m\), of \(X\) satisfies the equation \(m ^ { 4 } - 4 m ^ { 3 } + 13 \cdot 5 = 0\), and hence show that \(1.84 < m < 1.85\).
  5. What do these results suggest about the skewness of the distribution?
Edexcel S2 Q7
7. A corner-shop has weekly sales (in thousands of pounds), which can be modelled by the continuous random variable \(X\) with probability density function $$\begin{array} { l l } \mathrm { f } ( x ) = k ( x - 2 ) ( 10 - x ) & 2 \leq x \leq 10
\mathrm { f } ( x ) = 0 & \text { otherwise } . \end{array}$$
  1. Show that \(k = \frac { 3 } { 256 }\) and write down the mean of \(X\).
  2. Find the standard deviation of the weekly sales.
  3. Find the probability that the sales exceed \(\pounds 8000\) in any particular week. If the sales exceed \(\pounds 8000\) per week for 4 consecutive weeks, the manager gets a bonus.
  4. Find the probability that the manager gets a bonus in February.
Edexcel S2 Q1
\begin{enumerate} \item A company that makes ropes for mountaineering wants to assess the breaking strain of its ropes.
  1. Explain why a sample survey, and not a census, should be used.
  2. Suggest an appropriate sampling frame. \item It is thought that a random variable \(X\) has a Poisson distribution whose mean, \(\lambda\), is equal to 8 . Find the critical region to test the hypothesis \(\mathrm { H } _ { 0 } : \lambda = 8\) against the hypothesis \(\mathrm { H } _ { 1 } : \lambda < 8\), working at the \(1 \%\) significance level. \item A child cuts a 30 cm piece of string into two parts, cutting at a random point.
Edexcel S2 Q6
  1. When a park is redeveloped, it is claimed that \(70 \%\) of the local population approve of the new design. Assuming this to be true, find the probability that, in a group of 10 residents selected at random,
    1. 6 or more approve,
    2. exactly 7 approve.
    A conservation group, however, carries out a survey of 20 people, and finds that only 9 approve.
  2. Use this information to carry out a hypothesis test on the original claim, working at the \(5 \%\) significance level. State your conclusion clearly. If the conservationists are right, and only \(45 \%\) approve of the new park,
  3. use a suitable approximation to the binomial distribution to estimate the probability that in a larger survey, of 500 people, less than half will approve.
Edexcel S2 Q7
7. A continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by $$\begin{array} { l l } \mathrm { f } ( x ) = \frac { 2 x } { 3 } & 0 \leq x < 1
\mathrm { f } ( x ) = 1 - \frac { x } { 3 } & 1 \leq x \leq 3
\mathrm { f } ( x ) = 0 & \text { otherwise. } \end{array}$$
  1. Sketch the graph of \(\mathrm { f } ( x )\) for all \(x\).
  2. Find the mean of \(X\).
  3. Find the standard deviation of \(X\).
  4. Show that the cumulative distribution function of \(X\) is given by $$\mathrm { F } ( x ) = \frac { x ^ { 2 } } { 3 } \quad 0 \leq x < 1$$ and find \(\mathrm { F } ( x )\) for \(1 \leq x \leq 3\).
Edexcel S2 Q1
  1. (a) Briefly explain the difference between a one-tailed test and a two-tailed test.
    (b) State, with a reason, which type of test would be more appropriate to test the claim that this decade's average temperature is greater than the last decade's.
  2. (a) Give one advantage and one disadvantage of
    1. a sample survey,
    2. a census.
      (b) Suggest a situation in which each could be used.
    3. A pharmaceutical company produces an ointment for earache that, in \(80 \%\) of cases, relieves pain within 6 hours. A new drug is tried out on a sample of 25 people with earache, and 24 of them get better within 6 hours.
      (a) Test, at the \(5 \%\) significance level, the claim that the new treatment is better than the old one. State your hypotheses carefully.
    A rival company suggests that the sample does not give a conclusive result;
    (b) Might they be right, and how could a more conclusive statement be achieved?
Edexcel S2 Q4
4. A centre for receiving calls for the emergency services gets an average of \(3 \cdot 5\) emergency calls every minute. Assuming that the number of calls per minute follows a Poisson distribution,
  1. find the probability that more than 6 calls arrive in any particular minute. Each operator takes a mean time of 2 minutes to deal with each call, and therefore seven operators are necessary to cope with the average demand.
  2. Find how many operators are required for there to be a \(99 \%\) probability that a call can be dealt with immediately. It is found from experience that a major disaster creates a surge of emergency calls. Taking the null hypothesis \(\mathrm { H } _ { 0 }\) that there is no disaster,
  3. find the number of calls that need to be received in one minute to disprove \(\mathrm { H } _ { 0 }\) at the \(0.1 \%\) significance level.
Edexcel S2 Q5
5. The random variable \(X\) has a continuous uniform distribution on the interval \(a \leq X \leq 3 a\).
  1. Without assuming any standard results, prove that \(\mu\), the mean value of \(X\), is equal to \(2 a\) and derive an expression for \(\sigma ^ { 2 }\), the variance of \(X\), in terms of \(a\).
  2. Find the probability that \(| X - \mu | < \sigma\) and compare this with the same probability when \(x\) is modelled by a Normal distribution with the same mean and variance. \section*{STATISTICS 2 (A) TEST PAPER 8 Page 2}
Edexcel S2 Q6
  1. Two people are playing darts. Peg hits points randomly on the circular board, whose radius is \(a\). If the distance from the centre \(O\) of the point that she hits is modelled by the variable \(R\),
    1. explain why the cumulative distribution function \(\mathrm { F } ( r )\) is given by
    $$\begin{array} { l l } \mathrm { F } ( r ) = 0 & r < 0 ,
    \mathrm {~F} ( r ) = \frac { r ^ { 2 } } { a ^ { 2 } } & 0 \leq r \leq a ,
    \mathrm {~F} ( r ) = 1 & r > a . \end{array}$$
  2. By first finding the probability density function of \(R\), show that the mean distance from \(O\) of the points that Peg hits is \(\frac { 2 a } { 3 }\). Bob, a more experienced player, aims for \(O\), and his points have a distance \(X\) from \(O\) whose cumulative distribution function is $$\mathrm { F } ( x ) = 0 , x < 0 ; \quad \mathrm { F } ( x ) = \frac { x } { a } \left( 2 - \frac { x } { a } \right) , 0 \leq x \leq a ; \quad \mathrm { F } ( x ) = 1 , x > a .$$
  3. Find the probability density function of \(X\), and explain why it shows that Bob is aiming for \(O\).
Edexcel S2 Q7
7. In an orchard, all the trees are either apple or pear trees. There are four times as many apple trees as pear trees. Find the probability that, in a random sample of 10 trees, there are
  1. equal numbers of apple and pear trees,
  2. more than 7 apple trees. In a sample of 60 trees in the orchard,
  3. find the expected number of pear trees.
  4. Calculate the standard deviation of the number of pear trees and compare this result with the standard deviation of the number of apple trees.
  5. Find the probability that exactly 35 in the sample of 60 trees are pear trees.
  6. Find an approximate value for the probability that more than 15 of the 60 trees are pear trees.
Edexcel S2 Q1
  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
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.