Questions S2 (1690 questions)

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Edexcel S2 Specimen Q7
20 marks Standard +0.3
The continuous random variable \(X\) has probability density function f(\(x\)) given by $$\text{f}(x) = \begin{cases} \frac{1}{20}x^3, & 1 \leq x \leq 3 \\ 0, & \text{otherwise} \end{cases}$$
  1. Sketch f(\(x\)) for all values of \(x\). [3]
  2. Calculate E(\(X\)). [3]
  3. Show that the standard deviation of \(X\) is 0.459 to 3 decimal places. [3]
  4. Show that for \(1 \leq x \leq 3\), P(\(X \leq x\)) is given by \(\frac{1}{80}(x^4 - 1)\) and specify fully the cumulative distribution function of \(X\). [5]
  5. Find the interquartile range for the random variable \(X\). [4]
Some statisticians use the following formula to estimate the interquartile range: $$\text{interquartile range} = \frac{4}{3} \times \text{standard deviation}.$$
  1. Use this formula to estimate the interquartile range in this case, and comment. [2]
AQA S2 2010 June Q1
9 marks Moderate -0.3
Judith, the village postmistress, believes that, since moving the post office counter into the local pharmacy, the mean daily number of customers that she serves has increased from \(79\). In order to investigate her belief, she counts the number of customers that she serves on \(12\) randomly selected days, with the following results. \(88 \quad 81 \quad 84 \quad 89 \quad 90 \quad 77 \quad 72 \quad 80 \quad 82 \quad 81 \quad 75 \quad 85\) Stating a necessary distributional assumption, test Judith's belief at the \(5\%\) level of significance. [9 marks]
AQA S2 2010 June Q2
8 marks Standard +0.3
It is claimed that a new drug is effective in the prevention of sickness in holiday-makers. A sample of \(100\) holiday-makers was surveyed, with the following results.
SicknessNo sicknessTotal
Drug taken245680
No drug taken11920
Total3565100
Assuming that the \(100\) holiday-makers are a random sample, use a \(\chi^2\) test, at the \(5\%\) level of significance, to investigate the claim. [8 marks]
AQA S2 2010 June Q3
10 marks Moderate -0.8
The continuous random variable \(X\) has a rectangular distribution defined by $$f(x) = \begin{cases} k & -3k \leqslant x \leqslant k \\ 0 & \text{otherwise} \end{cases}$$
    1. Sketch the graph of f. [2 marks]
    2. Hence show that \(k = \frac{1}{2}\). [2 marks]
  2. Find the exact numerical values for the mean and the standard deviation of \(X\). [3 marks]
    1. Find \(\mathrm{P}\left(X \geqslant -\frac{1}{4}\right)\). [2 marks]
    2. Write down the value of \(\mathrm{P}\left(X \neq -\frac{1}{4}\right)\). [1 mark]
AQA S2 2010 June Q4
5 marks Standard +0.3
The error, \(X\) °C, made in measuring a patient's temperature at a local doctors' surgery may be modelled by a normal distribution with mean \(\mu\) and standard deviation \(\sigma\). The errors, \(x\) °C, made in measuring the temperature of each of a random sample of \(10\) patients are summarised below. $$\sum x = 0.35 \quad \text{and} \quad \sum(x - \bar{x})^2 = 0.12705$$ Construct a \(99\%\) confidence interval for \(\mu\), giving the limits to three decimal places. [5 marks]
AQA S2 2010 June Q5
13 marks Standard +0.3
The number of telephone calls received, during an \(8\)-hour period, by an IT company that request an urgent visit by an engineer may be modelled by a Poisson distribution with a mean of \(7\).
  1. Determine the probability that, during a given \(8\)-hour period, the number of telephone calls received that request an urgent visit by an engineer is:
    1. at most \(5\); [1 mark]
    2. exactly \(7\); [2 marks]
    3. at least \(5\) but fewer than \(10\). [3 marks]
  2. Write down the distribution for the number of telephone calls received each hour that request an urgent visit by an engineer. [1 mark]
  3. The IT company has \(4\) engineers available for urgent visits and it may be assumed that each of these engineers takes exactly \(1\) hour for each such visit. At \(10\)am on a particular day, all \(4\) engineers are available for urgent visits.
    1. State the maximum possible number of telephone calls received between \(10\)am and \(11\)am that request an urgent visit and for which an engineer is immediately available. [1 mark]
    2. Calculate the probability that at \(11\)am an engineer will not be immediately available to make an urgent visit. [4 marks]
  4. Give a reason why a Poisson distribution may not be a suitable model for the number of telephone calls per hour received by the IT company that request an urgent visit by an engineer. [1 mark]
AQA S2 2010 June Q6
18 marks Standard +0.3
  1. The number of strokes, \(R\), taken by the members of Duffers Golf Club to complete the first hole may be modelled by the following discrete probability distribution.
    \(r\)\(\leqslant 2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(\geqslant 9\)
    \(\mathrm{P}(R = r)\)\(0\)\(0.1\)\(0.2\)\(0.3\)\(0.25\)\(0.1\)\(0.05\)\(0\)
    1. Determine the probability that a member, selected at random, takes at least \(5\) strokes to complete the first hole. [1 mark]
    2. Calculate \(\mathrm{E}(R)\). [2 marks]
    3. Show that \(\mathrm{Var}(R) = 1.66\). [4 marks]
  2. The number of strokes, \(S\), taken by the members of Duffers Golf Club to complete the second hole may be modelled by the following discrete probability distribution.
    \(s\)\(\leqslant 2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(\geqslant 9\)
    \(\mathrm{P}(S = s)\)\(0\)\(0.15\)\(0.4\)\(0.3\)\(0.1\)\(0.03\)\(0.02\)\(0\)
    Assuming that \(R\) and \(S\) are independent:
    1. show that \(\mathrm{P}(R + S \leqslant 8) = 0.24\); [5 marks]
    2. calculate the probability that, when \(5\) members are selected at random, at least \(4\) of them complete the first two holes in fewer than \(9\) strokes; [3 marks]
    3. calculate \(\mathrm{P}(R = 4 \mid R + S \leqslant 8)\). [3 marks]
AQA S2 2010 June Q7
12 marks Standard +0.3
The random variable \(X\) has probability density function defined by $$f(x) = \begin{cases} \frac{1}{2} & 0 \leqslant x \leqslant 1 \\ \frac{1}{18}(x - 4)^2 & 1 \leqslant x \leqslant 4 \\ 0 & \text{otherwise} \end{cases}$$
  1. State values for the median and the lower quartile of \(X\). [2 marks]
  2. Show that, for \(1 \leqslant x \leqslant 4\), the cumulative distribution function, \(\mathrm{F}(x)\), of \(X\) is given by $$\mathrm{F}(x) = 1 + \frac{1}{54}(x - 4)^3$$ (You may assume that \(\int (x - 4)^2 \, dx = \frac{1}{3}(x - 4)^3 + c\).) [4 marks]
  3. Determine \(\mathrm{P}(2 \leqslant X \leqslant 3)\). [2 marks]
    1. Show that \(q\), the upper quartile of \(X\), satisfies the equation \((q - 4)^3 = -13.5\). [3 marks]
    2. Hence evaluate \(q\) to three decimal places. [1 mark]
AQA S2 2016 June Q1
13 marks Standard +0.3
The water in a pond contains three different species of a spherical green algae: Volvox globator, at an average rate of 4.5 spheres per 1 cm³; Volvox aureus, at an average rate of 2.3 spheres per 1 cm³; Volvox tertius, at an average rate of 1.2 spheres per 1 cm³. Individual Volvox spheres may be considered to occur randomly and independently of all other Volvox spheres. Random samples of water are collected from this pond. Find the probability that:
  1. a 1 cm³ sample contains no more than 5 Volvox globator spheres; [1 mark]
  2. a 1 cm³ sample contains at least 2 Volvox aureus spheres; [3 marks]
  3. a 5 cm³ sample contains more than 8 but fewer than 12 Volvox tertius spheres; [3 marks]
  4. a 0.1 cm³ sample contains a total of exactly 2 Volvox spheres; [3 marks]
  5. a 1 cm³ sample contains at least 1 sphere of each of the three different species of algae. [3 marks]
AQA S2 2016 June Q2
4 marks Moderate -0.3
A normally distributed variable, \(X\), has unknown mean \(\mu\) and unknown standard deviation \(\sigma\). A sample of 10 values of \(X\) was taken. From these 10 values, a 95% confidence interval for \(\mu\) was calculated to be $$(30.47, 32.93)$$ Use this confidence interval to find unbiased estimates for \(\mu\) and \(\sigma^2\). [4 marks]
AQA S2 2016 June Q3
13 marks Moderate -0.8
Members of a library may borrow up to 6 books. Past experience has shown that the number of books borrowed, \(X\), follows the distribution shown in the table.
\(x\)0123456
P(X = x)00.190.260.200.130.070.15
  1. Find the probability that a member borrows more than 3 books. [1 mark]
  2. Assume that the numbers of books borrowed by two particular members are independent. Find the probability that one of these members borrows more than 3 books and the other borrows fewer than 3 books. [3 marks]
  3. Show that the mean of \(X\) is 3.08, and calculate the variance of \(X\). [4 marks]
  4. One of the library staff notices that the values of the mean and the variance of \(X\) are similar and suggests that a Poisson distribution could be used to model \(X\). Without further calculations, give two reasons why a Poisson distribution would not be suitable to model \(X\). [2 marks]
  5. The library introduces a fee of 10 pence for each book borrowed. Assuming that the probabilities do not change, calculate:
    1. the mean amount that will be paid by a member;
    2. the standard deviation of the amount that will be paid by a member.
    [3 marks]
AQA S2 2016 June Q4
7 marks Moderate -0.8
A digital thermometer measures temperatures in degrees Celsius. The thermometer rounds down the actual temperature to one decimal place, so that, for example, 36.23 and 36.28 are both shown as 36.2. The error, \(X\) °C, resulting from this rounding down can be modelled by a rectangular distribution with the following probability density function. $$f(x) = \begin{cases} k & 0 \leqslant x \leqslant 0.1 \\ 0 & \text{otherwise} \end{cases}$$
  1. State the value of \(k\). [1 mark]
  2. Find the probability that the error resulting from this rounding down is greater than 0.03 °C. [1 mark]
    1. State the value for E(\(X\)).
    2. Use integration to find the value for E(\(X^2\)).
    3. Hence find the value for the standard deviation of \(X\).
    [5 marks]
AQA S2 2016 June Q5
13 marks Standard +0.3
A car manufacturer keeps a record of how many of the new cars that it has sold experience mechanical problems during the first year. The manufacturer also records whether the cars have a petrol engine or a diesel engine. Data for a random sample of 250 cars are shown in the table.
Problems during first 3 monthsProblems during first year but after first 3 monthsNo problems during first yearTotal
Petrol engine1035170215
Diesel engine482335
Total1443193250
  1. Use a \(\chi^2\)-test to investigate, at the 10% significance level, whether there is an association between the mechanical problems experienced by a new car from this manufacturer and the type of engine. [11 marks]
  2. Arisa is planning to buy a new car from this manufacturer. She would prefer to buy a car with a diesel engine, but a friend has told her that cars with diesel engines experience more mechanical problems. Based on your answer to part (a), state, with a reason, the advice that you would give to Arisa. [2 marks]
AQA S2 2016 June Q6
16 marks Standard +0.3
Gerald is a scientist who studies sand lizards. He believes that sand lizards on islands are, on average, shorter than those on the mainland. The population of sand lizards on the mainland has a mean length of 18.2 cm and a standard deviation of 1.8 cm. Gerald visited three islands, A, B and C, and measured the length, \(X\) centimetres, of each of a sample of \(n\) sand lizards on each island. The samples may be regarded as random. The data are shown in the table.
Island\(\sum x\)\(n\)
A1384.578
B116.97
C394.620
  1. Carry out a hypothesis test to investigate whether the data from Island A provide support for Gerald's belief at the 2% significance level. Assume that the standard deviation of the lengths of sand lizards on Island A is 1.8 cm. [7 marks]
  2. For Island B, it is also given that $$\sum(x - \bar{x})^2 = 22.64$$
    1. Construct a 95% confidence interval for \(\mu_B\), where \(\mu_B\) centimetres is the mean length of sand lizards on Island B. Assume that the lengths of sand lizards on Island B are normally distributed with unknown standard deviation.
    2. Comment on whether your confidence interval provides support for Gerald's belief.
    [7 marks]
  3. Comment on whether the data from Island C provide support for Gerald's belief. [2 marks]
AQA S2 2016 June Q7
9 marks Standard +0.3
The continuous random variable \(X\) has a cumulative distribution function F(\(x\)), where $$\text{F}(x) = \begin{cases} 0 & x < 1 \\ \frac{1}{4}(x - 1) & 1 \leqslant x < 4 \\ \frac{1}{16}(12x - x^2 - 20) & 4 \leqslant x \leqslant 6 \\ 1 & x > 6 \end{cases}$$
  1. Sketch the probability density function, f(\(x\)), on the grid below. [5 marks]
  2. Find the mean value of \(X\). [4 marks]
Edexcel S2 Q1
4 marks Easy -2.0
  1. Explain why it is often useful to take samples as a means of obtaining information. [2 marks]
  2. Briefly define the term sampling frame. [1 mark]
  3. Suggest a suitable sampling frame for a sample survey on a proposal to install speed humps on a road. [1 mark]
Edexcel S2 Q2
8 marks Standard +0.3
An insurance company conducts its business by using a Call Centre. The average number of calls per minute is 3.5. In the first minute after a TV advertisement is shown, the number of calls received is 7.
  1. Stating your hypotheses carefully, and working at the 5\% significance level, test whether the advertisement has had an effect. [5 marks]
  2. Find the number of calls that would be required in the first minute for the null hypothesis to be rejected at the 0.1\% significance level. [3 marks]
Edexcel S2 Q3
10 marks Standard +0.3
On average, 35\% of the candidates in a certain subject get an A or B grade in their exam. In a class of 20 students, find the probability that
  1. less than 5 get A or B grades, [2 marks]
  2. exactly 8 get A or B grades. [2 marks]
Five such classes of 20 students are combined to sit the exam.
  1. Use a suitable approximation to find the probability that less than a quarter of the total get A or B grades. [6 marks]
Edexcel S2 Q4
11 marks Standard +0.3
Light bulbs produced in a certain factory have lifetimes, in 100s of hours, whose distribution is modelled by the random variable \(X\) with probability density function $$f(x) = \frac{2x(3-x)}{9}, \quad 0 \leq x \leq 3;$$ $$f(x) = 0 \quad \text{otherwise}.$$
  1. Sketch \(f(x)\). [2 marks]
  2. Write down the mean lifetime of a bulb. [1 mark]
  3. Show that ten times as many bulbs fail before 200 hours as survive beyond 250 hours. [5 marks]
  4. Given that a bulb lasts for 200 hours, find the probability that it will then last for at least another 50 hours. [2 marks]
  5. State, with a reason, whether you consider that the density function \(f\) is a realistic model for the lifetimes of light bulbs. [1 mark]
Edexcel S2 Q5
11 marks Moderate -0.3
In a packet of 40 biscuits, the number of currants in each biscuit is as follows
Number of currants, \(x\)0123456
Number of biscuits49118431
  1. Find the mean and variance of the random variable \(X\) representing the number of currants per biscuit. [4 marks]
  2. State an appropriate model for the distribution of \(X\), giving two reasons for your answer. [2 marks]
Another machine produces biscuits with a mean of 1.9 currants per biscuit.
  1. Determine which machine is more likely to produce a biscuit with at least two currants. [5 marks]
Edexcel S2 Q6
12 marks Standard +0.3
A greengrocer sells apples from a barrel in his shop. He claims that no more than 5\% of the apples are of poor quality. When he takes 10 apples out for a customer, 2 of them are bad.
  1. Stating your hypotheses clearly, test his claim at the 1\% significance level. [5 marks]
  2. State an assumption that has been made about the selection of the apples. [1 mark]
  3. When five other customers also buy 10 apples each, the numbers of bad apples they get are 1, 3, 1, 2 and 1 respectively. By combining all six customers' results, and using a suitable approximation, test at the 1\% significance level whether the combined results provide evidence that the proportion of bad apples in the barrel is greater than 5\%. [5 marks]
  4. Comment briefly on your results in parts (a) and (c). [1 mark]
Edexcel S2 Q7
19 marks Standard +0.3
Some children are asked to mark the centre of a scale 10 cm long. The position they choose is indicated by the variable \(X\), where \(0 \leq X \leq 10\). Initially, \(X\) is modelled as a random variable with a continuous uniform distribution.
  1. Find the mean and the standard deviation of \(X\). [3 marks]
It is suggested that a better model would be the distribution with probability density function $$f(x) = cx, \quad 0 \leq x \leq 5, \quad f(x) = c(10-x), \quad 5 < x \leq 10, \quad f(x) = 0 \text{ otherwise}.$$
  1. Write down the mean of \(X\). [1 mark]
  2. Find \(c\), and hence find the standard deviation of \(X\) in this model. [7 marks]
  3. Find P(\(4 < X < 6\)). [3 marks]
It is then proposed that an even better model for \(X\) would be a Normal distribution with the mean and standard deviation found in parts (b) and (c).
  1. Use these results to find P(\(4 < X < 6\)) in the third model. [4 marks]
  2. Compare your answer with (d). Which model do you think is most appropriate? [1 mark]
Edexcel S2 Q1
3 marks Easy -1.8
  1. Explain briefly why it is often useful to take a sample from a population. [2 marks]
  2. Suggest a suitable sampling frame for a local council to use to survey attitudes towards a proposed new shopping centre. [1 mark]
Edexcel S2 Q2
6 marks Standard +0.3
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. [6 marks]
Edexcel S2 Q3
9 marks Standard +0.3
A random variable \(X\) has a Poisson distribution with a mean, \(\lambda\), which is assumed to equal 5.
  1. Find P\((X = 0)\). [1 mark]
  2. 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 \(\lambda < 5\). [8 marks]