Questions — Edexcel S2 (562 questions)

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Edexcel S2 2010 June Q4
10 marks Standard +0.3
The lifetime, \(X\), in tens of hours, of a battery has a cumulative distribution function F(x) given by $$\text{F}(x) = \begin{cases} 0 & x < 1 \\ \frac{4}{9}(x^2 + 2x - 3) & 1 \leqslant x \leqslant 1.5 \\ 1 & x > 1.5 \end{cases}$$
  1. Find the median of \(X\), giving your answer to 3 significant figures. [3]
  2. Find, in full, the probability density function of the random variable \(X\). [3]
  3. Find P(\(X \geqslant 1.2\)) [2]
A camping lantern runs on 4 batteries, all of which must be working. Four new batteries are put into the lantern.
  1. Find the probability that the lantern will still be working after 12 hours. [2]
Edexcel S2 2010 June Q5
15 marks Standard +0.3
A company has a large number of regular users logging onto its website. On average 4 users every hour fail to connect to the company's website at their first attempt.
  1. Explain why the Poisson distribution may be a suitable model in this case. [1]
Find the probability that, in a randomly chosen 2 hour period,
    1. all users connect at their first attempt,
    2. at least 4 users fail to connect at their first attempt.
    [5]
The company suffered from a virus infecting its computer system. During this infection it was found that the number of users failing to connect at their first attempt, over a 12 hour period, was 60.
  1. Using a suitable approximation, test whether or not the mean number of users per hour who failed to connect at their first attempt had increased. Use a 5\% level of significance and state your hypotheses clearly. [9]
Edexcel S2 2010 June Q6
15 marks Moderate -0.3
A company claims that a quarter of the bolts sent to them are faulty. To test this claim the number of faulty bolts in a random sample of 50 is recorded.
  1. Give two reasons why a binomial distribution may be a suitable model for the number of faulty bolts in the sample. [2]
  2. Using a 5\% significance level, find the critical region for a two-tailed test of the hypothesis that the probability of a bolt being faulty is \(\frac{1}{4}\). The probability of rejection in either tail should be as close as possible to 0.025 [3]
  3. Find the actual significance level of this test. [2]
In the sample of 50 the actual number of faulty bolts was 8.
  1. Comment on the company's claim in the light of this value. Justify your answer. [2]
The machine making the bolts was reset and another sample of 50 bolts was taken. Only 5 were found to be faulty.
  1. Test at the 1\% level of significance whether or not the probability of a faulty bolt has decreased. State your hypotheses clearly. [6]
Edexcel S2 2010 June Q7
15 marks Standard +0.3
The random variable \(Y\) has probability density function f(y) given by $$\text{f}(y) = \begin{cases} ky(a - y) & 0 \leqslant y \leqslant 3 \\ 0 & \text{otherwise} \end{cases}$$ where \(k\) and \(a\) are positive constants.
    1. Explain why \(a \geqslant 3\)
    2. Show that \(k = \frac{2}{9(a-2)}\)
    [6]
Given that E(Y) = 1.75
  1. show that \(a = 4\) and write down the value of \(k\). [6]
For these values of \(a\) and \(k\),
  1. sketch the probability density function, [2]
  2. write down the mode of \(Y\). [1]
Edexcel S2 2015 June Q1
11 marks Moderate -0.3
In a survey it is found that barn owls occur randomly at a rate of 9 per 1000 km\(^2\).
  1. Find the probability that in a randomly selected area of 1000 km\(^2\) there are at least 10 barn owls. [2]
  2. Find the probability that in a randomly selected area of 200 km\(^2\) there are exactly 2 barn owls. [3]
  3. Using a suitable approximation, find the probability that in a randomly selected area of 50000 km\(^2\) there are at least 470 barn owls. [6]
Edexcel S2 2015 June Q2
8 marks Standard +0.3
The proportion of houses in Radville which are unable to receive digital radio is 25\%. In a survey of a random sample of 30 houses taken from Radville, the number, \(X\), of houses which are unable to receive digital radio is recorded.
  1. Find P(5 \(\leq X < 11\)) [3]
A radio company claims that a new transmitter set up in Radville will reduce the proportion of houses which are unable to receive digital radio. After the new transmitter has been set up, a random sample of 15 houses is taken, of which 1 house is unable to receive digital radio.
  1. Test, at the 10\% level of significance, the radio company's claim. State your hypotheses clearly. [5]
Edexcel S2 2015 June Q3
14 marks Standard +0.3
A random variable \(X\) has probability density function given by $$f(x) = \begin{cases} kx^2 & 0 \leq x \leq 2 \\ k\left(1 - \frac{x}{6}\right) & 2 < x \leq 6 \\ 0 & \text{otherwise} \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = \frac{1}{4}\) [4]
  2. Write down the mode of \(X\). [1]
  3. Specify fully the cumulative distribution function F(\(x\)). [5]
  4. Find the upper quartile of \(X\). [4]
Edexcel S2 2015 June Q4
12 marks Moderate -0.3
The continuous random variable \(L\) represents the error, in metres, made when a machine cuts poles to a target length. The distribution of \(L\) is a continuous uniform distribution over the interval [0, 0.5]
  1. Find P(\(L < 0.4\)). [1]
  2. Write down E(\(L\)). [1]
  3. Calculate Var(\(L\)). [2]
A random sample of 30 poles cut by this machine is taken.
  1. Find the probability that fewer than 4 poles have an error of more than 0.4 metres from the target length. [3]
When a new machine cuts poles to a target length, the error, \(X\) metres, is modelled by the cumulative distribution function F(\(x\)) where $$\text{F}(x) = \begin{cases} 0 & x < 0 \\ 4x - 4x^2 & 0 \leq x \leq 0.5 \\ 1 & \text{otherwise} \end{cases}$$
  1. Using this model, find P(\(X > 0.4\)) [2]
A random sample of 100 poles cut by this new machine is taken.
  1. Using a suitable approximation, find the probability that at least 8 of these poles have an error of more than 0.4 metres. [3]
Edexcel S2 2015 June Q5
12 marks Standard +0.3
\emph{Liftsforall} claims that the lift they maintain in a block of flats breaks down at random at a mean rate of 4 times per month. To test this, the number of times the lift breaks down in a month is recorded.
  1. Using a 5\% level of significance, find the critical region for a two-tailed test of the null hypothesis that 'the mean rate at which the lift breaks down is 4 times per month'. The probability of rejection in each of the tails should be as close to 2.5\% as possible. [3]
Over a randomly selected 1 month period the lift broke down 3 times.
  1. Test, at the 5\% level of significance, whether \emph{Liftsforall}'s claim is correct. State your hypotheses clearly. [2]
  2. State the actual significance level of this test. [1]
The residents in the block of flats have a maintenance contract with \emph{Liftsforall}. The residents pay \emph{Liftsforall} £500 for every quarter (3 months) in which there are at most 3 breakdowns. If there are 4 or more breakdowns in a quarter then the residents do not pay for that quarter. \emph{Liftsforall} installs a new lift in the block of flats. Given that the new lift breaks down at a mean rate of 2 times per month,
  1. find the probability that the residents do not pay more than £500 to \emph{Liftsforall} in the next year. [6]
Edexcel S2 2015 June Q6
11 marks Moderate -0.3
A continuous random variable \(X\) has probability density function f(\(x\)) where $$f(x) = \begin{cases} kx^n & 0 \leq x \leq 1 \\ 0 & \text{otherwise} \end{cases}$$ where \(k\) and \(n\) are positive integers.
  1. Find \(k\) in terms of \(n\). [3]
  2. Find E(\(X\)) in terms of \(n\). [3]
  3. Find E(\(X^2\)) in terms of \(n\). [2]
Given that \(n = 2\)
  1. find Var(3\(X\)). [3]
Edexcel S2 2015 June Q7
7 marks Standard +0.8
A bag contains a large number of 10p, 20p and 50p coins in the ratio 1 : 2 : 2 A random sample of 3 coins is taken from the bag. Find the sampling distribution of the median of these samples. [7]
Edexcel S2 Specimen Q1
4 marks Easy -1.8
A school held a disco for years 9, 10 and 11 which was attended by 500 pupils. The pupils were registered as they entered the disco. The disco organisers were keen to assess the success of the event. They designed a questionnaire to obtain information from those who attended.
  1. State one advantage and one disadvantage of using a sample survey rather than a census. [2]
  2. Suggest a suitable sampling frame. [1]
  3. Identify the sampling units. [1]
Edexcel S2 Specimen Q2
7 marks Moderate -0.8
A piece of string \(AB\) has length 12 cm. A child cuts the string at a randomly chosen point \(P\), into two pieces. The random variable \(X\) represents the length, in cm, of the piece \(AP\).
  1. Suggest a suitable model for the distribution of \(X\) and specify it fully [2]
  2. Find the cumulative distribution function of \(X\). [4]
  3. Write down P(\(X < 4\)). [1]
Edexcel S2 Specimen Q3
7 marks Moderate -0.3
A manufacturer of chocolates produces 3 times as many soft centred chocolates as hard centred ones. Assuming that chocolates are randomly distributed within boxes of chocolates, find the probability that in a box containing 20 chocolates there are
  1. equal numbers of soft centred and hard centred chocolates, [3]
  2. fewer than 5 hard centred chocolates. [2]
A large box of chocolates contains 100 chocolates.
  1. Write down the expected number of hard centred chocolates in a large box. [2]
Edexcel S2 Specimen Q4
11 marks Standard +0.3
A company director monitored the number of errors on each page of typing done by her new secretary and obtained the following results:
No. of errors012345
No. of pages376560492712
  1. Show that the mean number of errors per page in this sample of pages is 2. [2]
  2. Find the variance of the number of errors per page in this sample. [2]
  3. Explain how your answers to parts (a) and (b) might support the director's belief that the number of errors per page could be modelled by a Poisson distribution. [1]
Some time later the director notices that a 4-page report which the secretary has just typed contains only 3 errors. The director wishes to test whether or not this represents evidence that the number of errors per page made by the secretary is now less than 2.
  1. Assuming a Poisson distribution and stating your hypothesis clearly, carry out this test. Use a 5\% level of significance. [6]
Edexcel S2 Specimen Q5
12 marks Standard +0.3
In Manuel's restaurant the probability of a customer asking for a vegetarian meal is 0.30. During one particular day in a random sample of 20 customers at the restaurant 3 ordered a vegetarian meal.
  1. Stating your hypotheses clearly, test, at the 5\% level of significance, whether or not the proportion of vegetarian meals ordered that day is unusually low. [5]
Manuel's chef believes that the probability of a customer ordering a vegetarian meal is 0.10. The chef proposes to take a random sample of 100 customers to test whether or not there is evidence that the proportion of vegetarian meals ordered is different from 0.10.
  1. Stating your hypotheses clearly, use a suitable approximation to find the critical region for this test. The probability for each tail of the region should be as close as possible to 2.5\%. [6]
  2. State the significance level of this test giving your answer to 2 significant figures. [1]
Edexcel S2 Specimen Q6
14 marks Standard +0.3
A biologist is studying the behaviour of sheep in a large field. The field is divided up into a number of equally sized squares and the average number of sheep per square is 2.25. The sheep are randomly spread throughout the field.
  1. Suggest a suitable model for the number of sheep in a square and give a value for any parameter or parameters required. [1]
Calculate the probability that a randomly selected sample square contains
  1. no sheep, [1]
  2. more than 2 sheep. [4]
A sheepdog has been sent into the field to round up the sheep.
  1. Explain why the model may no longer be applicable. [1]
In another field, the average number of sheep per square is 20 and the sheep are randomly scattered throughout the field.
  1. Using a suitable approximation, find the probability that a randomly selected square contains fewer than 15 sheep. [7]
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]
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]