Questions — Edexcel (10514 questions)

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Edexcel S2 Q2
5 marks Standard +0.3
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 \(H_0 : \lambda = 8\) against the hypothesis \(H_1 : \lambda < 8\), working at the 1\% significance level. [5 marks]
Edexcel S2 Q3
7 marks Moderate -0.3
A child cuts a 30 cm piece of string into two parts, cutting at a random point.
  1. Name the distribution of \(L\), the length of the longer part of string, and sketch the probability density function for \(L\). [4 marks]
  2. Find the probability that one part of the string is more than twice as long as the other. [3 marks]
Edexcel S2 Q4
9 marks Standard +0.3
A supplier of widgets claims that only 10\% of his widgets have faults.
  1. In a consignment of 50 widgets, 9 are faulty. Test, at the 5\% significance level, whether this suggests that the supplier's claim is false. [6 marks]
  2. Find how many faulty widgets would be needed to provide evidence against the claim at the 1\% significance level. [3 marks]
Edexcel S2 Q5
12 marks Moderate -0.3
In a survey of 22 families, the number of children, \(X\), in each family was given by the following table, where \(f\) denotes the frequency:
\(X\)012345
\(f\)385321
  1. Find the mean and variance of \(X\). [4 marks]
  2. Explain why these results suggest that \(X\) may follow a Poisson distribution. [1 mark]
  3. State another feature of the data that suggests a Poisson distribution. [1 mark]
It is sometimes suggested that the number of children in a family follows a Poisson distribution with mean 2ยท4. Assuming that this is correct,
  1. find the probability that a family has less than two children. [3 marks]
  2. Use this result to find the probability that, in a random sample of 22 families, exactly 11 of the families have less than two children. [3 marks]
Edexcel S2 Q6
18 marks Standard +0.3
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, [3 marks]
  2. exactly 7 approve. [3 marks]
A conservation group, however, carries out a survey of 20 people, and finds that only 9 approve.
  1. Use this information to carry out a hypothesis test on the original claim, working at the 5\% significance level. State your conclusion clearly. [5 marks]
If the conservationists are right, and only 45\% approve of the new park,
  1. 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. [7 marks]
Edexcel S2 Q7
21 marks Standard +0.3
A continuous random variable \(X\) has probability density function f(x) given by $$\text{f(x)} = \frac{2x}{3} \quad 0 \leq x < 1,$$ $$\text{f(x)} = 1 - \frac{x}{3} \quad 1 \leq x \leq 3,$$ $$\text{f(x)} = 0 \quad \text{otherwise}.$$
  1. Sketch the graph of f(x) for all \(x\). [3 marks]
  2. Find the mean of \(X\). [5 marks]
  3. Find the standard deviation of \(X\). [7 marks]
  4. Show that the cumulative distribution function of \(X\) is given by $$\text{F(x)} = \frac{x^2}{3} \quad 0 \leq x < 1,$$ and find F(x) for \(1 \leq x \leq 3\). [6 marks]
Edexcel S2 Q1
4 marks Easy -1.8
  1. Briefly explain the difference between a one-tailed test and a two-tailed test. [2 marks]
  2. 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 marks]
Edexcel S2 Q2
6 marks Easy -2.0
  1. Give one advantage and one disadvantage of
    1. a sample survey, [2 marks]
    2. a census. [2 marks]
  2. Suggest a situation in which each could be used. [2 marks]
Edexcel S2 Q3
9 marks Standard +0.3
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.
  1. Test, at the 5\% significance level, the claim that the new treatment is better than the old one. State your hypotheses carefully. [6 marks] A rival company suggests that the sample does not give a conclusive result;
  2. Might they be right, and how could a more conclusive statement be achieved? [3 marks]
Edexcel S2 Q4
9 marks Standard +0.3
A centre for receiving calls for the emergency services gets an average of 3.5 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. [3 marks] 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. [3 marks] It is found from experience that a major disaster creates a surge of emergency calls. Taking the null hypothesis \(H_0\) that there is no disaster,
  3. find the number of calls that need to be received in one minute to disprove \(H_0\) at the 0.1 \% significance level. [3 marks]
Edexcel S2 Q5
13 marks Standard +0.3
The random variable \(X\) has a continuous uniform distribution on the interval \(a \leq X \leq 3a\).
  1. Without assuming any standard results, prove that \(\mu\), the mean value of \(X\), is equal to \(2a\) and derive an expression for \(\sigma^2\), the variance of \(X\), in terms of \(a\). [7 marks]
  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. [6 marks]
Edexcel S2 Q6
16 marks Standard +0.3
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 \(F(r)\) is given by $$F(r) = 0 \quad r < 0,$$ $$F(r) = \frac{r^2}{a^2} \quad 0 \leq r \leq a,$$ $$F(r) = 1 \quad r > a.$$ [4 marks]
  2. By first finding the probability density function of \(R\), show that the mean distance from \(O\) of the points that Peg hits is \(\frac{2a}{3}\). [7 marks] Bob, a more experienced player, aims for \(O\), and his points have a distance \(X\) from \(O\) whose cumulative distribution function is $$F(x) = 0, \quad x < 0; \quad F(x) = \frac{x}{a}\left(2 - \frac{x}{a}\right), \quad 0 \leq x \leq a; \quad F(x) = 1, \quad x > a.$$
  3. Find the probability density function of \(X\), and explain why it shows that Bob is aiming for \(O\). [5 marks]
Edexcel S2 Q7
18 marks Standard +0.3
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, [3 marks]
  2. more than 7 apple trees. [3 marks]
In a sample of 60 trees in the orchard,
  1. find the expected number of pear trees. [1 mark]
  2. Calculate the standard deviation of the number of pear trees and compare this result with the standard deviation of the number of apple trees. [2 marks]
  3. Find the probability that exactly 35 in the sample of 60 trees are pear trees. [4 marks]
  4. Find an approximate value for the probability that more than 15 of the 60 trees are pear trees. [5 marks]
Edexcel S2 Q1
4 marks Easy -1.8
Briefly explain what is meant by
  1. a statistical model, [2 marks]
  2. a sampling frame, [1 mark]
  3. a sampling unit. [1 mark]
Edexcel S2 Q2
4 marks Moderate -0.8
  1. Explain what is meant by the critical region of a statistical test. [2 marks]
  2. Under a hypothesis \(H_0\), an event \(A\) can happen with probability \(4 \cdot 2\%\). The event \(A\) does then happen. State, with justification, whether \(H_0\) should be accepted or rejected at the \(5\%\) significance level. [2 marks]
Edexcel S2 Q3
11 marks Moderate -0.8
  1. Briefly describe the main features of a binomial distribution. [2 marks]
I conduct an experiment by randomly selecting 10 cards, without replacement, from a normal pack of 52.
  1. Explain why the distribution of \(X\), the number of hearts obtained, is not \(\text{B}(10, \frac{1}{4})\). [2 marks]
After making the appropriate adjustment to the experiment, which should be stated, so that the distribution is \(\text{B}(10, \frac{1}{4})\), find
  1. the probability of getting no hearts, [3 marks]
  2. the probability of getting 4 or more hearts. [2 marks]
  3. If the modified experiment is repeated 50 times, find the total number of hearts that you would you expect to have drawn. [2 marks]
Edexcel S2 Q4
11 marks Standard +0.3
A Geiger counter is observed in the presence of a radioactive source. In 100 one-minute intervals, the number of counts recorded are as follows:
No of counts, \(X\)0123456
Frequency102429161263
  1. Find the mean and variance of this data, and show that it supports the idea that the random variable \(X\) is following a Poisson distribution. [5 marks]
  2. Use a Poisson distribution with the mean found in part (a) to calculate, to 3 decimal places, the probability that more than 6 counts will be recorded in any particular minute. [4 marks]
  3. Find the number of one-minute intervals, in the sample of 100, in which more than 6 counts would be expected. [2 marks]
Edexcel S2 Q5
14 marks Standard +0.3
A continuous random variable \(X\) has the cumulative distribution function $$F(x) = 0 \quad x < 2,$$ $$F(x) = k(x - a)^2 \quad 2 \leq x \leq 6,$$ $$F(x) = 1 \quad x \geq 6.$$
  1. Find the values of the constants \(a\) and \(k\). [4 marks]
  2. Show that the median of \(X\) is \(2(1 + \sqrt{2})\). [4 marks]
  3. Given that \(X > 4\), find the probability that \(X > 5\). [6 marks]
Edexcel S2 Q6
14 marks Standard +0.3
A small opinion poll shows that the Trendies have a \(10\%\) lead over the Oldies. The poll is based on a survey of 20 voters, in which the Trendies got 11 and the Oldies 9. The Oldies spokesman says that the result is consistent with a \(10\%\) lead for the Oldies, whilst the Trendies spokesperson says that this is impossible.
  1. At the \(5\%\) significance level, test which is right, stating your null hypothesis carefully. [6 marks]
  2. If it is indeed true that the Trendies are supported by \(55\%\) of the population, use a suitable approximation to find the probability that in a random sample of 200 voters they would obtain less than half of the votes. [8 marks]
Edexcel S2 Q7
17 marks Standard +0.3
A continuous random variable \(X\) has the probability density function $$\text{f}(x) = \frac{6x}{175} \quad 0 \leq x < 5,$$ $$\text{f}(x) = \frac{6x(10-x)}{875} \quad 5 \leq x \leq 10,$$ $$\text{f}(x) = 0 \quad \text{otherwise}.$$
  1. Verify that f is a probability density function. [6 marks]
  2. Write down the probability that \(X < 1\). [2 marks]
  3. Find the cumulative distribution function of \(X\), carefully showing how it changes for different domains. [7 marks]
  4. Find the probability that \(2 < X < 7\). [2 marks]
Edexcel S2 Q1
5 marks Moderate -0.8
A golfer believes that the distance, in metres, that she hits a ball with a 5 iron, follows a continuous uniform distribution over the interval \([100, 150]\).
  1. Find the median and interquartile range of the distance she hits a ball, that would be predicted by this model. [3 marks]
  2. Explain why the continuous uniform distribution may not be a suitable model. [2 marks]
Edexcel S2 Q2
8 marks Moderate -0.3
The continuous random variable \(X\) has the following cumulative distribution function: $$F(x) = \begin{cases} 0, & x < 0, \\ \frac{1}{64}(16x - x^2), & 0 \leq x \leq 8, \\ 1, & x > 8. \end{cases}$$
  1. Find \(P(X > 5)\). [2 marks]
  2. Find and specify fully the probability density function \(f(x)\) of \(X\). [3 marks]
  3. Sketch \(f(x)\) for all values of \(x\). [3 marks]
Edexcel S2 Q3
10 marks Moderate -0.3
An electrician records the number of repairs of different types of appliances that he makes each day. His records show that over 40 working days he repaired a total of 180 CD players.
  1. Explain why a Poisson distribution may be suitable for modelling the number of CD players he repairs each day and find the parameter for this distribution. [4 marks]
  2. Find the probability that on one particular day he repairs
    1. no CD players,
    2. more than 6 CD players. [3 marks]
  3. Find the probability that over 10 working days he will repair more than 6 CD players on exactly 3 of the days. [3 marks]
Edexcel S2 Q4
10 marks Moderate -0.8
A teacher wants to investigate the sports played by students at her school in their free time. She decides to ask a random sample of 120 pupils to complete a short questionnaire.
  1. Give two reasons why the teacher might choose to use a sample survey rather than a census. [2 marks]
  2. Suggest a suitable sampling frame that she could use. [1 mark]
The teacher believes that 1 in 20 of the students play tennis in their free time. She uses the data collected from her sample to test if the proportion is different from this.
  1. Using a suitable approximation and stating the hypotheses that she should use, find the critical region for this test. The probability for each tail of the region should be as close as possible to 5\%. [6 marks]
  2. State the significance level of this test. [1 mark]
Edexcel S2 Q5
11 marks Standard +0.3
As part of a business studies project, 8 groups of students are each randomly allocated 10 different shares from a listing of over 300 share prices in a newspaper. Each group has to follow the changes in the price of their shares over a 3-month period. At the end of the 3 months, 35\% of all the shares in the listing have increased in price and the rest have decreased.
  1. Find the probability that, for the 10 shares of one group,
    1. exactly 6 have gone up in price,
    2. more than 5 have gone down in price. [5 marks]
  2. Using a suitable approximation, find the probability that of the 80 shares allocated in total to the groups, more than 35 will have decreased in value. [6 marks]