Questions — Edexcel S2 (562 questions)

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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]
Edexcel S2 Q4
13 marks Standard +0.3
The waiting time, in minutes, at a dentist is modelled by the continuous random variable \(T\) with probability density function $$f(t) = k(10 - t) \quad 0 \leq t \leq 10,$$ $$f(t) = 0 \quad \text{otherwise}.$$
  1. Sketch the graph of \(f(t)\) and find the value of \(k\). [4 marks]
  2. Find the mean value of \(T\). [4 marks]
  3. Find the 95th percentile of \(T\). [3 marks]
  4. 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. [2 marks]
Edexcel S2 Q5
13 marks Standard +0.3
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. [1 mark]
  2. Find the probability that a particular page has more than 2 misprints. [3 marks]
  3. Find the probability that Chapter 1, with 8 pages, has no misprints at all. [2 marks]
Chapter 2 is longer, at 20 pages.
  1. 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. [7 marks]
Edexcel S2 Q6
13 marks Moderate -0.3
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 2.4.
  1. Show that \(n = 15\). [2 marks]
  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, [3 marks]
    2. at least 5. [2 marks]
Ten samples of 15 bags each are tested. Find the probability that
  1. all these batches contain less than 5 underweight bags, [3 marks]
  2. the fourth batch tested is the first to contain less than 5 underweight bags. [3 marks]
Edexcel S2 Q7
18 marks Standard +0.3
A continuous random variable \(X\) has a probability density function given by $$f(x) = \frac{x^2}{312} \quad 4 \leq x \leq 10,$$ $$f(x) = 0 \quad \text{otherwise}.$$
  1. Find E\((X)\). [3 marks]
  2. Find the variance of \(X\). [4 marks]
  3. Find the cumulative distribution function F\((x)\), for all values of \(x\). [5 marks]
  4. Hence find the median value of \(X\). [3 marks]
  5. Write down the modal value of \(X\). [1 mark]
It is sometimes suggested that, for most distributions, $$2 \times (\text{median} - \text{mean}) \approx \text{mode} - \text{median}.$$
  1. Show that this result is not satisfied in this case, and suggest a reason why. [2 marks]
Edexcel S2 Q1
4 marks Easy -1.8
An insurance company is investigating how often its customers crash their cars.
  1. Suggest an appropriate sampling frame. [1 mark]
  2. Describe the sampling units. [1 mark]
  3. State the advantage of a sample survey over a census in this case. [2 marks]
Edexcel S2 Q2
4 marks Easy -1.2
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 \(000°\) to \(360°\).
  1. Specify a suitable model for the distribution of \(X\). [1 mark]
  2. Find the mean and the standard deviation of \(X\). [3 marks]
Edexcel S2 Q3
10 marks Standard +0.3
A secretarial agency carefully assesses the work of a new recruit, with the following results after 150 pages:
No of errors0123456
No of pages163841291772
  1. Find the mean and variance of the number of errors per page. [4 marks]
  2. Explain how these results support the idea that the number of errors per page follows a Poisson distribution. [1 mark]
  3. After two weeks at the agency, the secretary types a fresh piece of work, six pages long, which is found to contain 15 errors. The director suspects that the secretary was trying especially hard during the early period and that she is now less conscientious. Using a Poisson distribution with the mean found in part (a), test this hypothesis at the 5% significance level. [5 marks]
Edexcel S2 Q4
12 marks Standard +0.3
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 marks]
  2. he is late more than twice. [3 marks]
In a half term of seven weeks, lateness on more than ten occasions results in loss of privileges the following half term.
  1. Use the Normal approximation to estimate the probability that he loses his privileges. [7 marks]
Edexcel S2 Q5
12 marks Standard +0.3
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 cm³, and the number of bubbles per cm³ has a Poisson distribution. In an ingot of 40 cm³, find
  1. the probability that there are less than two bubbles, [3 marks]
  2. the probability that there are more than 3 but less than 10 bubbles. [3 marks]
A new machine is being considered. Its manufacturer claims that it produces fewer bubbles per cm³. In a sample ingot of 60 cm³, there is just one bubble.
  1. Carry out a hypothesis test at the 1% significance level to decide whether the new machine is better. State your hypotheses and conclusion carefully. [6 marks]
Edexcel S2 Q6
15 marks Standard +0.3
A random variable \(X\) has a probability density function given by $$f(x) = \frac{4x^2(3-x)}{27} \quad 0 \leq x \leq 3,$$ $$f(x) = 0 \quad \text{otherwise}.$$
  1. Find the mode of \(X\). [3 marks]
  2. Find the mean of \(X\). [3 marks]
  3. Specify completely the cumulative distribution function of \(X\). [4 marks]
  4. Deduce that the median, \(m\), of \(X\) satisfies the equation \(m^4 - 4m^3 + 13·5 = 0\), and hence show that \(1·84 < m < 1·85\). [4 marks]
  5. What do these results suggest about the skewness of the distribution? [1 mark]
Edexcel S2 Q7
18 marks Standard +0.3
A corner-shop has weekly sales (in thousands of pounds), which can be modelled by the continuous random variable \(X\) with probability density function $$f(x) = k(x-2)(10-x) \quad 2 \leq x \leq 10,$$ $$f(x) = 0 \quad \text{otherwise}.$$
  1. Show that \(k = \frac{3}{256}\) and write down the mean of \(X\). [6 marks]
  2. Find the standard deviation of the weekly sales. [6 marks]
  3. Find the probability that the sales exceed £8 000 in any particular week. [4 marks]
If the sales exceed £8 000 per week for 4 consecutive weeks, the manager gets a bonus.
  1. Find the probability that the manager gets a bonus in February. [2 marks]
Edexcel S2 Q1
3 marks Easy -1.8
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 marks]
  2. Suggest an appropriate sampling frame. [1 mark]
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 2·4. 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]