Questions — Edexcel (10514 questions)

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Edexcel S1 Q4
14 marks Moderate -0.8
A College offers evening classes in GCSE Mathematics and English. In order to assess which age groups were reluctant to use the classes, the College collected data on the age in completed years of those currently attending each course. The results are shown in this back-to-back stem and leaf diagram. \includegraphics{figure_4} Key: \(1 | 3 | 2\) means age 31 doing Mathematics and age 32 doing English
  1. Find the median and quartiles of the age in completed years of those attending the Mathematics classes. [4 marks]
  2. On graph paper, draw a box plot representing the data for the Mathematics class. [3 marks]
The median and quartiles of the age in completed years of those attending the English classes are 25, 41 and 57 years respectively.
  1. Draw a box plot representing the data for the English class using the same scale as for the data from the Mathematics class. [3 marks]
  2. Using your box plots, compare and contrast the ages of those taking each class. [4 marks]
Edexcel S1 Q5
16 marks Moderate -0.3
A netball team are in a league with three other teams from which one team will progress to the next stage of the competition. The team's coach estimates their chances of winning each of their three matches in the league to be 0.6, 0.5 and 0.3 respectively, and believes these probabilities to be independent of each other.
  1. Show that the probability of the team winning exactly two of their three matches is 0.36 [4 marks]
Let the random variable \(W\) be the number of matches that the team win in the league.
  1. Find the probability distribution of \(W\). [4 marks]
  2. Find E\((W)\) and Var\((W)\). [6 marks]
  3. Comment on the coach's assumption that the probabilities of success in each of the three matches are independent. [2 marks]
Edexcel S1 Q6
16 marks Moderate -0.3
The Principal of a school believes that more students are absent on days when the temperature is lower. Over a two-week period in December she records the percentage of students who are absent, \(A\%\), and the temperature, \(T°\)C, at 9 am each morning giving these results.
\(T\) (°C)4\(-3\)\(-2\)\(-6\)037\(-1\)32
\(A\) (\%)8.514.117.020.317.915.512.412.813.711.6
  1. Represent these data on a scatter diagram. [4 marks]
You may use $$\Sigma T = 7, \quad \Sigma A = 143.8, \quad \Sigma T^2 = 137, \quad \Sigma A^2 = 2172.66, \quad \Sigma TA = 20.7$$
  1. Calculate the product moment correlation coefficient for these data and comment on the Principal's hypothesis. [6 marks]
  2. Find an equation of the regression line of \(A\) on \(T\) in the form \(A = p + qT\). [4 marks]
  3. Draw the regression line on your scatter diagram. [2 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]
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]