Questions — Edexcel S2 (562 questions)

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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]
Edexcel S2 Q6
12 marks Moderate -0.3
A shoe shop sells on average 4 pairs of shoes per hour on a weekday morning.
  1. Suggest a suitable distribution for modelling the number of sales made per hour on a weekday morning and state the value of any parameters needed. [1 mark]
  2. Explain why this model might have to be modified for modelling the number of sales made per hour on a Saturday morning. [1 mark]
  3. Find the probability that on a weekday morning the shop sells
    1. more than 4 pairs in a one-hour period,
    2. no pairs in a half-hour period,
    3. more than 4 pairs during each hour from 9 am until noon. [6 marks]
The area manager visits the shop on a weekday morning, the day after an advert appears in a local paper. In a one-hour period the shop sells 7 pairs of shoes, leading the manager to believe that the advert has increased the shop's sales.
  1. Stating your hypotheses clearly, test at the 5\% level of significance whether or not there is evidence of an increase in sales following the appearance of the advert. [4 marks]
Edexcel S2 Q7
19 marks Moderate -0.3
The continuous random variable \(T\) has the following probability density function: $$f(t) = \begin{cases} k(t^2 + 2), & 0 \leq t \leq 3, \\ 0, & \text{otherwise}. \end{cases}$$
  1. Show that \(k = \frac{1}{15}\). [4 marks]
  2. Sketch \(f(t)\) for all values of \(t\). [3 marks]
  3. State the mode of \(T\). [1 mark]
  4. Find \(E(T)\). [5 marks]
  5. Show that the standard deviation of \(T\) is 0.798 correct to 3 significant figures. [6 marks]
Edexcel S2 Q1
5 marks Easy -1.8
  1. Explain what you understand by the term sampling frame when conducting a sample survey. [1 mark]
  2. Suggest a suitable sampling frame and identify the sampling units when using a sample survey to study
    1. the frequency with which cars break down in the first 3 months after being serviced at a particular garage,
    2. the weight loss of people involved in trials of a new dieting programme.
    [4 marks]
Edexcel S2 Q2
9 marks Moderate -0.8
An ornithologist believes that on average 4.2 different species of bird will visit a bird table in a rural garden when 50 g of breadcrumbs are spread on it.
  1. Suggest a suitable distribution for modelling the number of species that visit a bird table meeting these criteria. [1 mark]
  2. Explain why the parameter used with this model may need to be changed if
    1. 50 g of nuts are used instead of breadcrumbs,
    2. 100g of breadcrumbs are used.
    [2 marks]
A bird table in a rural garden has 50 g of breadcrumbs spread on it. Find the probability that
  1. exactly 6 different species visit the table, [2 marks]
  2. more than 2 different species visit the table. [4 marks]
Edexcel S2 Q3
13 marks Moderate -0.8
In a test studying reaction times, white dots appear at random on a black rectangular screen. The continuous random variable \(X\) represents the distance, in centimetres, of the dot from the left-hand edge of the screen. The distribution of \(X\) is rectangular over the interval \([0, 20]\).
  1. Find \(P(2 < X < 3.6)\). [2 marks]
  2. Find the mean and variance of \(X\). [3 marks]
The continuous random variable \(Y\) represents the distance, in centimetres, of the dot from the bottom edge of the screen. The distribution of \(Y\) is rectangular over the interval \([0, 16]\). Find the probability that a dot appears
  1. in a square of side 4 cm at the centre of the screen, [4 marks]
  2. within 2 cm of the edge of the screen. [4 marks]
Edexcel S2 Q4
13 marks Standard +0.3
It is believed that the number of sets of traffic lights that fail per hour in a particular large city follows a Poisson distribution with a mean of 3. Find the probability that
  1. there will be no failures in a one-hour period, [1 mark]
  2. there will be more than 4 failures in a 30-minute period. [3 marks]
Using a suitable approximation, find the probability that in a 24-hour period there will be
  1. less than 60 failures, [5 marks]
  2. exactly 72 failures. [4 marks]
Edexcel S2 Q5
17 marks Moderate -0.3
Six standard dice with faces numbered 1 to 6 are thrown together. Assuming that the dice are fair, find the probability that
  1. none of the dice show a score of 6, [3 marks]
  2. more than one of the dice shows a score of 6, [4 marks]
  3. there are equal numbers of odd and even scores showing on the dice. [3 marks]
One of the dice is suspected of being biased such that it shows a score of 6 more often than the other numbers. This die is thrown eight times and gives a score of 6 three times.
  1. Stating your hypotheses clearly, test at the 5% level of significance whether or not this die is biased towards scoring a 6. [7 marks]
Edexcel S2 Q6
18 marks Standard +0.3
The continuous random variable \(X\) has the following probability density function: $$f(x) = \begin{cases} \frac{1}{8}x, & 0 \leq x \leq 2, \\ \frac{1}{12}(6-x), & 2 \leq x \leq 6, \\ 0, & \text{otherwise}. \end{cases}$$
  1. Sketch \(f(x)\) for all values of \(x\). [4 marks]
  2. State the mode of \(X\). [1 mark]
  3. Define fully the cumulative distribution function \(F(x)\) of \(X\). [9 marks]
  4. Show that the median of \(X\) is 2.536, correct to 4 significant figures. [4 marks]
Edexcel S2 Q1
6 marks Easy -1.8
  1. Explain briefly what you understand by the terms
    1. population,
    2. sample.
    [2 marks]
  2. Giving a reason for each of your answers, state whether you would use a census or a sample survey to investigate
    1. the dietary requirements of people attending a 4-day residential course,
    2. the lifetime of a particular type of battery.
    [4 marks]
Edexcel S2 Q2
9 marks Moderate -0.3
The manager of a supermarket receives an average of 6 complaints per day from customers. Find the probability that on one day she receives
  1. 3 complaints, [3 marks]
  2. 10 or more complaints. [2 marks]
The supermarket is open on six days each week.
  1. Find the probability that the manager receives 10 or more complaints on no more than one day in a week. [4 marks]