Questions S2 (1690 questions)

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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]
Edexcel S2 Q3
10 marks Moderate -0.3
The sales staff at an insurance company make house calls to prospective clients. Past records show that 30% of the people visited will take out a new policy with the company. On a particular day, one salesperson visits 8 people. Find the probability that, of these,
  1. exactly 2 take out new policies, [3 marks]
  2. more than 4 take out new policies. [2 marks]
The company awards a bonus to any salesperson who sells more than 50 policies in a month.
  1. Using a suitable approximation, find the probability that a salesperson gets a bonus in a month in which he visits 150 prospective clients. [5 marks]
Edexcel S2 Q4
10 marks Standard +0.3
A rugby player scores an average of 0.4 tries per match in which he plays.
  1. Find the probability that he scores 2 or more tries in a match. [5 marks]
The team's coach moves the player to a different position in the team believing he will then score more frequently. In the next five matches he scores 6 tries.
  1. Stating your hypotheses clearly, test at the 5% level of significance whether or not there is evidence of an increase in the mean number of tries the player scores per match as a result of playing in a different position. [5 marks]
Edexcel S2 Q5
13 marks Standard +0.3
The continuous random variable \(X\) has the following cumulative distribution function: $$\text{F}(x) = \begin{cases} 0, & x < 0, \\ \frac{1}{432} x^2(x^2 - 16x + 72), & 0 \leq x \leq 6, \\ 1, & x > 6. \end{cases}$$
  1. Find P(\(X < 2\)). [2 marks]
  2. Find and specify fully the probability density function f(\(x\)) of \(X\). [4 marks]
  3. Show that the mode of \(X\) is 2. [6 marks]
  4. State, with a reason, whether the median of \(X\) is higher or lower than the mode of \(X\). [1 mark]
Edexcel S2 Q6
13 marks Moderate -0.8
A shop receives weekly deliveries of 120 eggs from a local farm. The proportion of eggs received from the farm that are broken is 0.008
  1. Explain why it is reasonable to use the binomial distribution to model the number of eggs that are broken in each delivery. [3 marks]
  2. Use the binomial distribution to calculate the probability that at most one egg in a delivery will be broken. [4 marks]
  3. State the conditions under which the binomial distribution can be approximated by the Poisson distribution. [1 mark]
  4. Using the Poisson approximation to the binomial, find the probability that at most one egg in a delivery will be broken. Comment on your answer. [5 marks]
Edexcel S2 Q7
14 marks Moderate -0.3
The random variable \(X\) follows a continuous uniform distribution over the interval \([2, 11]\).
  1. Write down the mean of \(X\). [1 mark]
  2. Find P(\(X \geq 8.6\)). [2 marks]
  3. Find P(\(|X - 5| < 2\)). [2 marks]
The random variable \(Y\) follows a continuous uniform distribution over the interval \([a, b]\).
  1. Show by integration that $$\text{E}(Y^2) = \frac{1}{3}(b^2 + ab + a^2).$$ [5 marks]
  2. Hence, prove that $$\text{Var}(Y) = \frac{1}{12}(b - a)^2.$$ You may assume that E(\(Y\)) = \(\frac{1}{2}(a + b)\). [4 marks]
Edexcel S2 Q1
4 marks Easy -2.0
  1. State one advantage and one disadvantage in using a census rather than a sample survey in statistical work. [2]
  2. Give an example of a situation in which you would choose to use a census rather than a sample survey and explain why. [2]
Edexcel S2 Q2
8 marks Standard +0.3
An advert for Tatty's Crisps claims that 1 in 10 bags contain a free scratchcard game. Tatty's Crisps can be bought in a Family Pack containing 10 bags. Find the probability that the bags in one of these Family Packs contain
  1. no scratchcards, [2]
  2. more than 2 scratchcards. [2]
Tatty's Crisps can also be bought wholesale in boxes containing 50 bags. A pub Landlord notices that her customers only found 2 scratchcards in the crisps from one of these boxes.
  1. Stating your hypotheses clearly, test at the 10\% level of significance whether or not this gives evidence of there being fewer free scratchcards than is claimed by the advert. [4]
Edexcel S2 Q3
10 marks Moderate -0.8
A class of children are each asked to draw a line that they think is 10 cm long without using a ruler. The teacher models how many centimetres each child's line is longer than 10 cm by the random variable \(X\) and believes that \(X\) has the following probability density function: $$f(x) = \begin{cases} \frac{1}{8}, & -4 \leq x \leq 4, \\ 0, & \text{otherwise}. \end{cases}$$
  1. Write down the name of this distribution. [1]
  2. Define fully the cumulative distribution function F(x) of \(X\). [4]
  3. Calculate the proportion of children making an error of less than 15\% according to this model. [3]
  4. Give two reasons why this may not be a very suitable model. [2]
Edexcel S2 Q4
12 marks Standard +0.3
A bag contains 40 beads of the same shape and size. The ratio of red to green to blue beads is \(1 : 3 : 4\) and there are no beads of any other colour. In an experiment, a bead is picked at random, its colour noted and the bead replaced in the bag. This is done ten times.
  1. Suggest a suitable distribution for modelling the number of times a blue bead is picked out and give the value of any parameters needed. [2]
  2. Explain why this distribution would not be suitable if the beads were not replaced in the bag. [1]
  3. Find the probability that of the ten beads picked out
    1. five are blue,
    2. at least one is red. [6]
The experiment is repeated, but this time a bead is picked out and replaced \(n\) times.
  1. Find in the form \(a^n < b\), where \(a\) and \(b\) are exact fractions, the condition which \(n\) must satisfy in order to have at least a 99\% chance of picking out at least one red bead. [3]
Edexcel S2 Q5
13 marks Moderate -0.3
A charity receives donations of more than £10000 at an average rate of 25 per year. Find the probability that the charity receives
  1. exactly 30 such donations in one year, [3]
  2. less than 3 such donations in one month. [5]
  3. Using a suitable approximation, find the probability that the charity receives more than 45 donations of more than £10000 in the next two years. [5]
Edexcel S2 Q6
14 marks Standard +0.3
The length of time, in tens of minutes, that patients spend waiting at a doctor's surgery is modelled by the continuous random variable \(T\), with the following cumulative distribution function: $$F(t) = \begin{cases} 0, & t < 0, \\ \frac{1}{135}(54t + 9t^2 - 4t^3), & 0 \leq t \leq 3, \\ 1, & t > 3. \end{cases}$$
  1. Find the probability that a patient waits for more than 20 minutes. [3]
  2. Show that the median waiting time is between 11 and 12 minutes. [3]
  3. Define fully the probability density function f(t) of \(T\). [3]
  4. Find the modal waiting time in minutes. [4]
  5. Give one reason why this model may need to be refined. [1]
Edexcel S2 Q7
14 marks Standard +0.3
A student collects data on the number of bicycles passing outside his house in 5-minute intervals during one morning.
  1. Suggest, with reasons, a suitable distribution for modelling this situation. [3]
The student's data is shown in the table below.
Number of bicycles0123456 or more
Frequency714102120
  1. Show that the mean and variance of these data are 1.5 and 1.58 (to 3 significant figures) respectively and explain how these values support your answer to part (a). [6]
An environmental organisation declares a "car free day" encouraging the public to leave their cars at home. The student wishes to test whether or not there are more bicycles passing along his road on this day and records 16 bicycles in a half-hour period during the morning.
  1. Stating your hypotheses clearly, test at the 5\% level of significance whether or not there are more than 1.5 bicycles passing along his road per 5-minute interval that morning. [5]