Questions S2 (1690 questions)

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Edexcel S2 2011 January Q2
6 marks Moderate -0.5
A student takes a multiple choice test. The test is made up of 10 questions each with 5 possible answers. The student gets 4 questions correct. Her teacher claims she was guessing the answers. Using a one tailed test, at the 5\% level of significance, test whether or not there is evidence to reject the teacher's claim. State your hypotheses clearly. [6]
Edexcel S2 2011 January Q3
11 marks Moderate -0.3
The continuous random variable \(X\) is uniformly distributed over the interval \([-1,3]\). Find
  1. E(\(X\)) [1]
  2. Var(\(X\)) [2]
  3. E(\(X^2\)) [2]
  4. P(\(X < 1.4\)) [1]
A total of 40 observations of \(X\) are made.
  1. Find the probability that at least 10 of these observations are negative. [5]
Edexcel S2 2011 January Q4
6 marks Standard +0.3
Richard regularly travels to work on a ferry. Over a long period of time, Richard has found that the ferry is late on average 2 times every week. The company buys a new ferry to improve the service. In the 4-week period after the new ferry is launched, Richard finds the ferry is late 3 times and claims the service has improved. Assuming that the number of times the ferry is late has a Poisson distribution, test Richard's claim at the 5\% level of significance. State your hypotheses clearly. [6]
Edexcel S2 2011 January Q5
13 marks Moderate -0.3
A continuous random variable \(X\) has the probability density function f(\(x\)) shown in Figure 1. \includegraphics{figure_1} Figure 1
  1. Show that f(\(x\)) = \(4 - 8x\) for \(0 \leqslant x \leqslant 0.5\) and specify f(\(x\)) for all real values of \(x\). [4]
  2. Find the cumulative distribution function F(\(x\)). [4]
  3. Find the median of \(X\). [3]
  4. Write down the mode of \(X\). [1]
  5. State, with a reason, the skewness of \(X\). [1]
Edexcel S2 2011 January Q6
16 marks Standard +0.3
Cars arrive at a motorway toll booth at an average rate of 150 per hour.
  1. Suggest a suitable distribution to model the number of cars arriving at the toll booth, \(X\), per minute. [2]
  2. State clearly any assumptions you have made by suggesting this model. [2]
Using your model,
  1. find the probability that in any given minute
    1. no cars arrive,
    2. more than 3 cars arrive.
    [3]
  2. In any given 4 minute period, find \(m\) such that P(\(X > m\)) = 0.0487 [3]
  3. Using a suitable approximation find the probability that fewer than 15 cars arrive in any given 10 minute period. [6]
Edexcel S2 2011 January Q7
13 marks Standard +0.3
The queuing time in minutes, \(X\), of a customer at a post office is modelled by the probability density function $$\text{f}(x) = \begin{cases} kx(81 - x^2) & 0 \leqslant x \leqslant 9 \\ 0 & \text{otherwise} \end{cases}$$
  1. Show that \(k = \frac{4}{6561}\). [3]
Using integration, find
  1. the mean queuing time of a customer, [4]
  2. the probability that a customer will queue for more than 5 minutes. [3]
Three independent customers shop at the post office.
  1. Find the probability that at least 2 of the customers queue for more than 5 minutes. [3]
Edexcel S2 2002 June Q2
5 marks Easy -1.2
A random sample \(X_1, X_2, \ldots, X_n\) is taken from a finite population. A statistic \(Y\) is based on this sample.
  1. Explain what you understand by the statistic \(Y\). [2]
  2. Give an example of a statistic. [1]
  3. Explain what you understand by the sampling distribution of \(Y\). [2]
Edexcel S2 2002 June Q3
9 marks Moderate -0.3
The continuous random variable \(R\) is uniformly distributed on the interval \(\alpha \leq R \leq \beta\). Given that \(\mathrm{E}(R) = 3\) and \(\mathrm{Var}(R) = \frac{25}{3}\), find
  1. the value of \(\alpha\) and the value of \(\beta\), [7]
  2. \(\mathrm{P}(R < 6.6)\). [2]
Edexcel S2 2002 June Q4
13 marks Standard +0.3
Past records show that 20\% of customers who buy crisps from a large supermarket buy them in single packets. During a particular day a random sample of 25 customers who had bought crisps was taken and 2 of them had bought them in single packets.
  1. Use these data to test, at the 5\% level of significance, whether or not the percentage of customers who bought crisps in single packets that day was lower than usual. State your hypotheses clearly. [6]
At the same supermarket, the manager thinks that the probability of a customer buying a bumper pack of crisps is 0.03. To test whether or not this hypothesis is true the manager decides to take a random sample of 300 customers.
  1. Stating your hypotheses clearly, find the critical region to enable the manager to test whether or not there is evidence that the probability is different from 0.03. The probability for each tail of the region should be as close as possible to 2.5\%. [6]
  2. Write down the significance level of this test. [1]
Edexcel S2 2002 June Q5
13 marks Standard +0.3
A garden centre sells canes of nominal length 150 cm. The canes are bought from a supplier who uses a machine to cut canes of length \(L\) where \(L \sim \mathrm{N}(\mu, 0.3^2)\).
  1. Find the value of \(\mu\), to the nearest 0.1 cm, such that there is only a 5\% chance that a cane supplied to the garden centre will have length less than 150 cm. [4]
A customer buys 10 of these canes from the garden centre.
  1. Find the probability that at most 2 of the canes have length less than 150 cm. [3]
Another customer buys 500 canes.
  1. Using a suitable approximation, find the probability that fewer than 35 of the canes will have length less than 150 cm. [6]
Edexcel S2 2002 June Q7
17 marks Moderate -0.3
The continuous random variable \(X\) has probability density function $$f(x) = \begin{cases} \frac{x}{15}, & 0 \leq x \leq 2, \\ \frac{2}{15}, & 2 < x < 7, \\ \frac{4}{9} - \frac{2x}{45}, & 7 \leq x \leq 10, \\ 0, & \text{otherwise}. \end{cases}$$
  1. Sketch \(f(x)\) for all values of \(x\). [3]
    1. Find expressions for the cumulative distribution function, \(\mathrm{F}(x)\), for \(0 \leq x \leq 2\) and for \(7 \leq x \leq 10\).
    2. Show that for \(2 < x < 7\), \(\mathrm{F}(x) = \frac{2x}{15} - \frac{2}{15}\).
    3. Specify \(\mathrm{F}(x)\) for \(x < 0\) and for \(x > 10\).
    [8]
  3. Find \(\mathrm{P}(X \leq 8.2)\). [2]
  4. Find, to 3 significant figures, \(\mathrm{E}(X)\). [4]
Edexcel S2 2003 June Q2
7 marks Moderate -0.3
  1. Write down the condition needed to approximate a Poisson distribution by a Normal distribution. [1]
The random variable \(Y \sim \text{Po}(30)\).
  1. Estimate P(\(Y > 28\)). [6]
Edexcel S2 2003 June Q3
9 marks Easy -1.3
In a town, 30\% of residents listen to the local radio station. Four residents are chosen at random.
  1. State the distribution of the random variable \(X\), the number of these four residents that listen to local radio. [2]
  2. On graph paper, draw the probability distribution of \(X\). [3]
  3. Write down the most likely number of these four residents that listen to the local radio station. [1]
  4. Find E(\(X\)) and Var (\(X\)). [3]
Edexcel S2 2003 June Q4
12 marks Moderate -0.3
  1. Write down the conditions under which the binomial distribution may be a suitable model to use in statistical work. [4]
A six-sided die is biased. When the die is thrown the number 5 is twice as likely to appear as any other number. All the other faces are equally likely to appear. The die is thrown repeatedly. Find the probability that
    1. the first 5 will occur on the sixth throw,
    2. in the first eight throws there will be exactly three 5s.
    [8]
Edexcel S2 2003 June Q5
13 marks Moderate -0.8
A drinks machine dispenses lemonade into cups. It is electronically controlled to cut off the flow of lemonade randomly between 180 ml and 200 ml. The random variable \(X\) is the volume of lemonade dispensed into a cup.
  1. Specify the probability density function of \(X\) and sketch its graph. [4]
  2. Find the probability that the machine dispenses
    1. less than 183 ml,
    2. exactly 183 ml.
    [3]
  3. Calculate the inter-quartile range of \(X\). [1]
  4. Determine the value of \(x\) such that P(\(X \geq x\)) = 2P(\(X \leq x\)). [3]
  5. Interpret in words your value of \(x\). [2]
Edexcel S2 2003 June Q7
15 marks Moderate -0.3
A continuous random variable \(X\) has probability density function f(\(x\)) where $$\text{f}(x) = \begin{cases} k(x^2 + 2x + 1) & -1 \leq x \leq 0, \\ 0, & \text{otherwise} \end{cases}$$ where \(k\) is a positive integer.
  1. Show that \(k = 3\). [4]
Find
  1. E(\(X\)), [4]
  2. the cumulative distribution function F(\(x\)), [4]
  3. P(\(-0.3 < X < 0.3\)). [3]
Edexcel S2 2004 June Q1
3 marks Easy -1.8
Explain briefly what you understand by
  1. a sampling frame, [1]
  2. a statistic. [2]
Edexcel S2 2004 June Q2
5 marks Easy -1.3
The continuous random variable \(X\) is uniformly distributed over the interval \([-1, 4]\). Find
  1. P\((X < 2.7)\), [1]
  2. E\((X)\), [2]
  3. Var \((X)\). [2]
Edexcel S2 2004 June Q3
7 marks Moderate -0.3
Brad planted 25 seeds in his greenhouse. He has read in a gardening book that the probability of one of these seeds germinating is 0.25. Ten of Brad's seeds germinated. He claimed that the gardening book had underestimated this probability. Test, at the 5% level of significance, Brad's claim. State your hypotheses clearly. [7]
Edexcel S2 2004 June Q4
13 marks Moderate -0.8
  1. State two conditions under which a random variable can be modelled by a binomial distribution. [2]
In the production of a certain electronic component it is found that 10% are defective. The component is produced in batches of 20.
  1. Write down a suitable model for the distribution of defective components in a batch. [1]
Find the probability that a batch contains
  1. no defective components, [2]
  2. more than 6 defective components. [2]
  3. Find the mean and the variance of the defective components in a batch. [2]
A supplier buys 100 components. The supplier will receive a refund if there are more than 15 defective components.
  1. Using a suitable approximation, find the probability that the supplier will receive a refund. [4]
Edexcel S2 2004 June Q5
15 marks Standard +0.3
  1. Explain what you understand by a critical region of a test statistic. [2]
The number of breakdowns per day in a large fleet of hire cars has a Poisson distribution with mean \(\frac{1}{7}\).
  1. Find the probability that on a particular day there are fewer than 2 breakdowns. [3]
  2. Find the probability that during a 14-day period there are at most 4 breakdowns. [3]
The cars are maintained at a garage. The garage introduced a weekly check to try to decrease the number of cars that break down. In a randomly selected 28-day period after the checks are introduced, only 1 hire car broke down.
  1. Test, at the 5% level of significance, whether or not the mean number of breakdowns has decreased. State your hypotheses clearly. [7]
Edexcel S2 2004 June Q6
12 marks Standard +0.3
Minor defects occur in a particular make of carpet at a mean rate of 0.05 per m\(^2\).
  1. Suggest a suitable model for the distribution of the number of defects in this make of carpet. Give a reason for your answer.
A carpet fitter has a contract to fit this carpet in a small hotel. The hotel foyer requires 30 m\(^2\) of this carpet. Find the probability that the foyer carpet contains
  1. exactly 2 defects, [3]
  2. more than 5 defects. [3]
The carpet fitter orders a total of 355 m\(^2\) of the carpet for the whole hotel.
  1. Using a suitable approximation, find the probability that this total area of carpet contains 22 or more defects. [6]
Edexcel S2 2004 June Q7
17 marks Standard +0.3
A random variable \(X\) has probability density function given by $$f(x) = \begin{cases} \frac{1}{3}, & 0 \leq x \leq 1, \\ \frac{8x^3}{45}, & 1 \leq x \leq 2, \\ 0, & \text{otherwise}. \end{cases}$$
  1. Calculate the mean of \(X\). [5]
  2. Specify fully the cumulative distribution function F\((x)\). [7]
  3. Find the median of \(X\). [3]
  4. Comment on the skewness of the distribution of \(X\). [2]
Edexcel S2 2006 June Q1
3 marks Easy -2.0
Before introducing a new rule the secretary of a golf club decided to find out how members might react to this rule.
  1. Explain why the secretary decided to take a random sample of club members rather than ask all the members. [1]
  2. Suggest a suitable sampling frame. [1]
  3. Identify the sampling units. [1]
Edexcel S2 2006 June Q2
7 marks Moderate -0.3
The continuous random variable \(L\) represents the error, in mm, made when a machine cuts rods to a target length. The distribution of \(L\) is continuous uniform over the interval \([-4.0, 4.0]\). Find
  1. P\((L < -2.6)\), [1]
  2. P\((L < -3.0 \text{ or } L > 3.0)\). [2]
A random sample of 20 rods cut by the machine was checked.
  1. Find the probability that more than half of them were within 3.0 mm of the target length. [4]