Questions S2 (1690 questions)

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Edexcel S2 Q6
20 marks Moderate -0.3
A magazine has a large number of subscribers who each pay a membership fee that is due on January 1st each year. Not all subscribers pay their fee by the due date. Based on correspondence from the subscribers, the editor of the magazine believes that 40\% of subscribers wish to change the name of the magazine. Before making this change the editor decides to carry out a sample survey to obtain the opinions of the subscribers. He uses only those members who have paid their fee on time.
  1. Define the population associated with the magazine. [1]
  2. Suggest a suitable sampling frame for the survey. [1]
  3. Identify the sampling units. [1]
  4. Give one advantage and one disadvantage that would have resulted from the editor using a census rather than a sample survey. [2]
As a pilot study the editor took a random sample of 25 subscribers.
  1. Assuming that the editor's belief is correct, find the probability that exactly 10 of these subscribers agreed with changing the name. [3]
In fact only 6 subscribers agreed to the name being changed.
  1. Stating your hypotheses clearly test, at the 5\% level of significance, whether or not the percentage agreeing to the change is less that the editor believes. [5]
The full survey is to be carried out using 200 randomly chosen subscribers.
  1. Again assuming the editor's belief to be correct and using a suitable approximation, find the probability that in this sample there will be least 71 but fewer than 83 subscribers who agree to the name being changed. [7]
Edexcel S2 Q1
4 marks Easy -1.2
Explain briefly what you understand by
  1. a statistic, [2]
  2. a sampling distribution. [2]
Edexcel S2 Q2
7 marks Moderate -0.8
  1. Write down the condition needed to approximate a Poisson distribution by a Normal distribution. [1]
The random variable Y ~ Po(30).
  1. Estimate P(Y > 28). [6]
Edexcel S2 Q3
9 marks Easy -1.2
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 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, [8]
  2. in the first eight throws there will be exactly three 5s.
Edexcel S2 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]
Find the probability that the machine dispenses
  1. less than 183 ml, [3]
  2. exactly 183 ml. [1]
  3. Calculate the inter-quartile range of X. [3]
  4. Determine the value of s such that P(X ≤ s) = 1 - 2P(X ≤ s). [2]
  5. Interpret in words your value of s.
Edexcel S2 Q6
15 marks Standard +0.3
A doctor expects to see, on average, 1 patient per week with a particular disease.
  1. Suggest a suitable model for the distribution of the number of times per week that the doctor sees a patient with the disease. Give a reason for your answer. [3]
  2. Using your model, find the probability that the doctor sees more than 3 patients with the disease in a 4 week period. [4]
The doctor decides to send information to his patients to try to reduce the number of patients he sees with the disease. In the first 6 weeks after the information is sent out, the doctor sees 2 patients with the disease.
  1. Test, at the 5\% level of significance, whether or not there is reason to believe that sending the information has reduced the number of times the doctor sees patients with the disease. State your hypotheses clearly. [6]
Medical research into the nature of the disease discovers that it can be passed from one patient to another.
  1. Explain whether or not this research supports your choice of model. Give a reason for your answer. [2]
Edexcel S2 Q7
15 marks Standard +0.3
A continuous random variable X has probability density function f(x) where $$f(x) = \begin{cases} k(x^3 + 2x + 1), & -1 \leq x \leq 0, \\ 0, & otherwise \end{cases}$$ 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 Q1
5 marks Easy -2.0
A large dental practice wishes to investigate the level of satisfaction of its patients.
  1. Suggest a suitable sampling frame for the investigation. [1]
  2. Identify the sampling units. [1]
  3. State one advantage and one disadvantage of using a sample survey rather than a census. [2]
  4. Suggest a problem that might arise with the sampling frame when selecting patients. [1]
Edexcel S2 Q2
7 marks Moderate -0.8
The random variable R has the binomial distribution B(12, 0.35).
  1. Find P(R ≥ 4). [2]
The random variable S has the Poisson distribution with mean 2.71.
  1. Find P(S ≤ 1). [3]
The random variable T has the normal distribution N(2.5, 5²).
  1. Find P(T ≤ 18). [2]
Edexcel S2 2004 January Q1
5 marks Easy -1.8
A large dental practice wishes to investigate the level of satisfaction of its patients.
  1. Suggest a suitable sampling frame for the investigation. [1]
  2. Identify the sampling units. [1]
  3. State one advantage and one disadvantage of using a sample survey rather than a census. [2]
  4. Suggest a problem that might arise with the sampling frame when selecting patients. [1]
Edexcel S2 2004 January Q2
7 marks Easy -1.3
The random variable \(R\) has the binomial distribution B(12, 0.35).
  1. Find P(\(R \geq 4\)). [2]
The random variable \(S\) has the Poisson distribution with mean 2.71.
  1. Find P(\(S \leq 1\)). [3]
The random variable \(T\) has the normal distribution N(25, \(5^2\)).
  1. Find P(\(T \leq 18\)). [2]
Edexcel S2 2004 January Q3
9 marks Moderate -0.3
The discrete random variable \(X\) is distributed B(\(n\), \(p\)).
  1. Write down the value of \(p\) that will give the most accurate estimate when approximating the binomial distribution by a normal distribution. [1]
  2. Give a reason to support your value. [1]
  3. Given that \(n = 200\) and \(p = 0.48\), find P(\(90 \leq X < 105\)). [7]
Edexcel S2 2004 January Q4
10 marks Moderate -0.8
  1. Write down two conditions needed to be able to approximate the binomial distribution by the Poisson distribution. [2]
A researcher has suggested that 1 in 150 people is likely to catch a particular virus. Assuming that a person catching the virus is independent of any other person catching it,
  1. find the probability that in a random sample of 12 people, exactly 2 of them catch the virus. [4]
  2. Estimate the probability that in a random sample of 1200 people fewer than 7 catch the virus. [4]
Edexcel S2 2004 January Q5
13 marks Moderate -0.3
Vehicles pass a particular point on a road at a rate of 51 vehicles per hour.
  1. Give two reasons to support the use of the Poisson distribution as a suitable model for the number of vehicles passing this point. [2]
Find the probability that in any randomly selected 10 minute interval
  1. exactly 6 cars pass this point, [3]
  2. at least 9 cars pass this point. [2]
After the introduction of a roundabout some distance away from this point it is suggested that the number of vehicles passing it has decreased. During a randomly selected 10 minute interval 4 vehicles pass the point.
  1. Test, at the 5\% level of significance, whether or not there is evidence to support the suggestion that the number of vehicles has decreased. State your hypotheses clearly. [6]
Edexcel S2 2004 January Q6
13 marks Standard +0.3
From past records a manufacturer of ceramic plant pots knows that 20\% of them will have defects. To monitor the production process, a random sample of 25 pots is checked each day and the number of pots with defects is recorded.
  1. Find the critical regions for a two-tailed test of the hypothesis that the probability that a plant pot has defects is 0.20. The probability of rejection in either tail should be as close as possible to 2.5\%. [5]
  2. Write down the significance level of the above test. [1]
A garden centre sells these plant pots at a rate of 10 per week. In an attempt to increase sales, the price was reduced over a six-week period. During this period a total of 74 pots was sold.
  1. Using a 5\% level of significance, test whether or not there is evidence that the rate of sales per week has increased during this six-week period. [7]
Edexcel S2 2004 January Q7
18 marks Moderate -0.3
The continuous random variable \(X\) has probability density function $$\text{f}(x) = \begin{cases} kx(5 - x), & 0 \leq x \leq 4, \\ 0, & \text{otherwise,} \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = \frac{3}{56}\). [3]
  2. Find the cumulative distribution function F(\(x\)) for all values of \(x\). [4]
  3. Evaluate E(\(X\)). [3]
  4. Find the modal value of \(X\). [3]
  5. Verify that the median value of \(X\) lies between 2.3 and 2.5. [3]
  6. Comment on the skewness of \(X\). Justify your answer. [2]
Edexcel S2 2009 January Q1
11 marks Standard +0.3
A botanist is studying the distribution of daisies in a field. The field is divided into a number of equal sized squares. The mean number of daisies per square is assumed to be 3. The daisies are distributed randomly throughout the field. Find the probability that, in a randomly chosen square there will be
  1. more than 2 daisies, [3]
  2. either 5 or 6 daisies. [2]
The botanist decides to count the number of daisies, \(x\), in each of 80 randomly selected squares within the field. The results are summarised below $$\sum x = 295 \quad \sum x^2 = 1386$$
  1. Calculate the mean and the variance of the number of daisies per square for the 80 squares. Give your answers to 2 decimal places. [3]
  2. Explain how the answers from part (c) support the choice of a Poisson distribution as a model. [1]
  3. Using your mean from part (c), estimate the probability that exactly 4 daisies will be found in a randomly selected square. [2]
Edexcel S2 2009 January Q2
9 marks Easy -1.2
The continuous random variable \(X\) is uniformly distributed over the interval \([-2, 7]\).
  1. Write down fully the probability density function f(x) of \(X\). [2]
  2. Sketch the probability density function f(x) of \(X\). [2]
Find
  1. E(\(X^2\)), [3]
  2. P(\(-0.2 < X < 0.6\)). [2]
Edexcel S2 2009 January Q3
7 marks Moderate -0.3
A single observation \(x\) is to be taken from a Binomial distribution B(20, \(p\)). This observation is used to test \(H_0 : p = 0.3\) against \(H_1 : p \neq 0.3\)
  1. Using a 5\% level of significance, find the critical region for this test. The probability of rejecting either tail should be as close as possible to 2.5\%. [3]
  2. State the actual significance level of this test. [2]
The actual value of \(x\) obtained is 3.
  1. State a conclusion that can be drawn based on this value giving a reason for your answer. [2]
Edexcel S2 2009 January Q4
12 marks Moderate -0.8
The length of a telephone call made to a company is denoted by the continuous random variable \(T\). It is modelled by the probability density function $$\text{f}(t) = \begin{cases} kt & 0 \leqslant t \leqslant 10 \\ 0 & \text{otherwise} \end{cases}$$
  1. Show that the value of \(k\) is \(\frac{1}{50}\). [3]
  2. Find P(\(T > 6\)). [2]
  3. Calculate an exact value for E(\(T\)) and for Var(\(T\)). [5]
  4. Write down the mode of the distribution of \(T\). [1]
It is suggested that the probability density function, f(\(t\)), is not a good model for \(T\).
  1. Sketch the graph of a more suitable probability density function for \(T\). [1]
Edexcel S2 2009 January Q5
9 marks Moderate -0.3
A factory produces components of which 1\% are defective. The components are packed in boxes of 10. A box is selected at random.
  1. Find the probability that the box contains exactly one defective component. [2]
  2. Find the probability that there are at least 2 defective components in the box. [3]
  3. Using a suitable approximation, find the probability that a batch of 250 components contains between 1 and 4 (inclusive) defective components. [4]
Edexcel S2 2009 January Q6
14 marks Standard +0.3
A web server is visited on weekdays, at a rate of 7 visits per minute. In a random one minute on a Saturday the web server is visited 10 times.
    1. Test, at the 10\% level of significance, whether or not there is evidence that the rate of visits is greater on a Saturday than on weekdays. State your hypotheses clearly.
    2. State the minimum number of visits required to obtain a significant result.
    [7]
  2. State an assumption that has been made about the visits to the server. [1]
In a random two minute period on a Saturday the web server is visited 20 times.
  1. Using a suitable approximation, test at the 10\% level of significance, whether or not the rate of visits is greater on a Saturday. [6]
Edexcel S2 2009 January Q7
13 marks Standard +0.3
A random variable \(X\) has probability density function given by $$\text{f}(x) = \begin{cases} -\frac{2}{9}x + \frac{8}{9} & 1 \leqslant x \leqslant 4 \\ 0 & \text{otherwise} \end{cases}$$
  1. Show that the cumulative distribution function F(x) can be written in the form \(ax^2 + bx + c\), for \(1 \leqslant x \leqslant 4\) where \(a\), \(b\) and \(c\) are constants. [3]
  2. Define fully the cumulative distribution function F(x). [2]
  3. Show that the upper quartile of \(X\) is 2.5 and find the lower quartile. [6]
Given that the median of \(X\) is 1.88
  1. describe the skewness of the distribution. Give a reason for your answer. [2]
Edexcel S2 2011 January Q1
10 marks Moderate -0.3
A disease occurs in 3\% of a population.
  1. State any assumptions that are required to model the number of people with the disease in a random sample of size \(n\) as a binomial distribution. [2]
  2. Using this model, find the probability of exactly 2 people having the disease in a random sample of 10 people. [3]
  3. Find the mean and variance of the number of people with the disease in a random sample of 100 people. [2]
A doctor tests a random sample of 100 patients for the disease. He decides to offer all patients a vaccination to protect them from the disease if more than 5 of the sample have the disease.
  1. Using a suitable approximation, find the probability that the doctor will offer all patients a vaccination. [3]