Questions S2 (1690 questions)

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OCR S2 2016 June Q1
4 marks Easy -1.2
The results of 14 observations of a random variable \(V\) are summarised by $$n = 14, \quad \sum v = 3752, \quad \sum v^2 = 1007448.$$ Calculate unbiased estimates of E\((V)\) and Var\((V)\). [4]
OCR S2 2016 June Q2
6 marks Moderate -0.3
The mass, in kilograms, of a packet of flour is a normally distributed random variable with mean 1.03 and variance \(\sigma^2\). Given that 5% of packets have mass less than 1.00 kg, find the percentage of packets with mass greater than 1.05 kg. [6]
OCR S2 2016 June Q3
7 marks Standard +0.3
The random variable \(F\) has the distribution B\((40, 0.65)\). Use a suitable approximation to find P\((F \leq 30)\), justifying your approximation. [7]
OCR S2 2016 June Q4
5 marks Moderate -0.8
It is given that \(Y \sim\) Po\((\lambda)\), where \(\lambda \neq 0\), and that P\((Y = 4) =\) P\((Y = 5)\). Write down an equation for \(\lambda\). Hence find the value of \(\lambda\) and the corresponding value of P\((Y = 5)\). [5]
OCR S2 2016 June Q5
8 marks Standard +0.3
55% of the pupils in a large school are girls. A member of the student council claims that the probability that a girl rather than a boy becomes Head Student is greater than 0.55. As evidence for his claim he says that 6 of the last 8 Head Students have been girls.
  1. Use an exact binomial distribution to test the claim at the 10% significance level. [7]
  2. A statistics teacher says that considering only the last 8 Head Students may not be satisfactory. Explain what needs to be assumed about the data for the test to be valid. [1]
OCR S2 2016 June Q6
12 marks Moderate -0.3
The number of cars passing a point on a single-track one-way road during a one-minute period is denoted by \(X\). Cars pass the point at random intervals and the expected value of \(X\) is denoted by \(\lambda\).
  1. State, in the context of the question, two conditions needed for \(X\) to be well modelled by a Poisson distribution. [2]
  2. At a quiet time of the day, \(\lambda = 6.50\). Assuming that a Poisson distribution is valid, calculate P\((4 \leq X < 8)\). [3]
  3. At a busy time of the day, \(\lambda = 30\).
    1. Assuming that a Poisson distribution is valid, use a suitable approximation to find P\((X > 35)\). Justify your approximation. [6]
    2. Give a reason why a Poisson distribution might not be valid in this context when \(\lambda = 30\). [1]
OCR S2 2016 June Q7
11 marks Standard +0.3
A continuous random variable \(X\) has probability density function $$\text{f}(x) = \begin{cases} ax^{-3} + bx^{-4} & x \geq 1, \\ 0 & \text{otherwise,} \end{cases}$$ where \(a\) and \(b\) are constants.
  1. Explain what the letter \(x\) represents. [1]
It is given that P\((X > 2) = \frac{3}{16}\).
  1. Show that \(a = 1\), and find the value of \(b\). [7]
  2. Find E\((X)\). [3]
OCR S2 2016 June Q8
13 marks Standard +0.3
It is known that the lifetime of a certain species of animal in the wild has mean 13.3 years. A zoologist reads a study of 50 randomly chosen animals of this species that have been kept in zoos. According to the study, for these 50 animals the sample mean lifetime is 12.48 years and the population variance is 12.25 years\(^2\).
  1. Test at the 5% significance level whether these results provide evidence that animals of this species that have been kept in zoos have a shorter expected lifetime than those in the wild. [7]
  2. Subsequently the zoologist discovered that there had been a mistake in the study. The quoted variance of 12.25 years\(^2\) was in fact the sample variance. Determine whether this makes a difference to the conclusion of the test. [5]
  3. Explain whether the Central Limit Theorem is needed in these tests. [1]
OCR S2 2016 June Q9
6 marks Challenging +1.3
The random variable \(R\) has the distribution Po\((\lambda)\). A significance test is carried out at the 1% level of the null hypothesis H\(_0\): \(\lambda = 11\) against H\(_1\): \(\lambda > 11\), based on a single observation of \(R\). Given that in fact the value of \(\lambda\) is 14, find the probability that the result of the test is incorrect, and give the technical name for such an incorrect outcome. You should show the values of any relevant probabilities. [6]
OCR MEI S2 2007 January Q1
18 marks Moderate -0.8
In a science investigation into energy conservation in the home, a student is collecting data on the time taken for an electric kettle to boil as the volume of water in the kettle is varied. The student's data are shown in the table below, where \(v\) litres is the volume of water in the kettle and \(t\) seconds is the time taken for the kettle to boil (starting with the water at room temperature in each case). Also shown are summary statistics and a scatter diagram on which the regression line of \(t\) on \(v\) is drawn.
\(v\)0.20.40.60.81.0
\(t\)4478114156172
\(n = 5\), \(\Sigma v = 3.0\), \(\Sigma t = 564\), \(\Sigma v^2 = 2.20\), \(\Sigma vt = 405.2\). \includegraphics{figure_1}
  1. Calculate the equation of the regression line of \(t\) on \(v\), giving your answer in the form \(t = a + bv\). [5]
  2. Use this equation to predict the time taken for the kettle to boil when the amount of water which it contains is
    1. 0.5 litres,
    2. 1.5 litres.
    Comment on the reliability of each of these predictions. [4]
  3. In the equation of the regression line found in part (i), explain the role of the coefficient of \(v\) in the relationship between time taken and volume of water. [2]
  4. Calculate the values of the residuals for \(v = 0.8\) and \(v = 1.0\). [4]
  5. Explain how, on a scatter diagram with the regression line drawn accurately on it, a residual could be measured and its sign determined. [3]
OCR MEI S2 2007 January Q2
18 marks Moderate -0.3
  1. A farmer grows Brussels sprouts. The diameter of sprouts in a particular batch, measured in mm, is Normally distributed with mean 28 and variance 16. Sprouts that are between 24 mm and 33 mm in diameter are sold to a supermarket.
    1. Find the probability that the diameter of a randomly selected sprout will be within this range. [4]
    2. The farmer sells the sprouts in this range to the supermarket for 10 pence per kilogram. The farmer sells sprouts under 24 mm in diameter to a frozen food factory for 5 pence per kilogram. Sprouts over 33 mm in diameter are thrown away. Estimate the total income received by the farmer for the batch, which weighs 25 500 kg. [3]
    3. By harvesting sprouts earlier, the mean diameter for another batch can be reduced to \(k\) mm. Find the value of \(k\) for which only 5\% of the sprouts will be above 33 mm in diameter. You may assume that the variance is still 16. [3]
  2. The farmer also grows onions. The weight in kilograms of the onions is Normally distributed with mean 0.155 and variance 0.005. He is trying out a new variety, which he hopes will yield a higher mean weight. In order to test this, he takes a random sample of 25 onions of the new variety and finds that their total weight is 4.77 kg. You should assume that the weight in kilograms of the new variety is Normally distributed with variance 0.005.
    1. Write down suitable null and alternative hypotheses for the test in terms of \(\mu\). State the meaning of \(\mu\) in this case. [2]
    2. Carry out the test at the 1\% level. [6]
OCR MEI S2 2007 January Q3
18 marks Standard +0.3
An electrical retailer gives customers extended guarantees on washing machines. Under this guarantee all repairs in the first 3 years are free. The retailer records the numbers of free repairs made to 80 machines.
Number of repairs0123\(>3\)
Frequency5320610
  1. Show that the sample mean is 0.4375. [1]
  2. The sample standard deviation \(s\) is 0.6907. Explain why this supports a suggestion that a Poisson distribution may be a suitable model for the distribution of the number of free repairs required by a randomly chosen washing machine. [2]
The random variable \(X\) denotes the number of free repairs required by a randomly chosen washing machine. For the remainder of this question you should assume that \(X\) may be modelled by a Poisson distribution with mean 0.4375.
  1. Find P\((X = 1)\). Comment on your answer in relation to the data in the table. [4]
  2. The manager decides to monitor 8 washing machines sold on one day. Find the probability that there are at least 12 free repairs in total on these 8 machines. You may assume that the 8 machines form an independent random sample. [3]
  3. A launderette with 8 washing machines has needed 12 free repairs. Why does your answer to part (iv) suggest that the Poisson model with mean 0.4375 is unlikely to be a suitable model for free repairs on the machines in the launderette? Give a reason why the model may not be appropriate for the launderette. [3]
The retailer also sells tumble driers with the same guarantee. The number of free repairs on a tumble drier in three years can be modelled by a Poisson distribution with mean 0.15. A customer buys a tumble drier and a washing machine.
  1. Assuming that free repairs are required independently, find the probability that
    1. the two appliances need a total of 3 free repairs between them,
    2. each appliance needs exactly one free repair.
    [5]
OCR MEI S2 2007 January Q4
18 marks Standard +0.3
Two educational researchers are investigating the relationship between personal ambitions and home location of students. The researchers classify students into those whose main personal ambition is good academic results and those who have some other ambition. A random sample of 480 students is selected.
  1. One researcher summarises the data as follows.
    \multirow{2}{*}{Observed}Home location
    \cline{2-3}CityNon-city
    \multirow{2}{*}{Ambition}Good results102147
    \cline{2-3}Other75156
    Carry out a test at the 5\% significance level to examine whether there is any association between home location and ambition. State carefully your null and alternative hypotheses. Your working should include a table showing the contributions of each cell to the test statistic. [9]
  2. The other researcher summarises the same data in a different way as follows.
    \multirow{2}{*}{Observed}Home location
    \cline{2-4}CityTownCountry
    \multirow{2}{*}{Ambition}Good results1028364
    \cline{2-4}Other756492
    1. Calculate the expected frequencies for both 'Country' cells. [2]
    2. The test statistic for these data is 10.94. Carry out a test at the 5\% level based on this table, using the same hypotheses as in part (i). [3]
    3. The table below gives the contribution of each cell to the test statistic. Discuss briefly how personal ambitions are related to home location. [2]
      \multirow{2}{*}{
      Contribution to the
      test statistic
      }      		Home location
      \cline{2-4}CityTownCountry
      \multirow{2}{*}{Ambition}Good results1.1290.5963.540
      \cline{2-4}Other1.2170.6433.816
  3. Comment briefly on whether the analysis in part (ii) means that the conclusion in part (i) is invalid. [2]
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]