Questions S2 (1597 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
CAIE S2 2017 June Q6
6 The number of sports injuries per month at a certain college has a Poisson distribution. In the past the mean has been 1.1 injuries per month. The principal recently introduced new safety guidelines and she decides to test, at the \(2 \%\) significance level, whether the mean number of sports injuries has been reduced. She notes the number of sports injuries during a 6-month period.
  1. Find the critical region for the test and state the probability of a Type I error.
  2. State what is meant by a Type I error in this context.
  3. During the 6 -month period there are a total of 2 sports injuries. Carry out the test.
  4. Assuming that the mean remains 1.1 , calculate the probability that there will be fewer than 30 sports injuries during a 36-month period.
CAIE S2 2017 June Q1
1 In a survey of 2000 randomly chosen adults, 1602 said that they owned a smartphone. Calculate an approximate \(95 \%\) confidence interval for the proportion of adults in the whole population who own a smartphone.
CAIE S2 2017 June Q2
2 Javier writes an article containing 52460 words. He plans to upload the article to his website, but he knows that this process sometimes introduces errors. He assumes that for each word in the uploaded version of his article, the probability that it contains an error is 0.00008 . The number of words containing an error is denoted by \(X\).
  1. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\), giving your answers correct to three decimal places.
    Javier wants to use the Poisson distribution as an approximating distribution to calculate the probability that there will be fewer than 5 words containing an error in his uploaded article.
  2. Explain how your answers to part (i) are consistent with the use of the Poisson distribution as an approximating distribution.
  3. Use the Poisson distribution to calculate \(\mathrm { P } ( X < 5 )\).
CAIE S2 2017 June Q3
3 Household incomes, in thousands of dollars, in a certain country are represented by the random variable \(X\) with mean \(\mu\) and standard deviation \(\sigma\). The incomes of a random sample of 400 households are found and the results are summarised below. $$n = 400 \quad \Sigma x = 923 \quad \Sigma x ^ { 2 } = 3170$$
  1. Calculate unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\).
  2. A random sample of 50 households in one particular region of the country is taken and the sample mean income, in thousands of dollars, is found to be 2.6 . Using your values from part (i), test at the \(5 \%\) significance level whether household incomes in this region are greater, on average, than in the country as a whole.
CAIE S2 2017 June Q4
4 It is claimed that 1 in every 4 packets of certain biscuits contains a free gift. Marisa and André both suspect that the true proportion is less than 1 in 4.
  1. Marisa chooses 20 packets at random. She decides that if fewer than 3 contain free gifts, she will conclude that the claim is not justified. Use a binomial distribution to find the probability of a Type I error.
  2. André chooses 25 packets at random. He decides to carry out a significance test at the \(1 \%\) level, using a binomial distribution. Given that only 1 of the 25 packets contains a free gift, carry out the test.
CAIE S2 2017 June Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{c06524f0-a981-48a6-9af0-c4a3474396b3-06_394_723_258_705} The diagram shows the graph of the probability density function, f , of a random variable \(X\) which takes values between 0 and \(a\) only. It is given that \(\mathrm { P } ( X < 1 ) = 0.25\).
  1. Find, in any order,
    (a) \(\mathrm { P } ( X < 2 )\),
    (b) the value of \(a\),
    (c) \(\mathrm { f } ( x )\).
  2. Find the median of \(X\).
CAIE S2 2017 June Q6
6 Old televisions arrive randomly and independently at a recycling centre at an average rate of 1.2 per day.
  1. Find the probability that exactly 2 televisions arrive in a 2-day period.
  2. Use an appropriate approximating distribution to find the probability that at least 55 televisions arrive in a 50-day period.
    Independently of televisions, old computers arrive randomly and independently at the same recycling centre at an average rate of 4 per 7-day week.
  3. Find the probability that the total number of televisions and computers that arrive at the recycling centre in a 3-day period is less than 4.
CAIE S2 2017 June Q7
7
  1. A random variable \(X\) is normally distributed with mean 4.2 and standard deviation 1.1. Find the probability that the sum of two randomly chosen values of \(X\) is greater than 10 .
  2. Each candidate's overall score for an essay is calculated as follows. The mark for creativity is denoted by \(C\), the penalty mark for spelling errors is denoted by \(S\) and the overall score is defined by \(C - \frac { 1 } { 2 } S\). The variables \(C\) and \(S\) are independent and have distributions \(\mathrm { N } ( 29,105 )\) and \(\mathrm { N } ( 17,15 )\) respectively. Find the proportion of candidates receiving a negative overall score.
CAIE S2 2017 June Q1
1 A residents' association has 654 members, numbered from 1 to 654 . The secretary wishes to send a questionnaire to a random sample of members. In order to choose the members for the sample she uses a table of random numbers. The first line in the table is as follows. $$\begin{array} { l l l l l l } 1096 & 4357 & 3765 & 0431 & 0928 & 9264 \end{array}$$ The numbers of the first two members in the sample are 109 and 643. Find the numbers of the next three members in the sample.
CAIE S2 2017 June Q2
2 In a random sample of 200 shareholders of a company, 103 said that they wanted a change in the management.
  1. Find an approximate \(92 \%\) confidence interval for the proportion, \(p\), of all shareholders who want a change in the management.
  2. State the probability that a \(92 \%\) confidence interval does not contain \(p\).
CAIE S2 2017 June Q3
3 The mass, in tonnes, of iron ore produced per day at a mine is normally distributed with mean 7.0 and standard deviation 0.46. Find the probability that the total amount of iron ore produced in 10 randomly chosen days is more than 71 tonnes.
CAIE S2 2017 June Q4
4 Last year the mean level of a certain pollutant in a river was found to be 0.034 grams per millilitre. This year the levels of pollutant, \(X\) grams per millilitre, were measured at a random sample of 200 locations in the river. The results are summarised below. $$n = 200 \quad \Sigma x = 6.7 \quad \Sigma x ^ { 2 } = 0.2312$$
  1. Calculate unbiased estimates of the population mean and variance.
  2. Test, at the \(10 \%\) significance level, whether the mean level of pollutant has changed.
CAIE S2 2017 June Q5
5
  1. A random variable \(X\) has the distribution \(\operatorname { Po } ( 42 )\).
    (a) Use an appropriate approximating distribution to find \(\mathrm { P } ( X \geqslant 40 )\).
    (b) Justify your use of the approximating distribution.
  2. A random variable \(Y\) has the distribution \(\mathrm { B } ( 60,0.02 )\).
    (a) Use an appropriate approximating distribution to find \(\mathrm { P } ( Y > 2 )\).
    (b) Justify your use of the approximating distribution.
CAIE S2 2017 June Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{395f7f2c-42db-4fb6-9b22-3b0f46ad16d3-08_355_670_260_735} The diagram shows the graph of the probability density function, f , of a continuous random variable \(X\), where f is defined by $$\mathrm { f } ( x ) = \begin{cases} k \left( x - x ^ { 2 } \right) & 0 \leqslant x \leqslant 1
0 & \text { otherwise } \end{cases}$$
  1. Show that the value of the constant \(k\) is 6 .
  2. State the value of \(\mathrm { E } ( X )\) and find \(\operatorname { Var } ( X )\).
  3. Find \(\mathrm { P } ( 0.4 < X < 2 )\).
CAIE S2 2017 June Q7
7 In the past the number of accidents per month on a certain road was modelled by a random variable with distribution \(\operatorname { Po } ( 0.47 )\). After the introduction of speed restrictions, the government wished to test, at the 5\% significance level, whether the mean number of accidents had decreased. They noted the number of accidents during the next 12 months. It is assumed that accidents occur randomly and that a Poisson model is still appropriate.
  1. Given that the total number of accidents during the 12 months was 2 , carry out the test.
  2. Explain what is meant by a Type II error in this context.
    It is given that the mean number of accidents per month is now in fact 0.05 .
  3. Using another random sample of 12 months the same test is carried out again, with the same significance level. Find the probability of a Type II error.
CAIE S2 2018 June Q1
1 A random variable \(X\) has the distribution \(\mathrm { B } ( 75,0.03 )\).
  1. Use the Poisson approximation to the binomial distribution to calculate \(\mathrm { P } ( X < 3 )\).
  2. Justify the use of the Poisson approximation.
CAIE S2 2018 June Q2
2 Amy has to choose a random sample from the 265 students in her year at college. She numbers the students from 1 to 265 and then uses random numbers generated by her calculator. The first two random numbers produced by her calculator are 0.213165448 and 0.073165196 .
  1. Use these figures to find the numbers of the first four students in her sample.
    There were 25 students in Amy's sample. She asked each of them how much money, \(
    ) x$, they earned in a week, on average. Her results are summarised below. $$n = 25 \quad \Sigma x = 510 \quad \Sigma x ^ { 2 } = 13225$$
  2. Find unbiased estimates of the population mean and variance.
  3. Explain briefly what is meant by 'population' in this question.
CAIE S2 2018 June Q3
3 A researcher wishes to estimate the proportion, \(p\), of houses in London Road that have only one occupant. He takes a random sample of 64 houses in London Road and finds that 8 houses in the sample have only one occupant. Using this sample, he calculates that an approximate \(\alpha \%\) confidence interval for \(p\) has width 0.130 . Find \(\alpha\) correct to the nearest integer.
CAIE S2 2018 June Q4
4 The numbers, \(M\) and \(F\), of male and female students who leave a particular school each year to study engineering have means 3.1 and 0.8 respectively.
  1. State, in context, one condition required for \(M\) to have a Poisson distribution.
    Assume that \(M\) and \(F\) can be modelled by independent Poisson distributions.
  2. Find the probability that the total number of students who leave to study engineering in a particular year is more than 3 .
  3. Given that the total number of students who leave to study engineering in a particular year is more than 3 , find the probability that no female students leave to study engineering in that year.
CAIE S2 2018 June Q5
5 The time taken for a particular train journey is normally distributed. In the past, the time had mean 2.4 hours and standard deviation 0.3 hours. A new timetable is introduced and on 30 randomly chosen occasions the time for this journey is measured. The mean time for these 30 occasions is found to be 2.3 hours.
  1. Stating any assumption(s), test, at the \(5 \%\) significance level, whether the mean time for this journey has changed.
  2. A similar test at the \(5 \%\) significance level was carried out using the times from another randomly chosen 30 occasions.
    (a) State the probability of a Type I error.
    (b) State what is meant by a Type II error in this context.
CAIE S2 2018 June Q6
6 The times, in minutes, taken to complete the two parts of a task are normally distributed with means 4.5 and 2.3 respectively and standard deviations 1.1 and 0.7 respectively.
  1. Find the probability that the total time taken for the task is less than 8.5 minutes.
  2. Find the probability that the time taken for the first part of the task is more than twice the time taken for the second part.
CAIE S2 2018 June Q7
7 A random variable \(X\) has probability density function defined by $$f ( x ) = \begin{cases} k \left( \frac { 1 } { x ^ { 2 } } + \frac { 1 } { x ^ { 3 } } \right) & 1 \leqslant x \leqslant 2
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = \frac { 8 } { 7 }\).
  2. Find \(\mathrm { E } ( X )\).
  3. Three values of \(X\) are chosen at random. Find the probability that one of these values is less than 1.5 and the other two are greater than 1.5.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE S2 2011 June Q1
1 On average, 2 people in every 10000 in the UK have a particular gene. A random sample of 6000 people in the UK is chosen. The random variable \(X\) denotes the number of people in the sample who have the gene. Use an approximating distribution to calculate the probability that there will be more than 2 people in the sample who have the gene.
CAIE S2 2011 June Q2
2
  1. The time taken by a worker to complete a task was recorded for a random sample of 50 workers. The sample mean was 41.2 minutes and an unbiased estimate of the population variance was 32.6 minutes \({ } ^ { 2 }\). Find a \(95 \%\) confidence interval for the mean time taken to complete the task.
  2. The probability that an \(\alpha \%\) confidence interval includes only values that are lower than the population mean is \(\frac { 1 } { 16 }\). Find the value of \(\alpha\).
CAIE S2 2011 June Q3
3 Past experience has shown that the heights of a certain variety of rose bush have been normally distributed with mean 85.0 cm . A new fertiliser is used and it is hoped that this will increase the heights. In order to test whether this is the case, a botanist records the heights, \(x \mathrm {~cm}\), of a large random sample of \(n\) rose bushes and calculates that \(\bar { x } = 85.7\) and \(s = 4.8\), where \(\bar { x }\) is the sample mean and \(s ^ { 2 }\) is an unbiased estimate of the population variance. The botanist then carries out an appropriate hypothesis test.
  1. The test statistic, \(z\), has a value of 1.786 correct to 3 decimal places. Calculate the value of \(n\).
  2. Using this value of the test statistic, carry out the test at the \(5 \%\) significance level.