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CAIE S2 2017 June Q2
6 marks Moderate -0.3
2 Past experience has shown that the heights of a certain variety of plant have mean 64.0 cm and standard deviation 3.8 cm . During a particularly hot summer, it was expected that the heights of plants of this variety would be less than usual. In order to test whether this was the case, a botanist recorded the heights of a random sample of 100 plants and found that the value of the sample mean was 63.3 cm . Stating a necessary assumption, carry out the test at the \(2.5 \%\) significance level.
CAIE S2 2017 June Q3
6 marks Standard +0.8
3
  1. The waiting time at a certain bus stop has variance 2.6 minutes \({ } ^ { 2 }\). For a random sample of 75 people, the mean waiting time was 7.1 minutes. Calculate a \(92 \%\) confidence interval for the population mean waiting time.
  2. A researcher used 3 random samples to calculate 3 independent \(92 \%\) confidence intervals. Find the probability that all 3 of these confidence intervals contain only values that are greater than the actual population mean.
  3. Another researcher surveyed the first 75 people who waited at a bus stop on a Monday morning. Give a reason why this sample is unsuitable for use in finding a confidence interval for the mean waiting time.
CAIE S2 2017 June Q4
9 marks Moderate -0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{332f0909-c192-40f7-88b7-7cfec2db2eef-06_428_773_260_685} The time, \(X\) minutes, taken by a large number of runners to complete a certain race has probability density function f given by $$f ( x ) = \begin{cases} \frac { k } { x ^ { 2 } } & 5 \leqslant x \leqslant 10 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant, as shown in the diagram.
  1. Without calculation, explain how you can tell that there were more runners whose times were below 7.5 minutes than above 7.5 minutes.
  2. Show that \(k = 10\).
  3. Find \(\mathrm { E } ( X )\).
  4. Find \(\operatorname { Var } ( X )\).
CAIE S2 2017 June Q5
10 marks Standard +0.3
5 Large packets of sugar are packed in cartons, each containing 12 packets. The weights of these packets are normally distributed with mean 505 g and standard deviation 3.2 g . The weights of the cartons, when empty, are independently normally distributed with mean 150 g and standard deviation 7 g .
  1. Find the probability that the total weight of a full carton is less than 6200 g .
    Small packets of sugar are packed in boxes. The total weight of a full box has a normal distribution with mean 3130 g and standard deviation 12.1 g .
  2. Find the probability that the weight of a randomly chosen full carton is less than double the weight of a randomly chosen full box.
CAIE S2 2017 June Q6
14 marks Standard +0.3
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
4 marks Easy -1.2
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
5 marks Moderate -0.3
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
8 marks Standard +0.3
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
7 marks Moderate -0.3
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
8 marks Standard +0.3
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
9 marks Standard +0.3
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
9 marks Standard +0.3
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
3 marks Easy -1.8
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
4 marks Moderate -0.8
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
5 marks Standard +0.3
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
8 marks Moderate -0.3
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
9 marks Standard +0.3
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
10 marks Moderate -0.3
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
11 marks Standard +0.3
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
4 marks Moderate -0.8
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
6 marks Easy -1.3
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
5 marks Standard +0.3
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
7 marks Standard +0.8
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
8 marks Standard +0.3
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
9 marks Standard +0.8
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.