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CAIE S2 2018 June Q7
11 marks Standard +0.3
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
4 marks Moderate -0.3
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
5 marks Standard +0.3
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
6 marks Standard +0.3
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.
CAIE S2 2011 June Q4
10 marks Moderate -0.5
4
  1. \includegraphics[max width=\textwidth, alt={}, center]{7c9a87ac-69c6-4850-82aa-8235bba581e8-2_611_712_1466_358} \includegraphics[max width=\textwidth, alt={}, center]{7c9a87ac-69c6-4850-82aa-8235bba581e8-2_618_716_1464_1155} The diagrams show the graphs of two functions, \(g\) and \(h\). For each of the functions \(g\) and \(h\), give a reason why it cannot be a probability density function.
  2. The distance, in kilometres, travelled in a given time by a cyclist is represented by the continuous random variable \(X\) with probability density function given by $$f ( x ) = \begin{cases} \frac { 30 } { x ^ { 2 } } & 10 \leqslant x \leqslant 15 \\ 0 & \text { otherwise } \end{cases}$$
    1. Show that \(\mathrm { E } ( X ) = 30 \ln 1.5\).
    2. Find the median of \(X\). Find also the probability that \(X\) lies between the median and the mean.
CAIE S2 2011 June Q5
11 marks Standard +0.3
5 Cans of drink are packed in boxes, each containing 4 cans. The weights of these cans are normally distributed with mean 510 g and standard deviation 14 g . The weights of the boxes, when empty, are independently normally distributed with mean 200 g and standard deviation 8 g .
  1. Find the probability that the total weight of a full box of cans is between 2200 g and 2300 g .
  2. Two cans of drink are chosen at random. Find the probability that they differ in weight by more than 20 g .
CAIE S2 2011 June Q6
14 marks Standard +0.3
6 The number of injuries per month at a certain factory has a Poisson distribution. In the past the mean was 2.1 injuries per month. New safety procedures are put in place and the management wishes to use the next 3 months to test, at the \(2 \%\) significance level, whether there are now fewer injuries than before, on average.
  1. Find the critical region for the test.
  2. Find the probability of a Type I error.
  3. During the next 3 months there are a total of 3 injuries. Carry out the test.
  4. Assuming that the mean remains 2.1 , calculate an estimate of the probability that there will be fewer than 20 injuries during the next 12 months.
CAIE S2 2011 June Q1
4 marks Moderate -0.8
1 The weights of bags of fuel have mean 3.2 kg and standard deviation 0.04 kg . The total weight of a random sample of three bags is denoted by \(T \mathrm {~kg}\). Find the mean and standard deviation of \(T\). \(2 X\) is a random variable having the distribution \(\mathrm { B } \left( 12 , \frac { 1 } { 4 } \right)\). A random sample of 60 values of \(X\) is taken. Find the probability that the sample mean is less than 2.8 .
CAIE S2 2011 June Q3
7 marks Easy -1.2
3 The number of goals scored per match by Everly Rovers is represented by the random variable \(X\) which has mean 1.8.
  1. State two conditions for \(X\) to be modelled by a Poisson distribution. Assume now that \(X \sim \operatorname { Po } ( 1.8 )\).
  2. Find \(\mathrm { P } ( 2 < X < 6 )\).
  3. The manager promises the team a bonus if they score at least 1 goal in each of the next 10 matches. Find the probability that they win the bonus.
CAIE S2 2011 June Q4
8 marks Standard +0.3
4 A doctor wishes to investigate the mean fat content in low-fat burgers. He takes a random sample of 15 burgers and sends them to a laboratory where the mass, in grams, of fat in each burger is determined. The results are as follows. $$\begin{array} { l l l l l l l l l l l l l l l } 9 & 7 & 8 & 9 & 6 & 11 & 7 & 9 & 8 & 9 & 8 & 10 & 7 & 9 & 9 \end{array}$$ Assume that the mass, in grams, of fat in low-fat burgers is normally distributed with mean \(\mu\) and that the population standard deviation is 1.3 .
  1. Calculate a \(99 \%\) confidence interval for \(\mu\).
  2. Explain whether it was necessary to use the Central Limit theorem in the calculation in part (i).
  3. The manufacturer claims that the mean mass of fat in burgers of this type is 8 g . Use your answer to part (i) to comment on this claim.
CAIE S2 2011 June Q5
9 marks Standard +0.3
5 The number of adult customers arriving in a shop during a 5-minute period is modelled by a random variable with distribution \(\operatorname { Po } ( 6 )\). The number of child customers arriving in the same shop during a 10 -minute period is modelled by an independent random variable with distribution \(\mathrm { Po } ( 4.5 )\).
  1. Find the probability that during a randomly chosen 2 -minute period, the total number of adult and child customers who arrive in the shop is less than 3 .
  2. During a sale, the manager claims that more adult customers are arriving than usual. In a randomly selected 30 -minute period during the sale, 49 adult customers arrive. Test the manager's claim at the 2.5\% significance level.
CAIE S2 2011 June Q6
8 marks Standard +0.8
6 Jeevan thinks that a six-sided die is biased in favour of six. In order to test this, Jeevan throws the die 10 times. If the die shows a six on at least 4 throws out of 10 , she will conclude that she is correct.
  1. State appropriate null and alternative hypotheses.
  2. Calculate the probability of a Type I error.
  3. Explain what is meant by a Type II error in this situation.
  4. If the die is actually biased so that the probability of throwing a six is \(\frac { 1 } { 2 }\), calculate the probability of a Type II error.
CAIE S2 2011 June Q7
9 marks Moderate -0.3
7 A random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} k ( 1 - x ) & - 1 \leqslant x \leqslant 1 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = \frac { 1 } { 2 }\).
  2. Find \(\mathrm { P } \left( X > \frac { 1 } { 2 } \right)\).
  3. Find the mean of \(X\).
  4. Find \(a\) such that \(\mathrm { P } ( X < a ) = \frac { 1 } { 4 }\).
CAIE S2 2012 June Q1
3 marks Moderate -0.8
1 The weights, in grams, of packets of sugar are distributed with mean \(\mu\) and standard deviation 23. A random sample of 150 packets is taken. The mean weight of this sample is found to be 494 g . Calculate a 98\% confidence interval for \(\mu\).
CAIE S2 2012 June Q2
3 marks Moderate -0.8
2 An examination consists of a written paper and a practical test. The written paper marks ( \(M\) ) have mean 54.8 and standard deviation 16.0. The practical test marks ( \(P\) ) are independent of the written paper marks and have mean 82.4 and standard deviation 4.8. The final mark is found by adding \(75 \%\) of \(M\) to \(25 \%\) of \(P\). Find the mean and standard deviation of the final marks for the examination. [3]
CAIE S2 2012 June Q3
5 marks Moderate -0.3
3 When the council published a plan for a new road, only \(15 \%\) of local residents approved the plan. The council then published a revised plan and, out of a random sample of 300 local residents, 60 approved the revised plan. Is there evidence, at the \(2.5 \%\) significance level, that the proportion of local residents who approve the revised plan is greater than for the original plan?
CAIE S2 2012 June Q4
7 marks Moderate -0.3
4 The random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { k } { ( x + 1 ) ^ { 2 } } & 0 \leqslant x \leqslant 1 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = 2\).
  2. Find \(a\) such that \(\mathrm { P } ( X < a ) = \frac { 1 } { 5 }\).
  3. \includegraphics[max width=\textwidth, alt={}, center]{18cef198-5ca2-4700-88e9-1a2bd55f841e-2_367_524_1548_849} The diagram shows the graph of \(y = \mathrm { f } ( x )\). The median of \(X\) is denoted by \(m\). Use the diagram to explain whether \(m < 0.5\), \(m = 0.5\) or \(m > 0.5\).
CAIE S2 2012 June Q5
10 marks Standard +0.3
5 A random variable \(X\) has the distribution \(\operatorname { Po } ( 3.2 )\).
  1. A random value of \(X\) is found.
    (a) Find \(\mathrm { P } ( X \geqslant 3 )\).
    (b) Find the probability that \(X = 3\) given that \(X \geqslant 3\).
  2. Random samples of 120 values of \(X\) are taken.
    (a) Describe fully the distribution of the sample mean.
    (b) Find the probability that the mean of a random sample of size 120 is less than 3.3.
CAIE S2 2012 June Q6
11 marks Standard +0.3
6 A survey taken last year showed that the mean number of computers per household in Branley was 1.66 . This year a random sample of 50 households in Branley answered a questionnaire with the following results.
Number of computers01234\(> 4\)
Number of households512181050
  1. Calculate unbiased estimates for the population mean and variance of the number of computers per household in Branley this year.
  2. Test at the \(5 \%\) significance level whether the mean number of computers per household has changed since last year.
  3. Explain whether it is possible that a Type I error may have been made in the test in part (ii).
  4. State what is meant by a Type II error in the context of the test in part (ii), and give the set of values of the test statistic that could lead to a Type II error being made.
CAIE S2 2012 June Q7
11 marks Standard +0.3
7 At work Jerry receives emails randomly at a constant average rate of 15 emails per hour.
  1. Find the probability that Jerry receives more than 2 emails during a 20 -minute period at work.
  2. Jerry's working day is 8 hours long. Find the probability that Jerry receives fewer than 110 emails per day on each of 2 working days.
  3. At work Jerry also receives texts randomly and independently at a constant average rate of 1 text every 10 minutes. Find the probability that the total number of emails and texts that Jerry receives during a 5 -minute period at work is more than 2 and less than 6 .
CAIE S2 2021 November Q1
5 marks Moderate -0.8
1 It is known that the height \(H\), in metres, of trees of a certain kind has the distribution \(\mathrm { N } ( 12.5,10.24 )\). A scientist takes a random sample of 25 trees of this kind and finds the sample mean, \(\bar { H }\), of the heights.
  1. State the distribution of \(\bar { H }\), giving the values of any parameters.
  2. Find \(\mathrm { P } ( 12 < \bar { H } < 13 )\).
CAIE S2 2021 November Q2
4 marks Moderate -0.3
2 The number of enquiries received per day at a customer service desk has a Poisson distribution with mean 45.2. If more than 60 enquiries are received in a day, the customer service desk cannot deal with them all. Use a suitable approximating distribution to find the probability that, on a randomly chosen day, the customer service desk cannot deal with all the enquiries that are received.
CAIE S2 2021 November Q3
5 marks Standard +0.3
3 A random sample of 75 students at a large college was selected for a survey. 15 of these students said that they owned a car. From this result an approximate \(\alpha \%\) confidence interval for the proportion of all students at the college who own a car was calculated. The width of this interval was found to be 0.162 . Calculate the value of \(\alpha\) correct to 2 significant figures.
CAIE S2 2021 November Q4
9 marks Standard +0.3
4 A random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { 1 } { 18 } \left( 9 - x ^ { 2 } \right) & 0 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$
  1. Find \(\mathrm { P } ( X < 1.2 )\).
  2. Find \(\mathrm { E } ( X )\).
    The median of \(X\) is \(m\).
  3. Show that \(m ^ { 3 } - 27 m + 27 = 0\).
CAIE S2 2021 November Q5
9 marks Moderate -0.5
5
  1. The proportion of people having a particular medical condition is 1 in 100000 . A random sample of 2500 people is obtained. The number of people in the sample having the condition is denoted by \(X\).
    1. State, with a justification, a suitable approximating distribution for \(X\), giving the values of any parameters.
    2. Use the approximating distribution to calculate \(\mathrm { P } ( X > 0 )\).
  2. The percentage of people having a different medical condition is thought to be \(30 \%\). A researcher suspects that the true percentage is less than \(30 \%\). In a medical trial a random sample of 28 people was selected and 4 people were found to have this condition. Use a binomial distribution to test the researcher's suspicion at the \(2 \%\) significance level.