Questions S2 (1690 questions)

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CAIE S2 2024 November Q5
11 marks Challenging +1.2
5 A machine puts sweets into bags at random. The numbers of lemon and orange sweets in a bag have the independent distributions \(\operatorname { Po } ( 3.7 )\) and \(\operatorname { Po } ( 2.6 )\) respectively. A bag of sweets is chosen at random.
  1. Find the probability that the number of lemon sweets in the bag is more than 2 but not more than 5 .
  2. Find the probability that the total number of lemon and orange sweets in the bag is less than 4 . \includegraphics[max width=\textwidth, alt={}, center]{9ac74d4c-f5e0-4c5d-ab25-5692dfb06f0b-06_2725_47_107_2002} \includegraphics[max width=\textwidth, alt={}, center]{9ac74d4c-f5e0-4c5d-ab25-5692dfb06f0b-07_2716_29_107_22} 10 bags of sweets are chosen at random.
  3. Use approximating distributions to find the probability that the total number of lemon sweets in the 10 bags is less than the total number of orange sweets in the 10 bags.
CAIE S2 2024 November Q6
11 marks Standard +0.3
6 The time, \(X\) hours, taken by a large number of people to complete a challenge is modelled by the probability density function given by $$f ( x ) = \left\{ \begin{array} { c l } \frac { 1 } { x ^ { 2 } } & a \leqslant x \leqslant b \\ 0 & \text { otherwise } \end{array} \right.$$ where \(a\) and \(b\) are constants.
  1. State what the constants \(a\) and \(b\) represent in this context.
  2. Show that \(a = \frac { b } { b + 1 }\).
    It is given that \(\mathrm { E } ( X ) = \ln 3\).
  3. Show that \(b = 2\) and find the value of \(a\). \includegraphics[max width=\textwidth, alt={}, center]{9ac74d4c-f5e0-4c5d-ab25-5692dfb06f0b-09_2726_35_97_20}
  4. Find the median of \(X\).
CAIE S2 2024 November Q7
9 marks Moderate -0.3
7 The heights of one-year-old trees of a certain variety are known to have mean 2.3 m . A scientist believes that, on average, trees of this age and variety in her region are slightly taller than in other places. She plans to carry out a hypothesis test, at the \(2 \%\) significance level, in order to test her belief.
  1. State the probability that she will make a Type I error.
    She takes a random sample of 100 such trees in her region and measures their heights, \(h \mathrm {~m}\). Her results are summarised below. $$n = 100 \quad \sum h = 238 \quad \sum h ^ { 2 } = 580$$
  2. Carry out the test at the \(2 \%\) significance level. \includegraphics[max width=\textwidth, alt={}, center]{9ac74d4c-f5e0-4c5d-ab25-5692dfb06f0b-10_2717_35_109_2012}
  3. The scientist carries out the test correctly, but another scientist claims that she has made a Type II error. Comment on this claim.
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE S2 2020 Specimen Q1
4 marks Moderate -0.5
1 Leaves from a certain type of tree have lengths that are distributed with standard deviation 3.2 cm . A random sample of 250 of these leaves is taken and the mean length of this sample is found to be 12.5 cm .
  1. Calculate a 99\% confidence interval for the population mean length.
  2. Write down the probability that the whole of a \(99 \%\) confidence interval will lie below the population mean.
CAIE S2 2020 Specimen Q2
3 marks Easy -1.8
2 Describe briefly how to use random numbers to choose a sample of 10 students from a year-group of 276 students.
CAIE S2 2020 Specimen Q3
10 marks Moderate -0.5
3 The number of calls received at a small call centre has a Poisson distribution with mean 2.4 calls per 5-minute period.
  1. Find the probability of exactly 4 calls in an 8 -minute period.
  2. Find the probability of at least 3 calls in a 3-minute period.
    The number of calls received at a large call centre has a Poisson distribution with mean 41 calls per 5-minute period.
  3. Use an approximating distribution to find the probability that the number of calls received in a 5 -minute period is between 41 and 59 inclusive.
CAIE S2 2020 Specimen Q4
10 marks Standard +0.8
4 The lifetimes, in hours, of Longlive light bulbs and Enerlow light bulbs have the independent distributions \(\mathrm { N } \left( 1020,45 ^ { 2 } \right)\) and \(\mathrm { N } \left( 2800,52 ^ { 2 } \right)\) respectively.
  1. Find the probability that the total of the lifetimes of five randomly chosen Longlive bulbs is less than 5200 hours.
  2. Find the probability that the lifetime of a randomly chosen Enerlow bulb is at least three times that of a randomly chosen Longlive bulb.
CAIE S2 2020 Specimen Q5
7 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{43403c12-93e6-44e4-b15e-e3c4363be5f9-08_254_634_260_717} The diagram shows the graph of the probability density function, f , of a random variable \(X\), where $$f ( x ) = \begin{cases} \frac { 2 } { 9 } \left( 3 x - x ^ { 2 } \right) & 0 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$
  1. State the value of \(\mathrm { E } ( X )\) and find \(\operatorname { Var } ( X )\).
  2. State the value of \(\mathrm { P } ( 1.5 \leqslant X \leqslant 4 )\).
  3. Given that \(\mathrm { P } ( 1 \leqslant X \leqslant 2 ) = \frac { 13 } { 27 }\), find \(\mathrm { P } ( X > 2 )\).
CAIE S2 2020 Specimen Q6
9 marks Standard +0.3
6 At a certain hospital it was found that the probability that a patient did not arrive for an appointment was 0.2 . The hospital carries out some publicity in the hope that this probability will be reduced. They wish to test whether the publicity has worked. A random sample of 30 appointments is selected and the number of patients that do not arrive is noted. This figure is used to carry out a test at the \(5 \%\) significance level.
  1. Explain why the test is one-tailed and state suitable null and alternative hypotheses.
  2. Use a binomial distribution to find the critical region, and find the probability of a Type I error.
  3. In fact 3 patients out of the 30 do not arrive. State the conclusion of the test, explaining your answer.
CAIE S2 2020 Specimen Q7
7 marks Standard +0.3
7 The mean weight of bags of carrots is \(\mu\) kilograms. An inspector wishes to test whether \(\mu = 2.0\). He weighs a random sample of 200 bags and his results are summarised as follows. $$\Sigma x = 430 \quad \Sigma x ^ { 2 } = 1290$$ Carry out the test at the 10\% significance level.
CAIE S2 2004 June Q1
5 marks Moderate -0.3
1 Each multiple choice question in a test has 4 suggested answers, exactly one of which is correct. Rehka knows nothing about the subject of the test, but claims that she has a special method for answering the questions that is better than just guessing. There are 60 questions in the test, and Rehka gets 22 correct.
  1. State null and alternative hypotheses for a test of Rehka's claim.
  2. Using a normal approximation, test at the \(5 \%\) significance level whether Rehka's claim is justified.
CAIE S2 2004 June Q2
6 marks Moderate -0.3
2 In athletics matches the triple jump event consists of a hop, followed by a step, followed by a jump. The lengths covered by Albert in each part are independent normal variables with means \(3.5 \mathrm {~m} , 2.9 \mathrm {~m}\), 3.1 m and standard deviations \(0.3 \mathrm {~m} , 0.25 \mathrm {~m} , 0.35 \mathrm {~m}\) respectively. The length of the triple jump is the sum of the three parts.
  1. Find the mean and standard deviation of the length of Albert's triple jumps.
  2. Find the probability that the mean of Albert's next four triple jumps is greater than 9 m .
CAIE S2 2004 June Q3
6 marks Moderate -0.8
3 The independent random variables \(X\) and \(Y\) are such that \(X\) has mean 8 and variance 4.8 and \(Y\) has a Poisson distribution with mean 6. Find
  1. \(\mathrm { E } ( 2 X - 3 Y )\),
  2. \(\operatorname { Var } ( 2 X - 3 Y )\).
CAIE S2 2004 June Q4
7 marks Moderate -0.8
4 Packets of cat food are filled by a machine.
  1. In a random sample of 10 packets, the weights, in grams, of the packets were as follows. \(\begin{array} { l l l l l l l l l l } 374.6 & 377.4 & 376.1 & 379.2 & 371.2 & 375.0 & 372.4 & 378.6 & 377.1 & 371.5 \end{array}\) Find unbiased estimates of the population mean and variance.
  2. In a random sample of 200 packets, 38 were found to be underweight. Calculate a \(96 \%\) confidence interval for the population proportion of underweight packets.
CAIE S2 2004 June Q5
8 marks Standard +0.3
5 The lectures in a mathematics department are scheduled to last 54 minutes, and the times of individual lectures may be assumed to have a normal distribution with mean \(\mu\) minutes and standard deviation 3.1 minutes. One of the students commented that, on average, the lectures seemed too short. To investigate this, the times for a random sample of 10 lectures were used to test the null hypothesis \(\mu = 54\) against the alternative hypothesis \(\mu < 54\) at the \(10 \%\) significance level.
  1. Show that the null hypothesis is rejected in favour of the alternative hypothesis if \(\bar { x } < 52.74\), where \(\bar { x }\) minutes is the sample mean.
  2. Find the probability of a Type II error given that the actual mean length of lectures is 51.5 minutes.
CAIE S2 2004 June Q6
8 marks Standard +0.8
6 At a certain airfield planes land at random times at a constant average rate of one every 10 minutes.
  1. Find the probability that exactly 5 planes will land in a period of one hour.
  2. Find the probability that at least 2 planes will land in a period of 16 minutes.
  3. Given that 5 planes landed in an hour, calculate the conditional probability that 1 plane landed in the first half hour and 4 in the second half hour.
CAIE S2 2004 June Q7
10 marks Standard +0.3
7 The queuing time, \(T\) minutes, for a person queuing at a supermarket checkout has probability density function given by $$f ( t ) = \begin{cases} c t \left( 25 - t ^ { 2 } \right) & 0 \leqslant t \leqslant 5 \\ 0 & \text { otherwise } \end{cases}$$ where \(c\) is a constant.
  1. Show that the value of \(c\) is \(\frac { 4 } { 625 }\).
  2. Find the probability that a person will have to queue for between 2 and 4 minutes.
  3. Find the mean queuing time.
CAIE S2 2005 June Q1
4 marks Moderate -0.8
1 Exam marks, \(X\), have mean 70 and standard deviation 8.7. The marks need to be scaled using the formula \(Y = a X + b\) so that the scaled marks, \(Y\), have mean 55 and standard deviation 6.96. Find the values of \(a\) and \(b\).
CAIE S2 2005 June Q2
6 marks Easy -1.3
2 Jenny has to do a statistics project at school on how much pocket money, in dollars, is received by students in her year group. She plans to take a sample of 7 students from her year group, which contains 122 students.
  1. Give a suitable method of taking this sample. Her sample gives the following results. $$\begin{array} { l l l l l l l } 13.40 & 10.60 & 26.50 & 20.00 & 14.50 & 15.00 & 16.50 \end{array}$$
  2. Find unbiased estimates of the population mean and variance.
  3. Is the estimated population variance more than, less than or the same as the sample variance?
  4. Describe what you understand by 'population' in this question.
CAIE S2 2005 June Q3
7 marks Standard +0.3
3 A survey of a random sample of \(n\) people found that 61 of them read The Reporter newspaper. A symmetric confidence interval for the true population proportion, \(p\), who read The Reporter is \(0.1993 < p < 0.2887\).
  1. Find the mid-point of this confidence interval and use this to find the value of \(n\).
  2. Find the confidence level of this confidence interval.
CAIE S2 2005 June Q4
7 marks Standard +0.3
4 A study of a large sample of books by a particular author shows that the number of words per sentence can be modelled by a normal distribution with mean 21.2 and standard deviation 7.3. A researcher claims to have discovered a previously unknown book by this author. The mean length of 90 sentences chosen at random in this book is found to be 19.4 words.
  1. Assuming the population standard deviation of sentence lengths in this book is also 7.3, test at the \(5 \%\) level of significance whether the mean sentence length is the same as the author's. State your null and alternative hypotheses.
  2. State in words relating to the context of the test what is meant by a Type I error and state the probability of a Type I error in the test in part (i).
CAIE S2 2005 June Q5
7 marks Moderate -0.3
5 A clock contains 4 new batteries each of which gives a voltage which is normally distributed with mean 1.54 volts and standard deviation 0.05 volts. The voltages of the batteries are independent. The clock will only work if the total voltage is greater than 5.95 volts.
  1. Find the probability that the clock will work.
  2. Find the probability that the average total voltage of the batteries of 20 clocks chosen at random exceeds 6.2 volts.
CAIE S2 2005 June Q6
9 marks Standard +0.3
6 At a petrol station cars arrive independently and at random times at constant average rates of 8 cars per hour travelling east and 5 cars per hour travelling west.
  1. Find the probability that, in a quarter-hour period,
    1. one or more cars travelling east and one or more cars travelling west will arrive,
    2. a total of 2 or more cars will arrive.
    3. Find the approximate probability that, in a 12 -hour period, a total of more than 175 cars will arrive.
CAIE S2 2005 June Q7
10 marks Standard +0.3
7 The random variable \(X\) denotes the number of hours of cloud cover per day at a weather forecasting centre. The probability density function of \(X\) is given by $$f ( x ) = \begin{cases} \frac { ( x - 18 ) ^ { 2 } } { k } & 0 \leqslant x \leqslant 24 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = 2016\).
  2. On how many days in a year of 365 days can the centre expect to have less than 2 hours of cloud cover?
  3. Find the mean number of hours of cloud cover per day.
CAIE S2 2006 June Q1
3 marks Moderate -0.8
1 Packets of fish food have weights that are distributed with standard deviation 2.3 g . A random sample of 200 packets is taken. The mean weight of this sample is found to be 99.2 g . Calculate a \(99 \%\) confidence interval for the population mean weight.