Questions S2 (1690 questions)

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CAIE S2 2003 June Q1
4 marks Easy -1.3
1 A fair coin is tossed 5 times and the number of heads is recorded.
  1. The random variable \(X\) is the number of heads. State the mean and variance of \(X\).
  2. The number of heads is doubled and denoted by the random variable \(Y\). State the mean and variance of \(Y\).
CAIE S2 2003 June Q2
5 marks Moderate -0.3
2 Before attending a basketball course, a player found that \(60 \%\) of his shots made a score. After attending the course the player claimed he had improved. In his next game he tried 12 shots and scored in 10 of them. Assuming shots to be independent, test this claim at the \(10 \%\) significance level.
CAIE S2 2003 June Q3
6 marks Moderate -0.8
3 A consumer group, interested in the mean fat content of a particular type of sausage, takes a random sample of 20 sausages and sends them away to be analysed. The percentage of fat in each sausage is as follows. $$\begin{array} { l l l l l l l l l l l l l l l l l l l l } 26 & 27 & 28 & 28 & 28 & 29 & 29 & 30 & 30 & 31 & 32 & 32 & 32 & 33 & 33 & 34 & 34 & 34 & 35 & 35 \end{array}$$ Assume that the percentage of fat is normally distributed with mean \(\mu\), and that the standard deviation is known to be 3 .
  1. Calculate a 98\% confidence interval for the population mean percentage of fat.
  2. The manufacturer claims that the mean percentage of fat in sausages of this type is 30 . Use your answer to part (i) to determine whether the consumer group should accept this claim.
CAIE S2 2003 June Q4
7 marks Moderate -0.8
4 A random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} 1 - \frac { 1 } { 2 } x & 0 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$
  1. Find \(\mathrm { P } ( X > 1.5 )\).
  2. Find the mean of \(X\).
  3. Find the median of \(X\).
CAIE S2 2003 June Q5
8 marks Standard +0.3
5 Over a long period of time it is found that the time spent at cash withdrawal points follows a normal distribution with mean 2.1 minutes and standard deviation 0.9 minutes. A new system is tried out, to speed up the procedure. The null hypothesis is that the mean time spent is the same under the new system as previously. It is decided to reject the null hypothesis and accept that the new system is quicker if the mean withdrawal time from a random sample of 20 cash withdrawals is less than 1.7 minutes. Assume that, for the new system, the standard deviation is still 0.9 minutes, and the time spent still follows a normal distribution.
  1. Calculate the probability of a Type I error.
  2. If the mean withdrawal time under the new system is actually 1.5 minutes, calculate the probability of a Type II error.
CAIE S2 2003 June Q6
10 marks Standard +0.3
6 Computer breakdowns occur randomly on average once every 48 hours of use.
  1. Calculate the probability that there will be fewer than 4 breakdowns in 60 hours of use.
  2. Find the probability that the number of breakdowns in one year (8760 hours) of use is more than 200.
  3. Independently of the computer breaking down, the computer operator receives phone calls randomly on average twice in every 24 -hour period. Find the probability that the total number of phone calls and computer breakdowns in a 60-hour period is exactly 4 .
CAIE S2 2003 June Q7
10 marks Standard +0.8
7 Machine \(A\) fills bags of fertiliser so that their weights follow a normal distribution with mean 20.05 kg and standard deviation 0.15 kg . Machine \(B\) fills bags of fertiliser so that their weights follow a normal distribution with mean 20.05 kg and standard deviation 0.27 kg .
  1. Find the probability that the total weight of a random sample of 20 bags filled by machine \(A\) is at least 2 kg more than the total weight of a random sample of 20 bags filled by machine \(B\). [6]
  2. A random sample of \(n\) bags filled by machine \(A\) is taken. The probability that the sample mean weight of the bags is greater than 20.07 kg is denoted by \(p\). Find the value of \(n\), given that \(p = 0.0250\) correct to 4 decimal places.
CAIE S2 2020 June Q1
3 marks Easy -1.2
1 A random sample of 100 values of a variable \(X\) is taken. These values are summarised below. $$n = 100 \quad \Sigma x = 1556 \quad \Sigma x ^ { 2 } = 29004$$ Calculate unbiased estimates of the population mean and variance of \(X\).
CAIE S2 2020 June Q2
5 marks Standard +0.3
2 Each day at the gym, Sarah completes three runs. The distances, in metres, that she completes in the three runs have the independent distributions \(W \sim \mathrm {~N} ( 1520,450 ) , X \sim \mathrm {~N} ( 2250,720 )\) and \(Y \sim \mathrm {~N} ( 3860,1050 )\). Find the probability that, on a particular day, \(Y\) is less than the total of \(W\) and \(X\).
CAIE S2 2020 June Q3
6 marks Standard +0.8
3 The number of customers who visit a particular shop between 9.00 am and 10.00 am has the distribution \(\operatorname { Po } ( \lambda )\). In the past the value of \(\lambda\) was 5.2. Following some new advertising, the manager wishes to test whether the value of \(\lambda\) has increased. He chooses a random sample of 20 days and finds that the total number of customers who visited the shop between 9.00 am and 10.00 am on those days is 125 . Use an approximating distribution to test at the \(2.5 \%\) significance level whether the value of \(\lambda\) has increased.
CAIE S2 2020 June Q4
8 marks Standard +0.3
4 The random variable \(A\) has the distribution \(\operatorname { Po } ( 1.5 ) . A _ { 1 }\) and \(A _ { 2 }\) are independent values of \(A\).
  1. Find \(\mathrm { P } \left( A _ { 1 } + A _ { 2 } < 2 \right)\).
  2. Given that \(A _ { 1 } + A _ { 2 } < 2\), find \(\mathrm { P } \left( A _ { 1 } = 1 \right)\).
  3. Give a reason why \(A _ { 1 } - A _ { 2 }\) cannot have a Poisson distribution.
CAIE S2 2020 June Q5
10 marks Moderate -0.3
5 Sunita has a six-sided die with faces marked \(1,2,3,4,5,6\). The probability that the die shows a six on any throw is \(p\). Sunita throws the die 500 times and finds that it shows a six 70 times.
  1. Calculate an approximate \(99 \%\) confidence interval for \(p\).
  2. Sunita believes that the die is fair. Use your answer to part (a) to comment on her belief.
  3. Sunita uses the result of her 500 throws to calculate an \(\alpha \%\) confidence interval for \(p\). This interval has width 0.04 . Find the value of \(\alpha\).
CAIE S2 2020 June Q6
9 marks Standard +0.3
6 The length, \(X\) centimetres, of worms of a certain type is modelled by the probability density function $$f ( x ) = \begin{cases} \frac { 6 } { 125 } ( 10 - x ) ( x - 5 ) & 5 \leqslant x \leqslant 10 \\ 0 & \text { otherwise } \end{cases}$$
  1. State the value of \(\mathrm { E } ( X )\).
  2. Find \(\operatorname { Var } ( X )\).
  3. Two worms of this type are chosen at random. Find the probability that exactly one of them has length less than 6 cm .
CAIE S2 2020 June Q7
9 marks Moderate -0.8
7 A market researcher is investigating the length of time that customers spend at an information desk. He plans to choose a sample of 50 customers on a particular day.
  1. He considers choosing the first 50 customers who visit the information desk. Explain why this method is unsuitable.
    The actual lengths of time, in minutes, that customers spend at the information desk may be assumed to have mean \(\mu\) and variance 4.8. The researcher knows that in the past the value of \(\mu\) was 6.0. He wishes to test, at the \(2 \%\) significance level, whether this is still true. He chooses a random sample of 50 customers and notes how long they each spend at the information desk.
  2. State the probability of making a Type I error and explain what is meant by a Type I error in this context.
  3. Given that the mean time spent at the information desk by the 50 customers is 6.8 minutes, carry out the test.
  4. Give a reason why it was necessary to use the Central Limit theorem in your answer to part (c).
    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 2021 June Q1
6 marks Moderate -0.3
1 In a game, a ball is thrown and lands in one of 4 slots, labelled \(A , B , C\) and \(D\). Raju wishes to test whether the probability that the ball will land in slot \(A\) is \(\frac { 1 } { 4 }\).
  1. State suitable null and alternative hypotheses for Raju's test.
    The ball is thrown 100 times and it lands in slot \(A 15\) times.
  2. Use a suitable approximating distribution to carry out the test at the \(2 \%\) significance level.
CAIE S2 2021 June Q2
7 marks Moderate -0.8
2 The random variable \(X\) has the distribution \(\mathrm { B } ( 400,0.01 )\).
  1. Find \(\operatorname { Var } ( 4 X + 2 )\).
    1. State an appropriate approximating distribution for \(X\), giving the values of any parameters. Justify your choice of approximating distribution.
    2. Use your approximating distribution to find \(\mathrm { P } ( 2 \leqslant X \leqslant 5 )\).
CAIE S2 2021 June Q3
7 marks Moderate -0.8
3 \includegraphics[max width=\textwidth, alt={}, center]{9fca48da-82c3-4ce1-9e0c-93eb9a920f9d-05_456_668_260_735} The random variable \(X\) takes values in the range \(1 \leqslant x \leqslant p\), where \(p\) is a constant. The graph of the probability density function of \(X\) is shown in the diagram.
  1. Show that \(p = 2\).
  2. Find \(\mathrm { E } ( X )\).
CAIE S2 2021 June Q4
8 marks Moderate -0.8
4 Wendy's journey to work consists of three parts: walking to the train station, riding on the train and then walking to the office. The times, in minutes, for the three parts of her journey are independent and have the distributions \(\mathrm { N } \left( 15.0,1.1 ^ { 2 } \right) , \mathrm { N } \left( 32.0,3.5 ^ { 2 } \right)\) and \(\mathrm { N } \left( 8.6,1.2 ^ { 2 } \right)\) respectively.
  1. Find the mean and variance of the total time for Wendy's journey.
    If Wendy's journey takes more than 60 minutes, she is late for work.
  2. Find the probability that, on a randomly chosen day, Wendy will be late for work.
  3. Find the probability that the mean of Wendy's journey times over 15 randomly chosen days will be less than 54.5 minutes.
CAIE S2 2021 June Q5
6 marks Standard +0.3
5 The time, in minutes, spent by customers at a particular gym has the distribution \(\mathrm { N } ( \mu , 38.2 )\). In the past the value of \(\mu\) has been 42.4. Following the installation of some new equipment the management wishes to test whether the value of \(\mu\) has changed.
  1. State what is meant by a Type I error in this context.
  2. The mean time for a sample of 20 customers is found to be 45.6 minutes. Test at the \(2.5 \%\) significance level whether the value of \(\mu\) has changed.
CAIE S2 2021 June Q6
8 marks Moderate -0.8
6 The heights, \(h\) centimetres, of a random sample of 100 fully grown animals of a certain species were measured. The results are summarised below. $$n = 100 \quad \Sigma h = 7570 \quad \Sigma h ^ { 2 } = 588050$$
  1. Find unbiased estimates of the population mean and variance.
  2. Calculate a \(99 \%\) confidence interval for the mean height of animals of this species.
    Four random samples were taken and a \(99 \%\) confidence interval for the population mean, \(\mu\), was found from each sample.
  3. Find the probability that all four of these confidence intervals contain the true value of \(\mu\).
CAIE S2 2021 June Q7
8 marks Moderate -0.3
7 Customers arrive at a particular shop at random times. It has been found that the mean number of customers who arrive during a 5 -minute interval is 2.1 .
  1. Find the probability that exactly 4 customers arrive during a 10 -minute interval.
  2. Find the probability that at least 4 customers arrive during a 20 -minute interval.
  3. Use a suitable approximating distribution to find the probability that fewer than 40 customers arrive during a 2-hour interval.
    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 2021 June Q1
5 marks Moderate -0.3
1 The number of goals scored by a team in a match is independent of other matches, and is denoted by the random variable \(X\), which has a Poisson distribution with mean 1.36. A supporter offers to make a donation of \(\\) 5$ to the team for each goal that they score in the next 10 matches. Find the expectation and standard deviation of the amount that the supporter will pay.
CAIE S2 2021 June Q2
8 marks Standard +0.3
2 In the past, the time, in hours, for a particular train journey has had mean 1.40 and standard deviation 0.12 . Following the introduction of some new signals, it is required to test whether the mean journey time has decreased.
  1. State what is meant by a Type II error in this context.
  2. The mean time for a random sample of 50 journeys is found to be 1.36 hours. Assuming that the standard deviation of journey times is still 0.12 hours, test at the \(2.5 \%\) significance level whether the population mean journey time has decreased.
  3. State, with a reason, which of the errors, Type I or Type II, might have been made in the test in part (b).
CAIE S2 2021 June Q3
6 marks Standard +0.3
3 The local council claims that the average number of accidents per year on a particular road is 0.8 . Jane claims that the true average is greater than 0.8 . She looks at the records for a random sample of 3 recent years and finds that the total number of accidents during those 3 years was 5 .
  1. Assume that the number of accidents per year follows a Poisson distribution.
    1. State null and alternative hypotheses for a test of Jane's claim.
    2. Test at the \(5 \%\) significance level whether Jane's claim is justified.
  2. Jane finds that the number of accidents per year has been gradually increasing over recent years. State how this might affect the validity of the test carried out in part (a)(ii).
CAIE S2 2021 June Q4
9 marks Standard +0.3
4 The masses, \(m\) kilograms, of flour in a random sample of 90 sacks of flour are summarised as follows. $$n = 90 \quad \Sigma m = 4509 \quad \Sigma m ^ { 2 } = 225950$$
  1. Find unbiased estimates of the population mean and variance.
  2. Calculate a \(98 \%\) confidence interval for the population mean.
  3. Explain why it was necessary to use the Central Limit theorem in answering part (b).
  4. Find the probability that the confidence interval found in part (b) is wholly above the true value of the population mean.