Questions S2 (1597 questions)

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CAIE S2 2003 June Q1
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
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
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
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
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
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
6 marks
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
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
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
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
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
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
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
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
1 Accidents at two factories occur randomly and independently. On average, the numbers of accidents per month are 3.1 at factory \(A\) and 1.7 at factory \(B\). Find the probability that the total number of accidents in the two factories during a \(2 -\) month period is more than 3 .
CAIE S2 2021 June Q2
2 The time, in minutes, taken by students to complete a test has the distribution \(\mathrm { N } ( 125,36 )\).
  1. Find the probability that the mean time taken to complete the test by a random sample of 40 students is less than 123 minutes.
  2. Explain whether it was necessary to use the Central Limit theorem in the solution to part (a).
CAIE S2 2021 June Q3
3 The graph of the probability density function of a random variable \(X\) is symmetrical about the line \(x = 4\). Given that \(\mathrm { P } ( X < 5 ) = \frac { 20 } { 27 }\), find \(\mathrm { P } ( 3 < X < 5 )\).
4100 randomly chosen adults each throw a ball once. The length, \(l\) metres, of each throw is recorded. The results are summarised below. $$n = 100 \quad \Sigma l = 3820 \quad \Sigma l ^ { 2 } = 182200$$ Calculate a \(94 \%\) confidence interval for the population mean length of throws by adults.
CAIE S2 2021 June Q5
5 On average, 1 in 75000 adults has a certain genetic disorder.
  1. Use a suitable approximating distribution to find the probability that, in a random sample of 10000 people, at least 1 has the genetic disorder.
  2. In a random sample of \(n\) people, where \(n\) is large, the probability that no-one has the genetic disorder is more than 0.9 . Find the largest possible value of \(n\).
CAIE S2 2021 June Q6
6 The probability density function, f , of a random variable \(X\) is given by $$\mathrm { f } ( x ) = \begin{cases} k \left( 6 x - x ^ { 2 } \right) & 0 \leqslant x \leqslant 6
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
State the value of \(\mathrm { E } ( X )\) and show that \(\operatorname { Var } ( X ) = \frac { 9 } { 5 }\).
CAIE S2 2021 June Q7
7 The masses, in kilograms, of large and small sacks of flour have the distributions \(\mathrm { N } \left( 55,3 ^ { 2 } \right)\) and \(\mathrm { N } \left( 27,2.5 ^ { 2 } \right)\) respectively.
  1. Some sacks are loaded onto a boat. The maximum load of flour that the boat can carry safely is 340 kg . Find the probability that the boat can carry safely 3 randomly chosen large sacks of flour and 6 randomly chosen small sacks of flour.
  2. Find the probability that the mass of a randomly chosen large sack of flour is greater than the total mass of two randomly chosen small sacks of flour.
CAIE S2 2021 June Q8
8 At a certain large school it was found that the proportion of students not wearing correct uniform was 0.15 . The school sent a letter to parents asking them to ensure that their children wear the correct uniform. The school now wishes to test whether the proportion not wearing correct uniform has been reduced.
  1. It is suggested that a random sample of the students in Grade 12 should be used for the test. Give a reason why this would not be an appropriate sample.
    A suitable sample of 50 students is selected and the number not wearing correct uniform is noted. This figure is used to carry out a test at the 5\% significance level.
  2. State suitable null and alternative hypotheses.
  3. Use a binomial distribution to find the probability of a Type I error. You must justify your answer fully.
  4. In fact 4 students out of the 50 are not wearing correct uniform. State the conclusion of the test, explaining your answer.
  5. State, with a reason, which of the errors, Type I or Type II, may have been made.
    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
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
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
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
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.