Questions S2 (1597 questions)

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CAIE S2 2023 March Q1
1 Anita carried out a survey of 140 randomly selected students at her college. She found that 49 of these students watched a TV programme called Bunch.
  1. Calculate an approximate \(98 \%\) confidence interval for the proportion, \(p\), of students at Anita's college who watch Bunch.
    Carlos says that the confidence interval found in (a) is not useful because it is too wide.
  2. Without calculation, explain briefly how Carlos can use the results of Anita's survey to find a narrower confidence interval for \(p\).
CAIE S2 2023 March Q2
4 marks
2 The number of orders arriving at a shop during an 8-hour working day is modelled by the random variable \(X\) with distribution \(\operatorname { Po } ( 25.2 )\).
  1. State two assumptions that are required for the Poisson model to be valid in this context.
    1. Find the probability that the number of orders that arrive in a randomly chosen 3-hour period is between 3 and 5 inclusive.
    2. Find the probability that, in two randomly chosen 1 -hour periods, exactly 1 order will arrive in one of the 1 -hour periods, and at least 2 orders will arrive in the other 1 -hour period. [4]
  3. The shop can only deal with a maximum of 120 orders during any 36-hour period. Use a suitable approximating distribution to find the probability that, in a randomly chosen 36-hour period, there will be too many orders for the shop to deal with.
CAIE S2 2023 March Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{823cc2e5-e408-4b81-ac4d-7e9f584107cc-06_558_1077_260_523} The diagram shows the graph of the probability density function, f, of a random variable \(X\) that takes values between \(x = 0\) and \(x = 3\) only. The graph is symmetrical about the line \(x = 1.5\).
  1. It is given that \(\mathrm { P } ( X < 0.6 ) = a\) and \(\mathrm { P } ( 0.6 < X < 1.2 ) = b\). Find \(\mathrm { P } ( 0.6 < X < 1.8 )\) in terms of \(a\) and \(b\).
  2. It is now given that the equation of the probability density function of \(X\) is $$f ( x ) = \begin{cases} k x ^ { 2 } ( 3 - x ) ^ { 2 } & 0 \leqslant x \leqslant 3
    0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
    1. Show that \(k = \frac { 10 } { 81 }\).
    2. Find \(\operatorname { Var } ( X )\).
CAIE S2 2023 March Q4
4 The number of accidents per 3-month period on a certain road has the distribution \(\operatorname { Po } ( \lambda )\). In the past the value of \(\lambda\) has been 5.7. Following some changes to the road, the council carries out a hypothesis test to determine whether the value of \(\lambda\) has decreased. If there are fewer than 3 accidents in a randomly chosen 3 -month period, the council will conclude that the value of \(\lambda\) has decreased.
  1. Find the probability of a Type I error.
  2. Find the probability of a Type II error if the mean number of accidents per 3-month period is now actually 0.9 .
CAIE S2 2023 March Q5
5 The masses, in grams, of large and small packets of Maxwheat cereal have the independent distributions \(\mathrm { N } \left( 410.0,3.6 ^ { 2 } \right)\) and \(\mathrm { N } \left( 206.0,3.7 ^ { 2 } \right)\) respectively.
  1. Find the probability that a randomly chosen large packet has a mass that is more than double the mass of a randomly chosen small packet.
    The packets are placed in boxes. The boxes are identical in appearance. \(60 \%\) of the boxes contain exactly 10 randomly chosen large packets. 40\% of the boxes contain exactly 20 randomly chosen small packets.
  2. Find the probability that a randomly chosen box contains packets with a total mass of more than 4080 grams.
CAIE S2 2023 March Q6
6 Last year, the mean time taken by students at a school to complete a certain test was 25 minutes. Akash believes that the mean time taken by this year's students was less than 25 minutes. In order to test this belief, he takes a large random sample of this year's students and he notes the time taken by each student. He carries out a test, at the \(2.5 \%\) significance level, for the population mean time, \(\mu\) minutes. Akash uses the null hypothesis \(\mathrm { H } _ { 0 } : \mu = 25\).
  1. Give a reason why Akash should use a one-tailed test.
    Akash finds that the value of the test statistic is \(z = - 2.02\).
  2. Explain what conclusion he should draw.
    In a different one-tailed hypothesis test the \(z\)-value was found to be 2.14 .
  3. Given that this value would lead to a rejection of the null hypothesis at the \(\alpha \%\) significance level, find the set of possible values of \(\alpha\).
    The population mean time taken by students at another school to complete a test last year was \(m\) minutes. Sorin carries out a one-tailed test to determine whether the population mean this year is less than \(m\), using a random sample of 100 students. He assumes that the population standard deviation of the times is 3.9 minutes. The sample mean is 24.8 minutes, and this result just leads to the rejection of the null hypothesis at the 5\% significance level.
  4. Find the value of \(m\).
    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 2020 June Q1
1 The lengths, \(X\) centimetres, of a random sample of 7 leaves from a certain variety of tree are as follows.
3.9
4.8
4.8
4.4
5.2
5.5
6.1
  1. Calculate unbiased estimates of the population mean and variance of \(X\).
    It is now given that the true value of the population variance of \(X\) is 0.55 , and that \(X\) has a normal distribution.
  2. Find a 95\% confidence interval for the population mean of \(X\).
CAIE S2 2020 June Q2
2 In the past the yield of a certain crop, in tonnes per hectare, had mean 0.56 and standard deviation 0.08 Following the introduction of a new fertilizer, the farmer intends to test at the \(2.5 \%\) significance level whether the mean yield has increased. He finds that the mean yield over 10 years is 0.61 tonnes per hectare.
  1. State two assumptions that are necessary for the test.
  2. Carry out the test.
CAIE S2 2020 June Q3
3 The masses, in kilograms, of large sacks of flour and small sacks of flour have the independent distributions \(\mathrm { N } \left( 40,1.5 ^ { 2 } \right)\) and \(\mathrm { N } \left( 12,0.7 ^ { 2 } \right)\) respectively.
  1. Find the probability that the total mass of 6 randomly chosen large sacks of flour is more than 245 kg .
  2. Find the probability that the mass of a randomly chosen large sack of flour is less than 4 times the mass of a randomly chosen small sack of flour.
CAIE S2 2020 June Q4
4 A fair spinner has five sides numbered \(1,2,3,4,5\). The score on one spin is denoted by \(X\).
  1. Show that \(\operatorname { Var } ( X ) = 2\).
    Fiona has another spinner, also with five sides numbered \(1,2,3,4,5\). She suspects that it is biased so that the expected score is less than 3 . In order to test her suspicion, she plans to spin her spinner 40 times. If the mean score is less than 2.6 she will conclude that her spinner is biased in this way.
  2. Find the probability of a Type I error.
  3. State what is meant by a Type II error in this context.
CAIE S2 2020 June Q5
5 Each week a sports team plays one home match and one away match. In their home matches they score goals at a constant average rate of 2.1 goals per match. In their away matches they score goals at a constant average rate of 0.8 goals per match. You may assume that goals are scored at random times and independently of one another.
  1. A week is chosen at random.
    1. Find the probability that the team scores a total of 4 goals in their two matches.
    2. Find the probability that the team scores a total of 4 goals, with more goals scored in the home match than in the away match.
  2. Use a suitable approximating distribution to find the probability that the team scores fewer than 25 goals in 10 randomly chosen weeks.
  3. Justify the use of the approximating distribution used in part (b).
CAIE S2 2020 June Q6
6 The length of time, \(T\) minutes, that a passenger has to wait for a bus at a certain bus stop is modelled by the probability density function given by $$\mathrm { f } ( t ) = \begin{cases} \frac { 3 } { 4000 } \left( 20 t - t ^ { 2 } \right) & 0 \leqslant t \leqslant 20
0 & \text { otherwise } \end{cases}$$
  1. Sketch the graph of \(y = \mathrm { f } ( t )\).
  2. Hence explain, without calculation, why \(\mathrm { E } ( T ) = 10\).
  3. Find \(\operatorname { Var } ( T )\).
  4. It is given that \(\mathrm { P } ( T < 10 + a ) = p\), where \(0 < a < 10\). Find \(\mathrm { P } ( 10 - a < T < 10 + a )\) in terms of \(p\).
  5. Find \(\mathrm { P } ( 8 < T < 12 )\).
  6. Give one reason why this model may be unrealistic.
    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 2020 June Q1
1 The masses, in grams, of plums of a certain type have the distribution \(\mathrm { N } \left( 40.4,5.2 ^ { 2 } \right)\). The plums are packed in bags, with each bag containing 6 randomly chosen plums. If the total weight of the plums in a bag is less than 220 g the bag is rejected. Find the percentage of bags that are rejected.
CAIE S2 2020 June Q2
2 A shop obtains apples from a certain farm. It has been found that 5\% of apples from this farm are Grade A. Following a change in growing conditions at the farm, the shop management plan to carry out a hypothesis test to find out whether the proportion of Grade A apples has increased. They select 25 apples at random. If the number of Grade A apples is more than 3 they will conclude that the proportion has increased.
  1. State suitable null and alternative hypotheses for the test.
  2. Find the probability of a Type I error.
    In fact 2 of the 25 apples were Grade A .
  3. Which of the errors, Type I or Type II, is possible? Justify your answer.
CAIE S2 2020 June Q3
3 In the data-entry department of a certain firm, it is known that \(0.12 \%\) of data items are entered incorrectly, and that these errors occur randomly and independently.
  1. A random sample of 3600 data items is chosen. The number of these data items that are incorrectly entered is denoted by \(X\).
    1. State the distribution of \(X\), including the values of any parameters.
    2. State an appropriate approximating distribution for \(X\), including the values of any parameters. Justify your choice of approximating distribution.
    3. Use your approximating distribution to find \(\mathrm { P } ( X > 2 )\).
  2. Another large random sample of \(n\) data items is chosen. The probability that the sample contains no data items that are entered incorrectly is more than 0.1 . Use an approximating distribution to find the largest possible value of \(n\).
CAIE S2 2020 June Q4
4 The score on one spin of a 5 -sided spinner is denoted by the random variable \(X\) with probability distribution as shown in the table.
\(x\)01234
\(\mathrm { P } ( X = x )\)0.10.20.40.20.1
  1. Show that \(\operatorname { Var } ( X ) = 1.2\).
    The spinner is spun 200 times. The score on each spin is noted and the mean, \(\bar { X }\), of the 200 scores is found.
  2. Given that \(\mathrm { P } ( \bar { X } > a ) = 0.1\), find the value of \(a\).
  3. Explain whether it was necessary to use the Central Limit theorem in your answer to part (b).
  4. Johann has another, similar, spinner. He suspects that it is biased so that the mean score is less than 2 . He spins his spinner 200 times and finds that the mean of the 200 scores is 1.86 . Given that the variance of the score on one spin of this spinner is also 1.2 , test Johann's suspicion at the 5\% significance level.
CAIE S2 2020 June Q5
5
  1. The random variable \(X\) has the distribution \(\operatorname { Po } ( \lambda )\).
    1. State the values that \(X\) can take.
      It is given that \(\mathrm { P } ( X = 1 ) = 3 \times \mathrm { P } ( X = 0 )\).
    2. Find \(\lambda\).
    3. Find \(\mathrm { P } ( 4 \leqslant X \leqslant 6 )\).
  2. The random variable \(Y\) has the distribution \(\operatorname { Po } ( \mu )\) where \(\mu\) is large. Using a suitable approximating distribution, it is found that \(\mathrm { P } ( Y < 46 ) = 0.0668\), correct to 4 decimal places. Find \(\mu\).
CAIE S2 2020 June Q6
6 A random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { k } { x ^ { 2 } } & 1 \leqslant x \leqslant a
0 & \text { otherwise } \end{cases}$$ where \(k\) and \(a\) are positive constants.
  1. Show that \(k = \frac { a } { a - 1 }\).
  2. Find \(\mathrm { E } ( X )\) in terms of \(a\).
  3. Find the 60th percentile of \(X\) in terms of \(a\).
    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 2002 June Q1
1 The result of a fitness trial is a random variable \(X\) which is normally distributed with mean \(\mu\) and standard deviation 2.4. A researcher uses the results from a random sample of 90 trials to calculate a \(98 \%\) confidence interval for \(\mu\). What is the width of this interval?
CAIE S2 2002 June Q2
2 The manager of a video hire shop wishes to estimate the proportion of videos damaged by customers. He takes a random sample of 120 videos and finds that 33 of them are damaged. Find a \(95 \%\) confidence interval for the true proportion of videos that are being damaged when hired from this shop.
CAIE S2 2002 June Q3
3 Mary buys 3 packets of sugar and 5 packets of coffee and puts them in her shopping basket, together with her purse which weighs 350 g . Weights of packets of sugar are normally distributed with mean 500 g and standard deviation 20 g . Weights of packets of coffee are normally distributed with mean 200 g and standard deviation 12 g . Find the probability that the total weight in the shopping basket is less than 2900 g .
CAIE S2 2002 June Q4
4 The mean time to mark a certain set of examination papers is estimated by the examination board to be 12 minutes per paper. A random sample of 150 examination papers gave \(\Sigma x = 2130\) and \(\Sigma x ^ { 2 } = 37746\), where \(x\) is the time in minutes to mark an examination paper.
  1. Calculate unbiased estimates of the population mean and variance.
  2. Stating the null and alternative hypotheses, use a \(10 \%\) significance level to test whether the examination board's estimated time is consistent with the data.
CAIE S2 2002 June Q5
5 To test whether a coin is biased or not, it is tossed 10 times. The coin will be considered biased if there are 9 or 10 heads, or 9 or 10 tails.
  1. Show that the probability of making a Type I error in this test is approximately 0.0215 .
  2. Find the probability of making a Type II error in this test when the probability of a head is actually 0.7.
CAIE S2 2002 June Q6
6 Between 7 p.m. and 11 p.m., arrivals of patients at the casualty department of a hospital occur at random at an average rate of 6 per hour.
  1. Find the probability that, during any period of one hour between 7 p.m. and 11 p.m., exactly 5 people will arrive.
  2. A patient arrives at exactly 10.15 p.m. Find the probability that at least one more patient arrives before 10.35 p.m.
  3. Use a suitable approximation to estimate the probability that fewer than 20 patients arrive at the casualty department between 7 p.m. and 11 p.m. on any particular night.
CAIE S2 2002 June Q7
7 A factory is supplied with grain at the beginning of each week. The weekly demand, \(X\) thousand tonnes, for grain from this factory is a continuous random variable having the probability density function given by $$f ( x ) = \begin{cases} 2 ( 1 - x ) & 0 \leqslant x \leqslant 1
0 & \text { otherwise } \end{cases}$$ Find
  1. the mean value of \(X\),
  2. the variance of \(X\),
  3. the quantity of grain in tonnes that the factory should have in stock at the beginning of a week, in order to be \(98 \%\) certain that the demand in that week will be met.