Questions S2 (1597 questions)

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CAIE S2 2023 November Q1
1
  1. A random variable \(X\) has the distribution \(\operatorname { Po } ( 25 )\).
    Use the normal approximation to the Poisson distribution to find \(\mathrm { P } ( X > 30 )\).
  2. A random variable \(Y\) has the distribution \(\mathrm { B } ( 100 , p )\) where \(p < 0.05\). Use the Poisson approximation to the binomial distribution to write down an expression, in terms of \(p\), for \(\mathrm { P } ( Y < 3 )\).
CAIE S2 2023 November Q2
2 The length, in minutes, of mathematics lectures at a certain college has mean \(\mu\) and standard deviation 8.3.
  1. The total length of a random sample of 85 lectures was 4590 minutes. Calculate a 95\% confidence interval for \(\mu\).
    The length, in minutes, of history lectures at the college has mean \(m\) and standard deviation \(s\).
  2. Using a random sample of 100 history lectures, a 95\% confidence interval for \(m\) was found to have width 2.8 minutes. Find the value of \(s\).
CAIE S2 2023 November Q3
3 A researcher read a magazine article which stated that boys aged 1 to 3 prefer green to orange. It claimed that, when offered a green cube and an orange cube to play with, a boy is more likely to choose the green one. The researcher disagrees with this claim. She believes that boys of this age are equally likely to choose either colour. In order to test her belief, the researcher carried out a hypothesis test at the 5\% significance level. She offered a green cube and an orange cube to each of 10 randomly chosen boys aged 1 to 3 , and recorded the number, \(X\), of boys who chose the green cube. Out of the 10 boys, 8 boys chose the green cube.
    1. Assuming that the researcher's belief that either colour cube is equally likely to be chosen is valid, a student correctly calculates that \(\mathrm { P } ( X = 8 ) = 0.0439\), correct to 3 significant figures. He says that, because this value is less than 0.05 , the null hypothesis should be rejected. Explain why this statement is incorrect.
    2. Carry out the test on the researcher's claim that either colour cube is equally likely to be chosen.
  2. Another researcher claims that a Type I error was made in carrying out the test. Explain why this cannot be true.
    A similar test, at the \(5 \%\) significance level, was carried out later using 10 other randomly chosen boys aged 1 to 3 .
  3. Find the probability of a Type I error.
CAIE S2 2023 November Q4
4 The height \(H\), in metres, of mature trees of a certain variety is normally distributed with standard deviation 0.67. In order to test whether the population mean of \(H\) is greater than 4.23, the heights of a random sample of 200 trees are measured.
  1. Write down suitable null and alternative hypotheses for the test.
    The sample mean height, \(\bar { h }\) metres, of the 200 trees is found and the test is carried out. The result of the test is to reject the null hypothesis at the 5\% significance level.
  2. Find the set of possible values of \(\bar { h }\).
  3. Ajit said, 'In (b) we had to assume that \(\bar { H }\) is normally distributed, so it was necessary to use the Central Limit Theorem.' Explain whether you agree with Ajit.
CAIE S2 2023 November Q5
5 The random variable \(X\) has probability density function, f, given by $$f ( x ) = \begin{cases} \frac { 1 } { x ^ { 2 } } & a < x < b
0 & \text { otherwise } \end{cases}$$ where \(a\) and \(b\) are positive constants.
  1. It is given that \(\mathrm { E } ( X ) = \ln 2\). Show that \(b = 2 a\).
  2. Show that \(a = \frac { 1 } { 2 }\).
  3. Find the median of \(X\).
CAIE S2 2023 November Q6
7 marks
6 A factory makes loaves of bread in batches. One batch of loaves contains \(X\) kilograms of dried yeast and \(Y\) kilograms of flour, where \(X\) and \(Y\) have the independent distributions \(\mathrm { N } \left( 0.7,0.02 ^ { 2 } \right)\) and \(\mathrm { N } \left( 100.0,3.0 ^ { 2 } \right)\) respectively. Dried yeast costs \(
) 13.50\( per kilogram and flour costs \)\\( 0.90\) per kilogram. For making one batch of bread the total of all other costs is \(
) 55\(. The factory sells each batch of bread for \)\\( 200\). Find the probability that the profit made on one randomly chosen batch of bread is greater than \(
) 40$. [7]
CAIE S2 2023 November Q7
7 A random variable \(X\) has the distribution \(\operatorname { Po } ( 2.4 )\).
  1. Find \(\mathrm { P } ( 2 \leqslant X < 4 )\).
  2. Two independent values of \(X\) are chosen. Find the probability that both of these values are greater than 1 .
  3. It is given that \(\mathrm { P } ( X = r ) < \mathrm { P } ( X = r + 1 )\).
    1. Find the set of possible values of \(r\).
    2. Hence find the value of \(r\) for which \(\mathrm { P } ( X = r )\) is greatest.
      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 2023 November Q1
1 A random variable \(X\) has the distribution \(\mathrm { N } ( 410,400 )\).
Find the probability that the mean of a random sample of 36 values of \(X\) is less than 405.
CAIE S2 2023 November Q4
4 The masses, in kilograms, of chemicals \(A\) and \(B\) produced per day by a factory are modelled by the independent random variables \(X\) and \(Y\) respectively, where \(X \sim \mathrm {~N} ( 10.3,5.76 )\) and \(Y \sim \mathrm {~N} ( 11.4,9.61 )\). The income generated by the chemicals is \(
) 2.50\( per kilogram for \)A\( and \)\\( 3.25\) per kilogram for \(B\).
  1. Find the mean and variance of the daily income generated by chemical \(A\).
  2. Find the probability that, on a randomly chosen day, the income generated by chemical \(A\) is greater than the income generated by chemical \(B\).
CAIE S2 2023 November Q5
5 In the past the number of enquiries per minute at a customer service desk has been modelled by a random variable with distribution \(\operatorname { Po } ( 0.31 )\). Following a change in the position of the desk, it is expected that the mean number of enquiries per minute will increase. In order to test whether this is the case, the total number of enquiries during a randomly chosen 5-minute period is noted. You should assume that a Poisson model is still appropriate. Given that the total number of enquiries is 5, carry out the test at the \(2.5 \%\) significance level.
CAIE S2 2024 November Q1
1 The heights of a certain species of deer are known to have standard deviation 0.35 m . A zoologist takes a random sample of 150 of these deer and finds that the mean height of the deer in the sample is 1.42 m .
  1. Calculate a 96\% confidence interval for the population mean height.
  2. Bubay says that \(96 \%\) of deer of this species are likely to have heights that are within this confidence interval. Explain briefly whether Bubay is correct.
CAIE S2 2024 November Q2
2 The masses, in kilograms, of small and large bags of wheat have the independent distributions \(\mathrm { N } ( 16.0,0.4 )\) and \(\mathrm { N } ( 51.0,0.9 )\) respectively. Find the probability that the total mass of 3 randomly chosen small bags is greater than the mass of one randomly chosen large bag.
\includegraphics[max width=\textwidth, alt={}, center]{acd6f1c9-bbaf-40ca-b5cb-8466ddb9f596-04_2720_38_109_2010}
CAIE S2 2024 November Q3
3 The times, \(T\) minutes, taken by a random sample of 75 students to complete a test were noted. The results were summarised by \(\Sigma t = 230\) and \(\Sigma t ^ { 2 } = 930\).
  1. Calculate unbiased estimates of the population mean and variance of \(T\).
    You should now assume that your estimates from part (a) are the true values of the population mean and variance of \(T\).
  2. The times taken by another random sample of 75 students were noted, and the sample mean, \(\bar { T }\), was found. Find the value of \(a\) such that \(P ( \bar { T } > a ) = 0.234\).
CAIE S2 2024 November Q4
4 A random variable \(X\) has probability density function f defined by $$f ( x ) = \begin{cases} \frac { a } { x ^ { 2 } } - \frac { 18 } { x ^ { 3 } } & 2 \leqslant x \leqslant 3
0 & \text { otherwise } \end{cases}$$ where \(a\) is a constant.
  1. Show that \(a = \frac { 27 } { 2 }\).
  2. Show that \(\mathrm { E } ( X ) = \frac { 27 } { 2 } \ln \frac { 3 } { 2 } - 3\).
CAIE S2 2024 November Q5
5 The lengths, in centimetres, of worms of a certain kind are normally distributed with mean \(\mu\) and standard deviation 2.3. An article in a magazine states that the value of \(\mu\) is 12.7 . A scientist wishes to test whether this value is correct. He measures the lengths, \(x \mathrm {~cm}\), of a random sample of 50 worms of this kind and finds that \(\sum x = 597.1\). He plans to carry out a test, at the \(1 \%\) significance level, of whether the true value of \(\mu\) is different from 12.7 .
  1. State, with a reason, whether he should use a one-tailed or a two-tailed test.
  2. Carry out the test.
CAIE S2 2024 November Q6
6 The numbers of customers arriving at service desks \(A\) and \(B\) during a 10 -minute period have the independent distributions \(\operatorname { Po } ( 1.8 )\) and \(\operatorname { Po } ( 2.1 )\) respectively.
  1. Find the probability that during a randomly chosen 15 -minute period more than 2 customers will arrive at \(\operatorname { desk } A\).
  2. Find the probability that during a randomly chosen 5-minute period the total number of customers arriving at both desks is less than 4 .
    \includegraphics[max width=\textwidth, alt={}, center]{acd6f1c9-bbaf-40ca-b5cb-8466ddb9f596-08_2720_35_109_2012}
  3. An inspector waits at desk \(B\). She wants to wait long enough to be \(90 \%\) certain of seeing at least one customer arrive at the desk. Find the minimum time for which she should wait, giving your answer correct to the nearest minute.
CAIE S2 2024 November Q7
7 The number of accidents per year on a certain road has the distribution \(\operatorname { Po } ( \lambda )\). In the past the value of \(\lambda\) was 3.3 . Recently, a new speed limit was imposed and the council wishes to test whether the value of \(\lambda\) has decreased. The council notes the total number, \(X\), of accidents during two randomly chosen years after the speed limit was introduced and it carries out a test at the \(5 \%\) significance level.
  1. Calculate the probability of a Type I error.
  2. Given that \(X = 2\), carry out the test.
    \includegraphics[max width=\textwidth, alt={}, center]{acd6f1c9-bbaf-40ca-b5cb-8466ddb9f596-10_2716_40_109_2010}
    \includegraphics[max width=\textwidth, alt={}, center]{acd6f1c9-bbaf-40ca-b5cb-8466ddb9f596-11_2716_29_107_22}
  3. The council decides to carry out another similar test at the \(5 \%\) significance level using the same hypotheses and two different randomly chosen years. Given that the true value of \(\lambda\) is 0.6 , calculate the probability of a Type II error.
  4. Using \(\lambda = 0.6\) and a suitable approximating distribution, find the probability that there will be more than 10 accidents in 30 years.
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE S2 2024 November Q1
1 A random variable \(X\) has the distribution \(\mathrm { B } \left( 4500000 , \frac { 1 } { 1000000 } \right)\).
Use a Poisson distribution to calculate an estimate of \(\mathrm { P } ( X \geqslant 4 )\).
\includegraphics[max width=\textwidth, alt={}, center]{9ac74d4c-f5e0-4c5d-ab25-5692dfb06f0b-03_2716_29_107_22}
CAIE S2 2024 November Q2
2 The lengths of a random sample of 50 roads in a certain region were measured.Using the results,a \(95 \%\) confidence interval for the mean length,in metres,of all roads in this region was found to be[245,263].
  1. Find the mean length of the 50 roads in the sample.
  2. Calculate an estimate of the standard deviation of the lengths of roads in this region.
  3. It is now given that the lengths of roads in this region are normally distributed.
    State,with a reason,whether this fact would make any difference to your calculation in part(b).
CAIE S2 2024 November Q3
3 A factory owner models the number of employees who use the factory canteen on any day by the distribution \(\mathrm { B } ( 25 , p )\). In the past the value of \(p\) was 0.8 . A new menu is introduced in the canteen and the owner wants to test whether the value of \(p\) has increased. On a randomly chosen day he notes that the number of employees who use the canteen is 23 .
  1. Use the binomial distribution to carry out the test at the \(10 \%\) significance level.
  2. Given that there are 30 employees at the factory comment on the suitability of the owner's model.
    \includegraphics[max width=\textwidth, alt={}, center]{9ac74d4c-f5e0-4c5d-ab25-5692dfb06f0b-04_2713_33_111_2016}
    \includegraphics[max width=\textwidth, alt={}, center]{9ac74d4c-f5e0-4c5d-ab25-5692dfb06f0b-05_2716_29_107_22}
CAIE S2 2024 November Q4
4 A population is normally distributed with mean 35 and standard deviation 8.1 . A random sample of size 140 is chosen from this population and the sample mean is denoted by \(\bar { X }\).
  1. Find \(\mathrm { P } ( \bar { X } > 36 )\).
  2. It is given that \(\mathrm { P } ( \bar { X } < a ) = 0.986\). Find the value of \(a\).
CAIE S2 2024 November Q5
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
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
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 2024 November Q2
2 The masses, in kilograms, of small and large bags of wheat have the independent distributions \(\mathrm { N } ( 16.0,0.4 )\) and \(\mathrm { N } ( 51.0,0.9 )\) respectively. Find the probability that the total mass of 3 randomly chosen small bags is greater than the mass of one randomly chosen large bag.
\includegraphics[max width=\textwidth, alt={}, center]{4f215475-30fd-47fb-aa77-0b53e339f50c-04_2716_38_108_2009}