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CAIE S2 2017 November Q8
12 marks Challenging +1.2
8 In order to test the effect of a drug, a researcher monitors the concentration, \(X\), of a certain protein in the blood stream of patients. For patients who are not taking the drug the mean value of \(X\) is 0.185 . A random sample of 150 patients taking the drug was selected and the values of \(X\) were found. The results are summarised below. $$n = 150 \quad \Sigma x = 27.0 \quad \Sigma x ^ { 2 } = 5.01$$ The researcher wishes to test at the \(1 \%\) significance level whether the mean concentration of the protein in the blood stream of patients taking the drug is less than 0.185 .
  1. Carry out the test.
  2. Given that, in fact, the mean concentration for patients taking the drug is 0.175 , find the probability of a Type II error occurring in the test.
CAIE S2 2017 November Q1
6 marks Moderate -0.8
1
    1. A random variable \(X\) has the distribution \(\mathrm { B } ( 2540,0.001 )\). Use the Poisson approximation to the binomial distribution to find \(\mathrm { P } ( X > 1 )\).
    2. Explain why the Poisson approximation is appropriate in this case.
  2. Two independent random variables, \(S\) and \(T\), have distributions \(\operatorname { Po } ( 2.1 )\) and \(\operatorname { Po } ( 3.5 )\) respectively. Find the mean and standard deviation of \(S + T\).
CAIE S2 2017 November Q2
6 marks Standard +0.3
2 The number of words in History essays by students at a certain college has mean \(\mu\) and standard deviation 1420.
  1. The mean number of words in a random sample of 125 History essays was found to be 4820 . Calculate a \(98 \%\) confidence interval for \(\mu\).
  2. Another random sample of \(n\) History essays was taken. Using this sample, a \(95 \%\) confidence interval for \(\mu\) was found to be 4700 to 4980 , both correct to the nearest integer. Find the value of \(n\).
CAIE S2 2017 November Q3
8 marks Standard +0.3
3 The masses, \(m \mathrm {~kg}\), of packets of flour are normally distributed. The mean mass is supposed to be 1.01 kg . A quality control officer measures the masses of a random sample of 100 packets. The results are summarised below. $$n = 100 \quad \Sigma m = 98.2 \quad \Sigma m ^ { 2 } = 104.52$$
  1. Test at the \(5 \%\) significance level whether the population mean mass is less than 1.01 kg .
  2. Explain whether it was necessary to use the Central Limit theorem in your answer to part (i).
CAIE S2 2017 November Q4
10 marks Standard +0.3
4 The random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { k } { \sqrt { } x } & 0 < x \leqslant a \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) and \(a\) are constants. It is given that \(\mathrm { E } ( X ) = 3\).
  1. Find the value of \(a\) and show that \(k = \frac { 1 } { 6 }\).
  2. Find the median of \(X\).
CAIE S2 2017 November Q5
10 marks Standard +0.3
5 The marks in paper 1 and paper 2 of an examination are denoted by \(X\) and \(Y\) respectively, where \(X\) and \(Y\) have the independent continuous distributions \(\mathrm { N } \left( 56,6 ^ { 2 } \right)\) and \(\mathrm { N } \left( 43,5 ^ { 2 } \right)\) respectively.
  1. Find the probability that a randomly chosen paper 1 mark is more than a randomly chosen paper 2 mark.
  2. Each candidate's overall mark is \(M\) where \(M = X + 1.5 Y\). The minimum overall mark for grade A is 135 . Find the proportion of students who gain a grade A .
CAIE S2 2017 November Q6
10 marks Standard +0.3
6 In a certain factory the number of items per day found to be defective has had the distribution \(\operatorname { Po } ( 1.03 )\). After the introduction of new quality controls, the management wished to test at the \(10 \%\) significance level whether the mean number of defective items had decreased. They noted the total number of defective items produced in 5 randomly chosen days. It is assumed that defective items occur randomly and that a Poisson model is still appropriate.
  1. Given that the total number of defective items produced during the 5 days was 2 , carry out the test.
  2. Using another random sample of 5 days the same test is carried out again, with the same significance level. Find the probability of a Type I error.
  3. Explain what is meant by a Type I error in this context.
CAIE S2 2018 November Q1
4 marks Moderate -0.8
1 The standard deviation of the heights of adult males is 7.2 cm . The mean height of a sample of 200 adult males is found to be 176 cm .
  1. Calculate a \(97.5 \%\) confidence interval for the mean height of adult males.
  2. State a necessary condition for the calculation in part (i) to be valid.
CAIE S2 2018 November Q2
4 marks Moderate -0.8
2 A headteacher models the number of children who bring a 'healthy' packed lunch to school on any day by the distribution \(\mathrm { B } ( 150 , p )\). In the past, she has found that \(p = \frac { 1 } { 3 }\). Following the opening of a fast food outlet near the school, she wishes to test, at the \(1 \%\) significance level, whether the value of \(p\) has decreased.
  1. State the null and alternative hypotheses for this test.
    On a randomly chosen day she notes the number, \(N\), of children who bring a 'healthy' packed lunch to school. She finds that \(N = 36\) and then, assuming that the null hypothesis is true, she calculates that \(\mathrm { P } ( N \leqslant 36 ) = 0.0084\).
  2. State, with a reason, the conclusion that the headteacher should draw from the test.
  3. According to the model, what is the largest number of children who might bring a packed lunch to school?
CAIE S2 2018 November Q3
4 marks Standard +0.3
3 A population has mean 12 and standard deviation 2.5. A large random sample of size \(n\) is chosen from this population and the sample mean is denoted by \(\bar { X }\). Given that \(\mathrm { P } ( \bar { X } < 12.2 ) = 0.975\), correct to 3 significant figures, find the value of \(n\).
CAIE S2 2018 November Q4
8 marks Standard +0.3
4 Small drops of two liquids, \(A\) and \(B\), are randomly and independently distributed in the air. The average numbers of drops of \(A\) and \(B\) per cubic centimetre of air are 0.25 and 0.36 respectively.
  1. A sample of \(10 \mathrm {~cm} ^ { 3 }\) of air is taken at random. Find the probability that the total number of drops of \(A\) and \(B\) in this sample is at least 4 .
  2. A sample of \(100 \mathrm {~cm} ^ { 3 }\) of air is taken at random. Use an approximating distribution to find the probability that the total number of drops of \(A\) and \(B\) in this sample is less than 60 .
CAIE S2 2018 November Q5
9 marks Standard +0.3
5 The times, in months, taken by a builder to build two types of house, \(P\) and \(Q\), are represented by the independent variables \(T _ { 1 } \sim \mathrm {~N} \left( 2.2,0.4 ^ { 2 } \right)\) and \(T _ { 2 } \sim \mathrm {~N} \left( 2.8,0.5 ^ { 2 } \right)\) respectively.
  1. Find the probability that the total time taken to build one house of each type is less than 6 months.
  2. Find the probability that the time taken to build a type \(Q\) house is more than 1.2 times the time taken to build a type \(P\) house.
CAIE S2 2018 November Q6
9 marks Standard +0.3
6 The random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} k x ^ { - 1 } & 2 \leqslant x \leqslant 6 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = \frac { 1 } { \ln 3 }\).
  2. Show that \(\mathrm { E } ( X ) = 3.64\), correct to 3 significant figures.
  3. Given that the median of \(X\) is \(m\), find \(\mathrm { P } ( m < X < \mathrm { E } ( X ) )\).
CAIE S2 2018 November Q7
12 marks Challenging +1.2
7 A mill owner claims that the mean mass of sacks of flour produced at his mill is 51 kg . A quality control officer suspects that the mean mass is actually less than 51 kg . In order to test the owner's claim she finds the mass, \(x \mathrm {~kg}\), of each of a random sample of 150 sacks and her results are summarised as follows. $$n = 150 \quad \Sigma x = 7480 \quad \Sigma x ^ { 2 } = 380000$$
  1. Carry out the test at the \(2.5 \%\) significance level.
    You may now assume that the population standard deviation of the masses of sacks of flour is 6.856 kg . The quality control officer weighs another random sample of 150 sacks and carries out another test at the 2.5\% significance level.
  2. Given that the population mean mass is 49 kg , find the probability of a Type II error.
    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 2018 November Q1
3 marks Easy -1.2
1 The random variable \(X\) has the distribution \(\operatorname { Po } ( 2.3 )\). Find \(\mathrm { P } ( 2 \leq X < 5 )\).
CAIE S2 2018 November Q2
4 marks Moderate -0.3
2 The standard deviation of the volume of drink in cans of Koola is 4.8 centilitres. A random sample of 180 cans is taken and the mean volume of drink in these 180 cans is found to be 330.1 centilitres.
  1. Calculate a \(95 \%\) confidence interval for the mean volume of drink in all cans of Koola. Give the end-points of your interval correct to 1 decimal place.
  2. Explain whether it was necessary to use the Central Limit theorem in your answer to part (i).
CAIE S2 2018 November Q3
5 marks Standard +0.3
3 Sugar and flour for making cakes are measured in cups. The mass, in grams, of one cup of sugar has the distribution \(\mathrm { N } ( 250,10 )\). The mass, in grams, of one cup of flour has the independent distribution \(\mathrm { N } ( 160,9 )\). Each cake contains 2 cups of sugar and 5 cups of flour. Find the probability that the total mass of sugar and flour in one cake exceeds 1310 grams.
CAIE S2 2018 November Q4
7 marks Standard +0.3
4 The time, \(X\) hours, taken by a large number of runners to complete a race is modelled by the probability density function given by $$f ( x ) = \begin{cases} \frac { k } { ( x + 1 ) ^ { 2 } } & 0 \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) and \(a\) are constants.
  1. Show that \(k = \frac { a + 1 } { a }\).
  2. State what the constant \(a\) represents in this context.
    Three quarters of the runners take half an hour or less to complete the race.
  3. Find the value of \(a\).
CAIE S2 2018 November Q5
10 marks Standard +0.3
5 The numbers of basketball courts in a random sample of 70 schools in South Mowland are summarised in the table.
Number of basketball courts01234\(> 4\)
Number of schools228261040
  1. Calculate unbiased estimates for the population mean and variance of the number of basketball courts per school in South Mowland.
    The mean number of basketball courts per school in North Mowland is 1.9 .
  2. Test at the \(5 \%\) significance level whether the mean number of basketball courts per school in South Mowland is less than the mean for North Mowland.
  3. State, with a reason, which of the errors, Type I or Type II, might have been made in the test in part (ii).
CAIE S2 2018 November Q6
10 marks Standard +0.3
6 In the past, Angus found that his train was late on \(15 \%\) of his daily journeys to work. Following a timetable change, Angus found that out of 60 randomly chosen days, his train was late on 6 days.
  1. Test at the \(10 \%\) significance level whether Angus' train is late less often than it was before the timetable change.
    Angus used his random sample to find an \(\alpha \%\) confidence interval for the proportion of days on which his train is late. The upper limit of his interval was 0.150 , correct to 3 significant figures.
  2. Calculate the value of \(\alpha\) correct to the nearest integer.
CAIE S2 2018 November Q7
11 marks Standard +0.8
7 The independent random variables \(X\) and \(Y\) have the distributions \(\operatorname { Po } ( 2.1 )\) and \(\operatorname { Po } ( 3.5 )\) respectively.
  1. Find \(\mathrm { P } ( X + Y = 3 )\).
  2. Given that \(X + Y = 3\), find \(\mathrm { P } ( X = 2 )\).
  3. A random sample of 100 values of \(X\) is taken. Find the probability that the sample mean is more than 2.2.
    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 2019 November Q1
7 marks Moderate -0.8
1 On average, 1 in 150 components made by a certain machine are faulty. The random variable \(X\) denotes the number of faulty components in a random sample of 500 components.
  1. Describe fully the distribution of \(X\).
  2. State a suitable approximating distribution for \(X\), giving a justification for your choice.
  3. Use your approximating distribution to find the probability that the sample will include at least 3 faulty components.
CAIE S2 2019 November Q2
8 marks Standard +0.3
2 The heights of a certain species of animal have been found to have mean 65.2 cm and standard deviation 7.1 cm . A researcher suspects that animals of this species in a certain region are shorter on average than elsewhere. She takes a large random sample of \(n\) animals of this species from this region and finds that their mean height is 63.2 cm . She then carries out an appropriate hypothesis test.
  1. She finds that the value of the test statistic \(z\) is - 2.182 , correct to 3 decimal places.
    (a) Stating a necessary assumption, calculate the value of \(n\).
    (b) Carry out the hypothesis test at the \(4 \%\) significance level.
  2. Explain why it was necessary to use the Central Limit theorem in carrying out the test.
CAIE S2 2019 November Q3
7 marks Moderate -0.3
3 The masses, in grams, of bags of flour are normally distributed with mean \(\mu\). The masses, \(m\) grams, of a random sample of 50 bags are summarised by \(\Sigma m = 25110\) and \(\Sigma m ^ { 2 } = 12610300\).
  1. Calculate a \(96 \%\) confidence interval for \(\mu\), giving the end-points correct to 1 decimal place.
    Another random sample of 50 bags of flour is taken and a \(99 \%\) confidence interval for \(\mu\) is calculated.
  2. Without calculation, state whether this confidence interval will be wider or narrower than the confidence interval found in part (i). Give a reason for your answer.
CAIE S2 2019 November Q4
9 marks Moderate -0.3
4 A random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { 1 } { 2 } x & 0 \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) is a constant.
  1. Find \(a\).
  2. Show that \(\mathrm { E } ( X ) = \frac { 4 } { 3 }\).
    The median of \(X\) is denoted by \(m\).
  3. Find \(\mathrm { P } ( \mathrm { E } ( X ) < X < m )\).