Questions — CAIE S2 (737 questions)

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CAIE S2 2020 November Q6
12 marks Standard +0.3
6 The time, in minutes, for Anjan's journey to work on Mondays has mean 38.4 and standard deviation 6.9.
  1. Find the probability that Anjan's mean journey time for a random sample of 30 Mondays is between 38 and 40 minutes.
    Anjan wishes to test whether his mean journey time is different on Tuesdays. He chooses a random sample of 30 Tuesdays and finds that his mean journey time for these 30 Tuesdays is 40.2 minutes. Assume that the standard deviation for his journey time on Tuesdays is 6.9 minutes.
    1. State, with a reason, whether Anjan should use a one-tail or a two-tail test.
    2. Carry out the test at the \(10 \%\) significance level.
    3. Explain whether it was necessary to use the Central Limit theorem in part (b)(ii).
      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 November Q1
3 marks Moderate -0.5
1 On average, 1 in 50000 people have a certain gene.
Use a suitable approximating distribution to find the probability that more than 2 people in a random sample of 150000 have the gene.
CAIE S2 2020 November Q2
4 marks Moderate -0.8
2 A six-sided die has faces marked \(1,2,3,4,5,6\). When the die is thrown 300 times it shows a six on 56 throws.
  1. Calculate an approximate \(96 \%\) confidence interval for the probability that the die shows a six on one throw.
  2. Maroulla claims that the die is biased. Use your answer to part (a) to comment on this claim.
CAIE S2 2020 November Q3
7 marks Moderate -0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{ec7cab36-683b-4022-9cac-fb3b4e64778a-04_332_1100_260_520} A random variable \(X\) takes values between 0 and 3 only and has probability density function as shown in the diagram, where \(c\) is a constant.
  1. Show that \(c = \frac { 2 } { 3 }\).
  2. Find \(\mathrm { P } ( X > 2 )\).
  3. Calculate \(\mathrm { E } ( X )\).
CAIE S2 2020 November Q4
8 marks Standard +0.3
4 The areas, \(X \mathrm {~cm} ^ { 2 }\), of petals of a certain kind of flower have mean \(\mu \mathrm { cm } ^ { 2 }\). In the past it has been found that \(\mu = 8.9\). Following a change in the climate, a botanist claims that the mean is no longer 8.9. The areas of a random sample of 200 petals from this kind of flower are measured, and the results are summarized by $$\Sigma x = 1850 , \quad \Sigma x ^ { 2 } = 17850 .$$ Test the botanist's claim at the \(2.5 \%\) significance level.
CAIE S2 2020 November Q5
9 marks Moderate -0.8
5 Customers arrive at a shop at a constant average rate of 2.3 per minute.
  1. State another condition for the number of customers arriving per minute to have a Poisson distribution.
    It is now given that the number of customers arriving per minute has the distribution \(\mathrm { Po } ( 2.3 )\).
  2. Find the probability that exactly 3 customers arrive during a 1 -minute period.
  3. Find the probability that more than 3 customers arrive during a 2 -minute period.
  4. Five 1-minute periods are chosen at random. Find the probability that no customers arrive during exactly 2 of these 5 periods.
CAIE S2 2020 November Q6
9 marks Standard +0.3
6 A biscuit manufacturer claims that, on average, 1 in 3 packets of biscuits contain a prize offer. Gerry suspects that the proportion of packets containing the prize offer is less than 1 in 3 . In order to test the manufacturer's claim, he buys 20 randomly selected packets. He finds that exactly 2 of these packets contain the prize offer.
  1. Carry out the test at the \(10 \%\) significance level.
  2. Maria also suspects that the proportion of packets containing the prize offer is less than 1 in 3 . She also carries out a significance test at the \(10 \%\) level using 20 randomly selected packets. She will reject the manufacturer's claim if she finds that there are 3 or fewer packets containing the prize offer. Find the probability of a Type II error in Maria's test if the proportion of packets containing the prize offer is actually 1 in 7 .
  3. Explain what is meant by a Type II error in this context.
CAIE S2 2020 November Q7
10 marks Standard +0.8
7 Before a certain type of book is published it is checked for errors, which are then corrected. For costing purposes each error is classified as either minor or major. The numbers of minor and major errors in a book are modelled by the independent distributions \(\mathrm { N } ( 380,140 )\) and \(\mathrm { N } ( 210,80 )\) respectively. You should assume that no continuity corrections are needed when using these models. A book of this type is chosen at random.
  1. Find the probability that the number of minor errors is at least 200 more than the number of major errors.
    The costs of correcting a minor error and a major error are 20 cents and 50 cents respectively.
  2. Find the probability that the total cost of correcting the errors in the book is less than \(\\) 190$.
    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 November Q3
6 marks Standard +0.8
3 The masses, in kilograms, of female and male animals of a certain species have the distributions \(\mathrm { N } \left( 102,27 ^ { 2 } \right)\) and \(\mathrm { N } \left( 170,55 ^ { 2 } \right)\) respectively. Find the probability that a randomly chosen female has a mass that is less than half the mass of a randomly chosen male. \includegraphics[max width=\textwidth, alt={}, center]{6346fd4b-7bc9-4205-94db-67368b9415fe-06_76_1659_484_244}
CAIE S2 2020 November Q4
5 marks Moderate -0.5
4 \includegraphics[max width=\textwidth, alt={}, center]{6346fd4b-7bc9-4205-94db-67368b9415fe-07_316_984_260_577} The diagram shows the probability density function, \(\mathrm { f } ( x )\), of a random variable \(X\). For \(0 \leqslant x \leqslant a\), \(\mathrm { f } ( x ) = k\); elsewhere \(\mathrm { f } ( x ) = 0\).
  1. Express \(k\) in terms of \(a\).
  2. Given that \(\operatorname { Var } ( X ) = 3\), find \(a\).
CAIE S2 2017 November Q1
3 marks Moderate -0.8
1 A random variable, \(X\), has the distribution \(\operatorname { Po } ( 31 )\). Use the normal approximation to the Poisson distribution to find \(\mathrm { P } ( X > 40 )\).
CAIE S2 2017 November Q2
4 marks Standard +0.3
2 An airline has found that, on average, 1 in 100 passengers do not arrive for each flight, and that this occurs randomly. For one particular flight the airline always sells 403 seats. The plane only has room for 400 passengers, so the flight is overbooked if the number of passengers who do not arrive is less than 3 . Use a suitable approximation to find the probability that the flight is overbooked.
CAIE S2 2017 November Q3
4 marks Standard +0.3
3 After an election 153 adults, from a random sample of 200 adults, said that they had voted. Using this information, an \(\alpha \%\) confidence interval for the proportion of all adults who voted in the election was found to be 0.695 to 0.835 , both correct to 3 significant figures. Find the value of \(\alpha\), correct to the nearest integer.
CAIE S2 2017 November Q4
4 marks Moderate -0.3
4 The lengths, in millimetres, of rods produced by a machine are normally distributed with mean \(\mu\) and standard deviation 0.9. A random sample of 75 rods produced by the machine has mean length 300.1 mm .
  1. Find a \(99 \%\) confidence interval for \(\mu\), giving your answer correct to 2 decimal places.
    The manufacturer claims that the machine produces rods with mean length 300 mm .
  2. Use the confidence interval found in part (i) to comment on this claim.
CAIE S2 2017 November Q5
6 marks Moderate -0.3
5 A continuous random variable, \(X\), has probability density function given by $$f ( x ) = \begin{cases} \frac { 1 } { 4 } ( x + 1 ) & 0 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$
  1. Find \(\mathrm { E } ( X )\).
    ................................................................................................................................. .
  2. Find the median of \(X\).
CAIE S2 2017 November Q6
8 marks Moderate -0.3
6 The numbers of barrels of oil, in millions, extracted per day in two oil fields \(A\) and \(B\) are modelled by the independent random variables \(X\) and \(Y\) respectively, where \(X \sim \mathrm {~N} \left( 3.2,0.4 ^ { 2 } \right)\) and \(Y \sim \mathrm {~N} \left( 4.3,0.6 ^ { 2 } \right)\). The income generated by the oil from the two fields is \(\\) 90\( per barrel for \)A\( and \)\\( 95\) per barrel for \(B\).
  1. Find the mean and variance of the daily income, in millions of dollars, generated by field \(A\). [3]
  2. Find the probability that the total income produced by the two fields in a day is at least \(\\) 670$ million.
CAIE S2 2017 November Q7
9 marks Standard +0.3
7 In the past the number of cars sold per day at a showroom has been modelled by a random variable with distribution \(\operatorname { Po } ( 0.7 )\). Following an advertising campaign, it is hoped that the mean number of sales per day will increase. In order to test at the \(10 \%\) significance level whether this is the case, the total number of sales during the first 5 days after the campaign is noted. You should assume that a Poisson model is still appropriate.
  1. Given that the total number of cars sold during the 5 days is 5 , carry out the test.
    The number of cars sold per day at another showroom has the independent distribution \(\operatorname { Po } ( 0.6 )\). Assume that the distribution for the first showroom is still \(\operatorname { Po } ( 0.7 )\).
  2. Find the probability that the total number of cars sold in the two showrooms during 3 days is exactly 2 .
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.