Questions S2 (1690 questions)

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CAIE S2 2012 June Q5
10 marks Standard +0.3
5 A random variable \(X\) has the distribution \(\operatorname { Po } ( 3.2 )\).
  1. A random value of \(X\) is found.
    1. Find \(\mathrm { P } ( X \geqslant 3 )\).
    2. Find the probability that \(X = 3\) given that \(X \geqslant 3\).
    3. Random samples of 120 values of \(X\) are taken.
      (a) Describe fully the distribution of the sample mean.
      (b) Find the probability that the mean of a random sample of size 120 is less than 3.3.
CAIE S2 2012 June Q6
11 marks Standard +0.3
6 A survey taken last year showed that the mean number of computers per household in Branley was 1.66 . This year a random sample of 50 households in Branley answered a questionnaire with the following results.
Number of computers01234\(> 4\)
Number of households512181050
  1. Calculate unbiased estimates for the population mean and variance of the number of computers per household in Branley this year.
  2. Test at the \(5 \%\) significance level whether the mean number of computers per household has changed since last year.
  3. Explain whether it is possible that a Type I error may have been made in the test in part (ii).
  4. State what is meant by a Type II error in the context of the test in part (ii), and give the set of values of the test statistic that could lead to a Type II error being made.
CAIE S2 2012 June Q7
11 marks Standard +0.3
7 At work Jerry receives emails randomly at a constant average rate of 15 emails per hour.
  1. Find the probability that Jerry receives more than 2 emails during a 20 -minute period at work.
  2. Jerry's working day is 8 hours long. Find the probability that Jerry receives fewer than 110 emails per day on each of 2 working days.
  3. At work Jerry also receives texts randomly and independently at a constant average rate of 1 text every 10 minutes. Find the probability that the total number of emails and texts that Jerry receives during a 5 -minute period at work is more than 2 and less than 6 .
CAIE S2 2021 November Q1
5 marks Moderate -0.8
1 It is known that the height \(H\), in metres, of trees of a certain kind has the distribution \(\mathrm { N } ( 12.5,10.24 )\). A scientist takes a random sample of 25 trees of this kind and finds the sample mean, \(\bar { H }\), of the heights.
  1. State the distribution of \(\bar { H }\), giving the values of any parameters.
  2. Find \(\mathrm { P } ( 12 < \bar { H } < 13 )\).
CAIE S2 2021 November Q2
4 marks Moderate -0.3
2 The number of enquiries received per day at a customer service desk has a Poisson distribution with mean 45.2. If more than 60 enquiries are received in a day, the customer service desk cannot deal with them all. Use a suitable approximating distribution to find the probability that, on a randomly chosen day, the customer service desk cannot deal with all the enquiries that are received.
CAIE S2 2021 November Q3
5 marks Standard +0.3
3 A random sample of 75 students at a large college was selected for a survey. 15 of these students said that they owned a car. From this result an approximate \(\alpha \%\) confidence interval for the proportion of all students at the college who own a car was calculated. The width of this interval was found to be 0.162 . Calculate the value of \(\alpha\) correct to 2 significant figures.
CAIE S2 2021 November Q4
9 marks Standard +0.3
4 A random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { 1 } { 18 } \left( 9 - x ^ { 2 } \right) & 0 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$
  1. Find \(\mathrm { P } ( X < 1.2 )\).
  2. Find \(\mathrm { E } ( X )\).
    The median of \(X\) is \(m\).
  3. Show that \(m ^ { 3 } - 27 m + 27 = 0\).
CAIE S2 2021 November Q5
9 marks Moderate -0.5
5
  1. The proportion of people having a particular medical condition is 1 in 100000 . A random sample of 2500 people is obtained. The number of people in the sample having the condition is denoted by \(X\).
    1. State, with a justification, a suitable approximating distribution for \(X\), giving the values of any parameters.
    2. Use the approximating distribution to calculate \(\mathrm { P } ( X > 0 )\).
  2. The percentage of people having a different medical condition is thought to be \(30 \%\). A researcher suspects that the true percentage is less than \(30 \%\). In a medical trial a random sample of 28 people was selected and 4 people were found to have this condition. Use a binomial distribution to test the researcher's suspicion at the \(2 \%\) significance level.
CAIE S2 2021 November Q6
8 marks Standard +0.3
6 The random variable \(T\) denotes the time, in seconds, for 100 m races run by Tania. \(T\) is normally distributed with mean \(\mu\) and variance \(\sigma ^ { 2 }\). A random sample of 40 races run by Tania gave the following results. $$n = 40 \quad \Sigma t = 560 \quad \Sigma t ^ { 2 } = 7850$$
  1. Calculate unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\).
    The random variable \(S\) denotes the time, in seconds, for 100 m races run by Suki. \(S\) has the independent distribution \(\mathrm { N } ( 14.2,0.3 )\).
  2. Using your answers to part (a), find the probability that, in a randomly chosen 100 m race, Suki's time will be at least 0.1 s more than Tania's time.
CAIE S2 2021 November Q7
10 marks Standard +0.3
7 The masses, in grams, of apples from a certain farm have mean \(\mu\) and standard deviation 5.2. The farmer says that the value of \(\mu\) is 64.6. A quality control inspector claims that the value of \(\mu\) is actually less than 64.6. In order to test his claim he chooses a random sample of 100 apples from the farm.
  1. The mean mass of the 100 apples is found to be 63.5 g . Carry out the test at the \(2.5 \%\) significance level.
  2. Later another test of the same hypotheses at the \(2.5 \%\) significance level, with another random sample of 100 apples from the same farm, is carried out. Given that the value of \(\mu\) is in fact 62.7 , calculate 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 2021 November Q1
6 marks Moderate -0.8
1 The mass, in kilograms, of a block of cheese sold in a supermarket is denoted by the random variable \(M\). The masses of a random sample of 40 blocks are summarised as follows. $$n = 40 \quad \Sigma m = 20.50 \quad \Sigma m ^ { 2 } = 10.7280$$
  1. Calculate unbiased estimates of the population mean and variance of \(M\).
  2. The price, \(\\) P\(, of a block of cheese of mass \)M \mathrm {~kg}\( is found using the formula \)P = 11 M + 0.50\(. Find estimates of the population mean and variance of \)P$.
CAIE S2 2021 November Q2
3 marks Easy -1.8
2 Andy and Jessica are doing a survey about musical preferences. They plan to choose a representative sample of six students from the 256 students at their college.
  1. Andy suggests that they go to the music building during the lunch hour and choose six students at random from the students who are there. Give a reason why this method is unsatisfactory.
  2. Jessica decides to use another method. She numbers all the students in the college from 1 to 256. Then she uses her calculator and generates the following random numbers. $$\begin{array} { l l l l l } 204393 & 162007 & 204028 & 587119 & 207395 \end{array}$$ From these numbers, she obtains six student numbers. The first three of her student numbers are 204, 162 and 7. Continue Jessica's method to obtain the next three student numbers.
CAIE S2 2021 November Q3
6 marks Standard +0.8
3 The probability that a certain spinner lands on red on any spin is \(p\). The spinner is spun 140 times and it lands on red 35 times.
  1. Find an approximate \(96 \%\) confidence interval for \(p\).
    From three further experiments, Jack finds a 90\% confidence interval, a 95\% confidence interval and a 99\% confidence interval for \(p\).
  2. Find the probability that exactly two of these confidence intervals contain the true value of \(p\).
CAIE S2 2021 November Q4
7 marks Moderate -0.3
4 A certain kind of firework is supposed to last for 30 seconds, on average, after it is lit. An inspector suspects that the fireworks actually last a shorter time than this, on average. He takes a random sample of 100 fireworks of this kind. Each firework in the sample is lit and the time it lasts is noted.
  1. Give a reason why it is necessary to take a sample rather than testing all the fireworks of this kind.
    It is given that the population standard deviation of the times that fireworks of this kind last is 5 seconds.
  2. The mean time lasted by the 100 fireworks in the sample is found to be 29 seconds. Test the inspector's suspicion at the \(1 \%\) significance level.
  3. State with a reason whether the Central Limit theorem was needed in the solution to part (b).
CAIE S2 2021 November Q5
9 marks Moderate -0.3
5 In a certain large document, typing errors occur at random and at a constant mean rate of 0.2 per page.
  1. Find the probability that there are fewer than 3 typing errors in 10 randomly chosen pages.
  2. Use an approximating distribution to find the probability that there are more than 50 typing errors in 200 randomly chosen pages.
    In the same document, formatting errors occur at random and at a constant mean rate of 0.3 per page.
  3. Find the probability that the total number of typing and formatting errors in 20 randomly chosen pages is between 8 and 11 inclusive.
CAIE S2 2021 November Q6
10 marks Standard +0.3
6 A machine is supposed to produce random digits. Bob thinks that the machine is not fair and that the probability of it producing the digit 0 is less than \(\frac { 1 } { 10 }\). In order to test his suspicion he notes the number of times the digit 0 occurs in 30 digits produced by the machine. He carries out a test at the \(10 \%\) significance level.
  1. State suitable null and alternative hypotheses.
  2. Find the rejection region for the test.
  3. State the probability of a Type I error.
    It is now given that the machine actually produces a 0 once in every 40 digits, on average.
  4. Find the probability of a Type II error.
  5. Explain the meaning of a Type II error in this context.
CAIE S2 2021 November Q7
9 marks Standard +0.3
7
  1. The probability density function of the random variable \(X\) is given by $$f ( x ) = \begin{cases} k x ( 4 - x ) & 0 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
    1. Show that \(k = \frac { 3 } { 16 }\).
    2. Find \(\mathrm { E } ( X )\).
  2. The random variable \(Y\) has the following properties.
    Given that \(\mathrm { P } ( Y < a ) = 0.2\), find \(\mathrm { P } ( 2.5 < Y < 5 - a )\) illustrating your method with a sketch on the axes provided. \includegraphics[max width=\textwidth, alt={}, center]{cea87af9-4b2a-4297-91e9-4eb5744b9e48-11_369_837_621_694}
    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 2022 November Q1
3 marks Easy -1.2
1 The heights, in metres, of a random sample of 10 mature trees of a certain variety are given below. \(\begin{array} { l l l l l l l l l l } 5.9 & 6.5 & 6.7 & 5.9 & 6.9 & 6.0 & 6.4 & 6.2 & 5.8 & 5.8 \end{array}\) Find unbiased estimates of the population mean and variance of the heights of all mature trees of this variety.
CAIE S2 2022 November Q2
8 marks Moderate -0.3
2 A spinner has five sectors, each printed with a different colour. Susma and Sanjay both wish to test whether the spinner is biased so that it lands on red on fewer spins than it would if it were fair. Susma spins the spinner 40 times. She finds that it lands on red exactly 4 times.
  1. Use a binomial distribution to carry out the test at the \(5 \%\) significance level.
    Sanjay also spins the spinner 40 times. He finds that it lands on red \(r\) times.
  2. Use a binomial distribution to find the largest value of \(r\) that lies in the rejection region for the test at the 5\% significance level.
CAIE S2 2022 November Q3
7 marks Standard +0.3
3 Drops of water fall randomly from a leaking tap at a constant average rate of 5.2 per minute.
  1. Find the probability that at least 3 drops fall during a randomly chosen 30 -second period.
  2. Use a suitable approximating distribution to find the probability that at least 650 drops fall during a randomly chosen 2-hour period.
CAIE S2 2022 November Q4
8 marks Standard +0.3
4 Each month a company sells \(X \mathrm {~kg}\) of brown sugar and \(Y \mathrm {~kg}\) of white sugar, where \(X\) and \(Y\) have the independent distributions \(\mathrm { N } \left( 2500,120 ^ { 2 } \right)\) and \(\mathrm { N } \left( 3700,130 ^ { 2 } \right)\) respectively.
  1. Find the mean and standard deviation of the total amount of sugar that the company sells in 3 randomly chosen months.
    The company makes a profit of \(\\) 1.50\( per kilogram of brown sugar sold and makes a loss of \)\\( 0.20\) per kilogram of white sugar sold.
  2. Find the probability that, in a randomly chosen month, the total profit is less than \(\\) 3000$.
CAIE S2 2022 November Q5
7 marks Moderate -0.5
5 A builders' merchant sells stones of different sizes.
  1. The masses of size \(A\) stones have standard deviation 6 grams. The mean mass of a random sample of 200 size \(A\) stones is 45 grams. Find a 95\% confidence interval for the population mean mass of size \(A\) stones.
  2. The masses of size \(B\) stones have standard deviation 11 grams. Using a random sample of size 200, an \(\alpha \%\) confidence interval for the population mean mass is found to have width 4 grams. Find \(\alpha\).
CAIE S2 2022 November Q6
7 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{f45a7c6f-ebb4-43a2-8751-11aab3561d3c-08_490_1195_255_475} The diagram shows the graph of the probability density function of a random variable \(X\) that takes values between - 1 and 3 only. It is given that the graph is symmetrical about the line \(x = 1\). Between \(x = - 1\) and \(x = 3\) the graph is a quadratic curve. The random variable \(S\) is such that \(\mathrm { E } ( S ) = 2 \times \mathrm { E } ( X )\) and \(\operatorname { Var } ( S ) = \operatorname { Var } ( X )\).
  1. On the grid below, sketch a quadratic graph for the probability density function of \(S\). \includegraphics[max width=\textwidth, alt={}, center]{f45a7c6f-ebb4-43a2-8751-11aab3561d3c-08_490_1191_1169_479} The random variable \(T\) is such that \(\mathrm { E } ( T ) = \mathrm { E } ( X )\) and \(\operatorname { Var } ( T ) = \frac { 1 } { 4 } \operatorname { Var } ( X )\).
  2. On the grid below, sketch a quadratic graph for the probability density function of \(T\). \includegraphics[max width=\textwidth, alt={}, center]{f45a7c6f-ebb4-43a2-8751-11aab3561d3c-08_488_1187_1996_479} It is now given that $$f ( x ) = \begin{cases} \frac { 3 } { 32 } \left( 3 + 2 x - x ^ { 2 } \right) & - 1 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$
  3. Given that \(\mathrm { P } ( 1 - a < X < 1 + a ) = 0.5\), show that \(a ^ { 3 } - 12 a + 8 = 0\).
  4. Hence verify that \(0.69 < a < 0.70\).
CAIE S2 2022 November Q7
10 marks Standard +0.3
7 In the past Laxmi's time, in minutes, for her journey to college had mean 32.5 and standard deviation 3.1. After a change in her route, Laxmi wishes to test whether the mean time has decreased. She notes her journey times for a random sample of 50 journeys and she finds that the sample mean is 31.8 minutes. You should assume that the standard deviation is unchanged.
  1. Carry out a hypothesis test, at the \(8 \%\) significance level, of whether Laxmi's mean journey time has decreased.
    Later Laxmi carries out a similar test with the same hypotheses, at the \(8 \%\) significance level, using another random sample of size 50 .
  2. Given that the population mean is now 31.5, 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 2022 November Q6
7 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{8f9e5f25-05c1-4a4e-9094-2a1b42416588-08_490_1195_255_475} The diagram shows the graph of the probability density function of a random variable \(X\) that takes values between - 1 and 3 only. It is given that the graph is symmetrical about the line \(x = 1\). Between \(x = - 1\) and \(x = 3\) the graph is a quadratic curve. The random variable \(S\) is such that \(\mathrm { E } ( S ) = 2 \times \mathrm { E } ( X )\) and \(\operatorname { Var } ( S ) = \operatorname { Var } ( X )\).
  1. On the grid below, sketch a quadratic graph for the probability density function of \(S\). \includegraphics[max width=\textwidth, alt={}, center]{8f9e5f25-05c1-4a4e-9094-2a1b42416588-08_490_1191_1169_479} The random variable \(T\) is such that \(\mathrm { E } ( T ) = \mathrm { E } ( X )\) and \(\operatorname { Var } ( T ) = \frac { 1 } { 4 } \operatorname { Var } ( X )\).
  2. On the grid below, sketch a quadratic graph for the probability density function of \(T\). \includegraphics[max width=\textwidth, alt={}, center]{8f9e5f25-05c1-4a4e-9094-2a1b42416588-08_488_1187_1996_479} It is now given that $$f ( x ) = \begin{cases} \frac { 3 } { 32 } \left( 3 + 2 x - x ^ { 2 } \right) & - 1 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$
  3. Given that \(\mathrm { P } ( 1 - a < X < 1 + a ) = 0.5\), show that \(a ^ { 3 } - 12 a + 8 = 0\).
  4. Hence verify that \(0.69 < a < 0.70\).