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CAIE S2 2014 June Q5
9 marks Moderate -0.3
5 Mahmoud throws a coin 400 times and finds that it shows heads 184 times. The probability that the coin shows heads on any throw is denoted by \(p\).
  1. Calculate an approximate \(95 \%\) confidence interval for \(p\).
  2. Mahmoud claims that the coin is not fair. Use your answer to part (i) to comment on this claim.
  3. Mahmoud's result of 184 heads in 400 throws gives an \(\alpha \%\) confidence interval for \(p\) with width 0.1 . Calculate the value of \(\alpha\).
CAIE S2 2014 June Q6
10 marks Standard +0.3
6 The time, \(T\) hours, spent by people on a visit to a museum has probability density function $$\mathrm { f } ( t ) = \begin{cases} k t \left( 16 - t ^ { 2 } \right) & 0 \leqslant t \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = \frac { 1 } { 64 }\).
  2. Calculate the probability that two randomly chosen people each spend less than 1 hour on a visit to the museum.
  3. Find the mean time spent on a visit to the museum.
CAIE S2 2014 June Q7
10 marks Standard +0.3
7 A researcher is investigating the actual lengths of time that patients spend with the doctor at their appointments. He plans to choose a sample of 12 appointments on a particular day.
  1. Which of the following methods is preferable, and why?
    • Choose the first 12 appointments of the day.
    • Choose 12 appointments evenly spaced throughout the day.
    Appointments are scheduled to last 10 minutes. The actual lengths of time, in minutes, that patients spend with the doctor may be assumed to have a normal distribution with mean \(\mu\) and standard deviation 3.4. The researcher suspects that the actual time spent is more than 10 minutes on average. To test this suspicion, he recorded the actual times spent for a random sample of 12 appointments and carried out a hypothesis test at the 1\% significance level.
  2. State the probability of making a Type I error and explain what is meant by a Type I error in this context.
  3. Given that the total length of time spent for the 12 appointments was 147 minutes, carry out the test.
  4. Give a reason why the Central Limit theorem was not needed in part (iii).
CAIE S2 2015 June Q1
5 marks Easy -1.8
1 Jyothi wishes to choose a representative sample of 5 students from the 82 members of her school year.
  1. She considers going into the canteen and choosing a table with five students from her year sitting at it, and using these five people as her sample. Give two reasons why this method is unsatisfactory.
  2. Jyothi decides to use another method. She numbers all the students in her year from 1 to 82 . Then she uses her calculator and generates the following random numbers. $$231492 \quad 762305 \quad 346280$$ From these numbers, she obtains the student numbers \(23,14,76,5,34\) and 62 . Explain how Jyothi obtained these student numbers from the list of random numbers.
CAIE S2 2015 June Q2
5 marks Moderate -0.5
2 Marie claims that she can predict the winning horse at the local races. There are 8 horses in each race. Nadine thinks that Marie is just guessing, so she proposes a test. She asks Marie to predict the winners of the next 10 races and, if she is correct in 3 or more races, Nadine will accept Marie's claim.
  1. State suitable null and alternative hypotheses.
  2. Calculate the probability of a Type I error.
  3. State the significance level of the test.
CAIE S2 2015 June Q3
6 marks Challenging +1.2
3 A die is biased so that the probability that it shows a six on any throw is \(p\).
  1. In an experiment, the die shows a six on 22 out of 100 throws. Find an approximate \(97 \%\) confidence interval for \(p\).
  2. The experiment is repeated and another \(97 \%\) confidence interval is found. Find the probability that exactly one of the two confidence intervals includes the true value of \(p\).
CAIE S2 2015 June Q4
6 marks Moderate -0.8
4 The marks, \(x\), of a random sample of 50 students in a test were summarised as follows. $$n = 50 \quad \Sigma x = 1508 \quad \Sigma x ^ { 2 } = 51825$$
  1. Calculate unbiased estimates of the population mean and variance.
  2. Each student's mark is scaled using the formula \(y = 1.5 x + 10\). Find estimates of the population mean and variance of the scaled marks, \(y\).
CAIE S2 2015 June Q5
7 marks Moderate -0.8
5 The mean breaking strength of cables made at a certain factory is supposed to be 5 tonnes. The quality control department wishes to test whether the mean breaking strength of cables made by a particular machine is actually less than it should be. They take a random sample of 60 cables. For each cable they find the breaking strength by gradually increasing the tension in the cable and noting the tension when the cable breaks.
  1. Give a reason why it is necessary to take a sample rather then testing all the cables produced by the machine.
  2. The mean breaking strength of the 60 cables in the sample is found to be 4.95 tonnes. Given that the population standard deviation of breaking strengths is 0.15 tonnes, test at the \(1 \%\) significance level whether the population mean breaking strength is less than it should be.
  3. Explain whether it was necessary to use the Central Limit theorem in the solution to part (ii).
CAIE S2 2015 June Q6
10 marks Standard +0.3
6 People arrive at a checkout in a store at random, and at a constant mean rate of 0.7 per minute. Find the probability that
  1. exactly 3 people arrive at the checkout during a 5 -minute period,
  2. at least 30 people arrive at the checkout during a 1-hour period. People arrive independently at another checkout in the store at random, and at a constant mean rate of 0.5 per minute.
  3. Find the probability that a total of more than 3 people arrive at this pair of checkouts during a 2-minute period.
CAIE S2 2015 June Q7
11 marks Moderate -0.3
7 The probability density function of the random variable \(X\) is given by $$f ( x ) = \begin{cases} \frac { 3 } { 4 } x ( c - x ) & 0 \leqslant x \leqslant c \\ 0 & \text { otherwise } \end{cases}$$ where \(c\) is a constant.
  1. Show that \(c = 2\).
  2. Sketch the graph of \(y = \mathrm { f } ( x )\) and state the median of \(X\).
  3. Find \(\mathrm { P } ( X < 1.5 )\).
  4. Hence write down the value of \(\mathrm { P } ( 0.5 < X < 1 )\).
CAIE S2 2016 June Q1
5 marks Moderate -0.8
1 A six-sided die shows a six on 25 throws out of 200 throws. Test at the \(10 \%\) significance level the null hypothesis: P (throwing a six) \(= \frac { 1 } { 6 }\), against the alternative hypothesis: P (throwing a six) \(< \frac { 1 } { 6 }\).
CAIE S2 2016 June Q2
5 marks Easy -1.3
2 A researcher is investigating the lengths, in kilometres, of the journeys to work of the employees at a certain firm. She takes a random sample of 10 employees.
  1. State what is meant by 'random' in this context. The results of her sample are as follows. $$\begin{array} { l l l l l l l l l l } 1.5 & 2.0 & 3.6 & 5.9 & 4.8 & 8.7 & 3.5 & 2.9 & 4.1 & 3.0 \end{array}$$
  2. Find unbiased estimates of the population mean and variance.
  3. State what is meant by 'population' in this context.
CAIE S2 2016 June Q3
5 marks Moderate -0.8
3 Based on a random sample of 700 people living in a certain area, a confidence interval for the proportion, \(p\), of all people living in that area who had travelled abroad was found to be \(0.5672 < p < 0.6528\).
  1. Find the proportion of people in the sample who had travelled abroad.
  2. Find the confidence level of this confidence interval. Give your answer correct to the nearest integer.
CAIE S2 2016 June Q4
6 marks Moderate -0.3
4 In the past, the time spent by customers in a certain shop had mean 12.5 minutes and standard deviation 4.2 minutes. Following a change of layout in the shop, the mean time spent in the shop by a random sample of 50 customers is found to be 13.5 minutes.
  1. Assuming that the standard deviation remains at 4.2 minutes, test at the \(5 \%\) significance level whether the mean time spent by customers in the shop has changed.
  2. Another random sample of 50 customers is chosen and a similar test at the \(5 \%\) significance level is carried out. State the probability of a Type I error.
CAIE S2 2016 June Q5
9 marks Standard +0.8
5 The thickness of books in a large library is normally distributed with mean 2.4 cm and standard deviation 0.3 cm .
  1. Find the probability that the total thickness of 6 randomly chosen books is more than 16 cm .
  2. Find the probability that the thickness of a book chosen at random is less than 1.1 times the thickness of a second book chosen at random.
CAIE S2 2016 June Q6
10 marks Moderate -0.3
6 In each turn of a game, a coin is pushed and slides across a table. The distance, \(X\) metres, travelled by the coin has probability density function given by $$f ( x ) = \begin{cases} k x ^ { 2 } ( 2 - x ) & 0 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. State the greatest possible distance travelled by the coin in one turn.
  2. Show that \(k = \frac { 3 } { 4 }\).
  3. Find the mean distance travelled by the coin in one turn.
  4. Out of 400 turns, find the expected number of turns in which the distance travelled by the coin is less than 1 metre.
CAIE S2 2016 June Q7
10 marks Standard +0.3
7
  1. A large number of spoons and forks made in a factory are inspected. It is found that \(1 \%\) of the spoons and \(1.5 \%\) of the forks are defective. A random sample of 140 items, consisting of 80 spoons and 60 forks, is chosen. Use the Poisson approximation to the binomial distribution to find the probability that the sample contains
    1. at least 1 defective spoon and at least 1 defective fork,
    2. fewer than 3 defective items.
  2. The random variable \(X\) has the distribution \(\operatorname { Po } ( \lambda )\). It is given that $$\mathrm { P } ( X = 1 ) = p \quad \text { and } \quad \mathrm { P } ( X = 2 ) = 1.5 p$$ where \(p\) is a non-zero constant. Find the value of \(\lambda\) and hence find the value of \(p\).
CAIE S2 2016 June Q1
4 marks Moderate -0.5
1 The length of time, in minutes, taken by people to complete a task has mean 53.0 and standard deviation 6.2. Find the probability that the mean time taken to complete the task by a random sample of 50 people is more than 51 minutes.
CAIE S2 2016 June Q2
4 marks Moderate -0.3
2 Jacques is a chef. He claims that \(90 \%\) of his customers are satisfied with his cooking. Marie suspects that the true percentage is lower than \(90 \%\). She asks a random sample of 15 of Jacques' customers whether they are satisfied. She then performs a hypothesis test of the null hypothesis \(p = 0.9\) against the alternative hypothesis \(p < 0.9\), where \(p\) is the population proportion of customers who are satisfied. She decides to reject the null hypothesis if fewer than 12 customers are satisfied.
  1. In the context of the question, explain what is meant by a Type I error.
  2. Find the probability of a Type I error in Marie's test.
CAIE S2 2016 June Q3
5 marks Moderate -0.8
3
  1. Give a reason for using a sample rather than the whole population in carrying out a statistical investigation.
  2. Tennis balls of a certain brand are known to have a mean height of bounce of 64.7 cm , when dropped from a height of 100 cm . A change is made in the manufacturing process and it is required to test whether this change has affected the mean height of bounce. 100 new tennis balls are tested and it is found that their mean height of bounce when dropped from a height of 100 cm is 65.7 cm and the unbiased estimate of the population variance is \(15 \mathrm {~cm} ^ { 2 }\).
    (a) Calculate a \(95 \%\) confidence interval for the population mean.
    (b) Use your answer to part (ii) (a) to explain what conclusion can be drawn about whether the change has affected the mean height of bounce.
CAIE S2 2016 June Q4
6 marks Standard +0.3
4 At a certain company, computer faults occur randomly and at a constant mean rate. In the past this mean rate has been 2.1 per week. Following an update, the management wish to determine whether the mean rate has changed. During 20 randomly chosen weeks it is found that 54 computer faults occur. Use a suitable approximation to test at the \(5 \%\) significance level whether the mean rate has changed.
CAIE S2 2016 June Q5
10 marks Standard +0.8
5 Each box of Fruity Flakes contains \(X\) grams of flakes and \(Y\) grams of fruit, where \(X\) and \(Y\) are independent random variables, having distributions \(\mathrm { N } ( 400,50 )\) and \(\mathrm { N } ( 100,20 )\) respectively. The weight of each box, when empty, is exactly 20 grams. A full box of Fruity Flakes is chosen at random.
  1. Find the probability that the total weight of the box and its contents is less than 530 grams.
  2. Find the probability that the weight of flakes in the box is more than 4.1 times the weight of fruit in the box.
CAIE S2 2016 June Q6
10 marks Moderate -0.3
6 At a certain shop the demand for hair dryers has a Poisson distribution with mean 3.4 per week.
  1. Find the probability that, in a randomly chosen two-week period, the demand is for exactly 5 hair dryers.
  2. At the beginning of a week the shop has a certain number of hair dryers for sale. Find the probability that the shop has enough hair dryers to satisfy the demand for the week if
    (a) they have 4 hair dryers in the shop,
    (b) they have 5 hair dryers in the shop.
  3. Find the smallest number of hair dryers that the shop needs to have at the beginning of a week so that the probability of being able to satisfy the demand that week is at least 0.9 .
CAIE S2 2016 June Q7
11 marks Moderate -0.3
7
  1. \includegraphics[max width=\textwidth, alt={}, center]{1060d9f5-cf40-419e-b212-7266885c6617-3_465_1127_954_550} The diagram shows the graph of the probability density function of a variable \(X\). Given that the graph is symmetrical about the line \(x = 1\) and that \(\mathrm { P } ( 0 < X < 2 ) = 0.6\), find \(\mathrm { P } ( X > 0 )\).
  2. A flower seller wishes to model the length of time that tulips last when placed in a jug of water. She proposes a model using the random variable \(X\) (in hundreds of hours) with probability density function given by $$f ( x ) = \begin{cases} k \left( 2.25 - x ^ { 2 } \right) & 0 \leqslant x \leqslant 1.5 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
    1. Show that \(k = \frac { 4 } { 9 }\).
    2. Use this model to find the mean number of hours that a tulip lasts in a jug of water. The flower seller wishes to create a similar model for daffodils. She places a large number of daffodils in jugs of water and the longest time that any daffodil lasts is found to be 290 hours.
    3. Give a reason why \(\mathrm { f } ( x )\) would not be a suitable model for daffodils.
    4. The flower seller considers a model for daffodils of the form $$g ( x ) = \begin{cases} c \left( a ^ { 2 } - x ^ { 2 } \right) & 0 \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) and \(c\) are constants. State a suitable value for \(a\). (There is no need to evaluate \(c\).)
CAIE S2 2017 June Q1
5 marks Moderate -0.8
1 On average, 1 clover plant in 10000 has four leaves instead of three.
  1. Use an approximating distribution to calculate the probability that, in a random sample of 2000 clover plants, more than 2 will have four leaves.
  2. Justify your approximating distribution.