Questions — CAIE S2 (737 questions)

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CAIE S2 2015 November Q3
5 marks Standard +0.3
3 From a random sample of 65 people in a certain town, the proportion who own a bicycle was noted. From this result an \(\alpha \%\) confidence interval for the proportion, \(p\), of all people in the town who own a bicycle was calculated to be \(0.284 < p < 0.516\).
  1. Find the proportion of people in the sample who own a bicycle.
  2. Calculate the value of \(\alpha\) correct to 2 significant figures.
CAIE S2 2015 November Q4
8 marks Moderate -0.3
4 A random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} k \left( 4 - x ^ { 2 } \right) & - 2 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = \frac { 3 } { 32 }\).
  2. Sketch the graph of \(y = \mathrm { f } ( x )\) and hence write down the value of \(\mathrm { E } ( X )\).
  3. Find \(\mathrm { P } ( X < 1 )\).
CAIE S2 2015 November Q5
9 marks Standard +0.3
5
  1. Narika has a die which is known to be biased so that the probability of throwing a 6 on any throw is \(\frac { 1 } { 100 }\). She uses an approximating distribution to calculate the probability of obtaining no 6s in 450 throws. Find the percentage error in using the approximating distribution for this calculation.
  2. Johan claims that a certain six-sided die is biased so that it shows a 6 less often than it would if the die were fair. In order to test this claim, the die is thrown 25 times and it shows a 6 on only 2 throws. Test at the \(10 \%\) significance level whether Johan's claim is justified.
CAIE S2 2015 November Q6
9 marks Standard +0.8
6 Parcels arriving at a certain office have weights \(W \mathrm {~kg}\), where the random variable \(W\) has mean \(\mu\) and standard deviation 0.2 . The value of \(\mu\) used to be 2.60 , but there is a suspicion that this may no longer be true. In order to test at the 5\% significance level whether the value of \(\mu\) has increased, a random sample of 75 parcels is chosen. You may assume that the standard deviation of \(W\) is unchanged.
  1. The mean weight of the 75 parcels is found to be 2.64 kg . Carry out the test.
  2. Later another test of the same hypotheses at the \(5 \%\) significance level, with another random sample of 75 parcels, is carried out. Given that the value of \(\mu\) is now 2.68 , calculate the probability of a Type II error.
CAIE S2 2015 November Q7
9 marks Standard +0.3
7 The diameter, in cm, of pistons made in a certain factory is denoted by \(X\), where \(X\) is normally distributed with mean \(\mu\) and variance \(\sigma ^ { 2 }\). The diameters of a random sample of 100 pistons were measured, with the following results. $$n = 100 \quad \Sigma x = 208.7 \quad \Sigma x ^ { 2 } = 435.57$$
  1. Calculate unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\). The pistons are designed to fit into cylinders. The internal diameter, in cm , of the cylinders is denoted by \(Y\), where \(Y\) has an independent normal distribution with mean 2.12 and variance 0.000144 . A piston will not fit into a cylinder if \(Y - X < 0.01\).
  2. Using your answers to part (i), find the probability that a randomly chosen piston will not fit into a randomly chosen cylinder.
CAIE S2 2016 November Q1
3 marks Easy -1.3
1 The weights, in kilograms, of a random sample of eight 16-year old males are given below. $$\begin{array} { l l l l l l l l } 58.9 & 63.5 & 62.7 & 59.4 & 66.9 & 68.0 & 60.4 & 68.2 \end{array}$$ Find unbiased estimates of the population mean and variance of the weights of all 16 -year old males.
CAIE S2 2016 November Q2
5 marks Standard +0.3
2 A die has six faces numbered \(1,2,3,4,5,6\). Manjit suspects that the die is biased so that it shows a six on fewer throws than it would if it were fair. In order to test her suspicion, she throws the die a certain number of times and counts the number of sixes.
  1. State suitable null and alternative hypotheses for Manjit's test.
  2. There are no sixes in the first 15 throws. Show that this result is not significant at the \(5 \%\) level.
  3. Find the smallest value of \(n\) such that, if there are no sixes in the first \(n\) throws, this result is significant at the 5\% level.
CAIE S2 2016 November Q3
7 marks Standard +0.3
3 Particles are emitted randomly from a radioactive substance at a constant average rate of 3.6 per minute. Find the probability that
  1. more than 3 particles are emitted during a 20 -second period,
  2. more than 240 particles are emitted during a 1-hour period.
CAIE S2 2016 November Q4
7 marks Standard +0.3
4 Each week a farmer sells \(X\) litres of milk and \(Y \mathrm {~kg}\) of cheese, where \(X\) and \(Y\) have the independent distributions \(\mathrm { N } \left( 1520,53 ^ { 2 } \right)\) and \(\mathrm { N } \left( 175,12 ^ { 2 } \right)\) respectively.
  1. Find the mean and standard deviation of the total amount of milk that the farmer sells in 4 randomly chosen weeks. During a year when milk prices are low, the farmer makes a loss of 2 cents per litre on milk and makes a profit of 21 cents per kg on cheese, so the farmer's overall weekly profit is \(( 21 Y - 2 X )\) cents.
  2. Find the probability that, in a randomly chosen week, the farmer's overall profit is positive.
CAIE S2 2016 November Q5
8 marks Standard +0.3
5
  1. The masses, in grams, of certain tomatoes are normally distributed with standard deviation 9 grams. A random sample of 100 tomatoes has a sample mean of 63 grams. Find a \(90 \%\) confidence interval for the population mean mass of these tomatoes.
  2. The masses, in grams, of certain potatoes are normally distributed with known population standard deviation but unknown population mean. A random sample of potatoes is taken in order to find a confidence interval for the population mean. Using a sample of size 50 , a \(95 \%\) confidence interval is found to have width 8 grams.
    1. Using another sample of size 50 , an \(\alpha \%\) confidence interval has width 4 grams. Find \(\alpha\).
    2. Find the sample size \(n\), such that a \(95 \%\) confidence interval has width 4 grams.
CAIE S2 2016 November Q6
9 marks Moderate -0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{e0ad3268-117e-4a0c-942d-84ee148d8907-3_371_504_260_534} \includegraphics[max width=\textwidth, alt={}, center]{e0ad3268-117e-4a0c-942d-84ee148d8907-3_373_495_260_1123} \includegraphics[max width=\textwidth, alt={}, center]{e0ad3268-117e-4a0c-942d-84ee148d8907-3_371_497_776_534} \includegraphics[max width=\textwidth, alt={}, center]{e0ad3268-117e-4a0c-942d-84ee148d8907-3_367_488_778_1128} The diagrams show the probability density functions of four random variables \(W , X , Y\) and \(Z\). Each of the four variables takes values between 0 and 3 only, and their medians are \(m _ { W } , m _ { X } , m _ { Y }\) and \(m _ { Z }\) respectively.
  1. List \(m _ { W } , m _ { X } , m _ { Y }\) and \(m _ { Z }\) in order of size, starting with the largest.
  2. The probability density function of \(X\) is given by $$f ( x ) = \begin{cases} \frac { 4 } { 81 } x ^ { 3 } & 0 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$
    1. Show that \(\mathrm { E } ( X ) = \frac { 12 } { 5 }\).
    2. Calculate \(\mathrm { P } ( X > \mathrm { E } ( X ) )\).
    3. Write down the value of \(\mathrm { P } ( X < 2 \mathrm { E } ( X ) )\).
CAIE S2 2016 November Q7
11 marks Standard +0.3
7 In the past the time, in minutes, taken for a particular rail journey has been found to have mean 20.5 and standard deviation 1.2. Some new railway signals are installed. In order to test whether the mean time has decreased, a random sample of 100 times for this journey are noted. The sample mean is found to be 20.3 minutes. You should assume that the standard deviation is unchanged.
  1. Carry out a significance test, at the \(4 \%\) level, of whether the population mean time has decreased. Later another significance test of the same hypotheses, using another random sample of size 100 , is carried out at the \(4 \%\) level.
  2. Given that the population mean is now 20.1, find the probability of a Type II error.
  3. State what is meant by a Type II error in this context.
CAIE S2 2016 November Q6
9 marks Moderate -0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{1e20bcc7-a501-4df0-9d49-cca2db4c279a-3_371_504_260_534} \includegraphics[max width=\textwidth, alt={}, center]{1e20bcc7-a501-4df0-9d49-cca2db4c279a-3_373_495_260_1123} \includegraphics[max width=\textwidth, alt={}, center]{1e20bcc7-a501-4df0-9d49-cca2db4c279a-3_371_497_776_534} \includegraphics[max width=\textwidth, alt={}, center]{1e20bcc7-a501-4df0-9d49-cca2db4c279a-3_367_488_778_1128} The diagrams show the probability density functions of four random variables \(W , X , Y\) and \(Z\). Each of the four variables takes values between 0 and 3 only, and their medians are \(m _ { W } , m _ { X } , m _ { Y }\) and \(m _ { Z }\) respectively.
  1. List \(m _ { W } , m _ { X } , m _ { Y }\) and \(m _ { Z }\) in order of size, starting with the largest.
  2. The probability density function of \(X\) is given by $$f ( x ) = \begin{cases} \frac { 4 } { 81 } x ^ { 3 } & 0 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$
    1. Show that \(\mathrm { E } ( X ) = \frac { 12 } { 5 }\).
    2. Calculate \(\mathrm { P } ( X > \mathrm { E } ( X ) )\).
    3. Write down the value of \(\mathrm { P } ( X < 2 \mathrm { E } ( X ) )\).
CAIE S2 2016 November Q1
3 marks Easy -1.2
1 The random variable \(X\) has the distribution \(\operatorname { Po } ( 3.5 )\). Find \(\mathrm { P } ( X < 3 )\).
CAIE S2 2016 November Q2
3 marks Easy -1.8
2 Dominic wishes to choose a random sample of five students from the 150 students in his year. He numbers the students from 1 to 150 . Then he uses his calculator to generate five random numbers between 0 and 1 . He multiplies each random number by 150 and rounds up to the next whole number to give a student number.
  1. Dominic's first random number is 0.392 . Find the student number that is produced by this random number.
  2. Dominic's second student number is 104 . Find a possible random number that would produce this student number.
  3. Explain briefly why five random numbers may not be enough to produce a sample of five student numbers.
CAIE S2 2016 November Q3
5 marks Standard +0.3
3 A men's triathlon consists of three parts: swimming, cycling and running. Competitors' times, in minutes, for the three parts can be modelled by three independent normal variables with means 34.0, 87.1 and 56.9, and standard deviations 3.2, 4.1 and 3.8, respectively. For each competitor, the total of his three times is called the race time. Find the probability that the mean race time of a random sample of 15 competitors is less than 175 minutes.
CAIE S2 2016 November Q4
5 marks Challenging +1.2
4 The manufacturer of a tablet computer claims that the mean battery life is 11 hours. A consumer organisation wished to test whether the mean is actually greater than 11 hours. They invited a random sample of members to report the battery life of their tablets. They then calculated the sample mean. Unfortunately a fire destroyed the records of this test except for the following partial document. \includegraphics[max width=\textwidth, alt={}, center]{c460afa4-1387-421d-87ac-74a64be99714-2_467_593_1612_776} Given that the population of battery lives is normally distributed with standard deviation 1.6 hours, find the set of possible values of the sample size, \(n\).
CAIE S2 2016 November Q5
8 marks Moderate -0.3
5 It is claimed that \(30 \%\) of packets of Froogum contain a free gift. Andre thinks that the actual proportion is less than \(30 \%\) and he decides to carry out a hypothesis test at the \(5 \%\) significance level. He buys 20 packets of Froogum and notes the number of free gifts he obtains.
  1. State null and alternative hypotheses for the test.
  2. Use a binomial distribution to find the probability of a Type I error. Andre finds that 3 of the 20 packets contain free gifts.
  3. Carry out the test.
CAIE S2 2016 November Q6
8 marks Standard +0.3
6 A variable \(X\) takes values \(1,2,3,4,5\), and these values are generated at random by a machine. Each value is supposed to be equally likely, but it is suspected that the machine is not working properly. A random sample of 100 values of \(X\), generated by the machine, gives the following results. $$n = 100 \quad \Sigma x = 340 \quad \Sigma x ^ { 2 } = 1356$$
  1. Find a 95\% confidence interval for the population mean of the values generated by the machine.
  2. Use your answer to part (i) to comment on whether the machine may be working properly.
CAIE S2 2016 November Q7
9 marks Standard +0.3
7 Men arrive at a clinic independently and at random, at a constant mean rate of 0.2 per minute. Women arrive at the same clinic independently and at random, at a constant mean rate of 0.3 per minute.
  1. Find the probability that at least 2 men and at least 3 women arrive at the clinic during a 5 -minute period.
  2. Find the probability that fewer than 36 people arrive at the clinic during a 1-hour period.
CAIE S2 2016 November Q8
9 marks Standard +0.3
8 \includegraphics[max width=\textwidth, alt={}, center]{c460afa4-1387-421d-87ac-74a64be99714-4_302_517_276_427} \includegraphics[max width=\textwidth, alt={}, center]{c460afa4-1387-421d-87ac-74a64be99714-4_304_508_274_1215} \includegraphics[max width=\textwidth, alt={}, center]{c460afa4-1387-421d-87ac-74a64be99714-4_305_506_717_431} \includegraphics[max width=\textwidth, alt={}, center]{c460afa4-1387-421d-87ac-74a64be99714-4_302_504_717_1217} The diagrams show the probability density functions of four random variables \(W , X , Y\) and \(Z\). Each of the four variables takes values between - 3 and 3 only, and their standard deviations are \(\sigma _ { W } , \sigma _ { X } , \sigma _ { Y }\) and \(\sigma _ { Z }\) respectively.
  1. List \(\sigma _ { W } , \sigma _ { X } , \sigma _ { Y }\) and \(\sigma _ { Z }\) in order of size, starting with the largest.
  2. The probability density function of \(X\) is given by $$f ( x ) = \begin{cases} \frac { 1 } { 18 } x ^ { 2 } & - 3 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$
    1. Show that \(\sigma _ { X } = 2.32\) correct to 3 significant figures.
    2. Calculate \(\mathrm { P } \left( X > \sigma _ { X } \right)\).
    3. Write down the value of \(\mathrm { P } \left( X > 2 \sigma _ { X } \right)\).
CAIE S2 2019 June Q6
9 marks Standard +0.3
  1. Show that \(b = \frac { a } { a - 1 }\).
  2. Given that the median of \(X\) is \(\frac { 3 } { 2 }\), find the values of \(a\) and \(b\).
  3. Use your values of \(a\) and \(b\) from part (ii) to find \(\mathrm { E } ( X )\).
CAIE S2 2021 June Q1
4 marks Standard +0.3
Accidents at two factories occur randomly and independently. On average, the numbers of accidents per month are 3.1 at factory \(A\) and 1.7 at factory \(B\). Find the probability that the total number of accidents in the two factories during a 2-month period is more than 3. [4]
CAIE S2 2021 June Q2
4 marks Moderate -0.8
The time, in minutes, taken by students to complete a test has the distribution \(\text{N}(125, 36)\).
  1. Find the probability that the mean time taken to complete the test by a random sample of 40 students is less than 123 minutes. [3]
  2. Explain whether it was necessary to use the Central Limit theorem in the solution to part (a). [1]
CAIE S2 2021 June Q3
2 marks Moderate -0.8
The graph of the probability density function of a random variable \(X\) is symmetrical about the line \(x = 4\). Given that \(\text{P}(X < 5) = \frac{20}{39}\), find \(\text{P}(3 < X < 5)\). [2]