Questions — CAIE S2 (717 questions)

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CAIE S2 2016 November Q2
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
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
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
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
6
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\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 $$f ( x ) = \begin{cases} \frac { 4 } { 81 } x ^ { 3 } & 0 \leqslant x \leqslant 3
    0 & \text { otherwise } \end{cases}$$ (a) Show that \(\mathrm { E } ( X ) = \frac { 12 } { 5 }\).
    (b) Calculate \(\mathrm { P } ( X > \mathrm { E } ( X ) )\).
    (c) Write down the value of \(\mathrm { P } ( X < 2 \mathrm { E } ( X ) )\).
CAIE S2 2016 November Q7
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
6
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\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 $$f ( x ) = \begin{cases} \frac { 4 } { 81 } x ^ { 3 } & 0 \leqslant x \leqslant 3
    0 & \text { otherwise } \end{cases}$$ (a) Show that \(\mathrm { E } ( X ) = \frac { 12 } { 5 }\).
    (b) Calculate \(\mathrm { P } ( X > \mathrm { E } ( X ) )\).
    (c) Write down the value of \(\mathrm { P } ( X < 2 \mathrm { E } ( X ) )\).
CAIE S2 2016 November Q1
1 The random variable \(X\) has the distribution \(\operatorname { Po } ( 3.5 )\). Find \(\mathrm { P } ( X < 3 )\).
CAIE S2 2016 November Q2
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
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
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
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
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
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
8
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\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 $$f ( x ) = \begin{cases} \frac { 1 } { 18 } x ^ { 2 } & - 3 \leqslant x \leqslant 3
    0 & \text { otherwise } \end{cases}$$ (a) Show that \(\sigma _ { X } = 2.32\) correct to 3 significant figures.
    (b) Calculate \(\mathrm { P } \left( X > \sigma _ { X } \right)\).
    (c) Write down the value of \(\mathrm { P } \left( X > 2 \sigma _ { X } \right)\).
CAIE S2 2016 June Q6
  1. Find \(\mathrm { P } ( X + Y = 4 )\). A random sample of 75 values of \(X\) is taken.
  2. State the approximate distribution of the sample mean, \(\bar { X }\), including the values of the parameters.
  3. Hence find the probability that the sample mean is more than 1.7.
  4. Explain whether the Central Limit theorem was needed to answer part (ii).
CAIE S2 2019 June Q6
  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 \(\mathrm { E } ( X )\).