Questions — CAIE S2 (717 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
CAIE S2 2008 November Q7
7 The time in hours taken for clothes to dry can be modelled by the continuous random variable with probability density function given by $$f ( t ) = \begin{cases} k \sqrt { } t & 1 \leqslant t \leqslant 4
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = \frac { 3 } { 14 }\).
  2. Find the mean time taken for clothes to dry.
  3. Find the median time taken for clothes to dry.
  4. Find the probability that the time taken for clothes to dry is between the mean time and the median time.
CAIE S2 2009 November Q2
2 The lengths of sewing needles in travel sewing kits are distributed normally with mean \(\mu \mathrm { mm }\) and standard deviation 1.5 mm . A random sample of \(n\) needles is taken. Find the smallest value of \(n\) such that the width of a \(95 \%\) confidence interval for the population mean is at most 1 mm .
CAIE S2 2009 November Q3
3 The weights of pebbles on a beach are normally distributed with mean 48.5 grams and standard deviation 12.4 grams.
  1. Find the probability that the mean weight of a random sample of 5 pebbles is greater than 51 grams.
  2. The probability that the mean weight of a random sample of \(n\) pebbles is less than 51.6 grams is 0.9332 . Find the value of \(n\).
CAIE S2 2009 November Q4
4 The number of severe floods per year in a certain country over the last 100 years has followed a Poisson distribution with mean 1.8. Scientists suspect that global warming has now increased the mean. A hypothesis test, at the \(5 \%\) significance level, is to be carried out to test this suspicion. The number of severe floods, \(X\), that occur next year will be used for the test.
  1. Show that the rejection region for the test is \(X > 4\).
  2. Find the probability of making a Type II error if the mean number of severe floods is now actually 2.3.
CAIE S2 2009 November Q5
5 The continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} k \cos x & 0 \leqslant x \leqslant \frac { 1 } { 4 } \pi
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = \sqrt { } 2\).
  2. Find \(\mathrm { P } ( X > 0.4 )\).
  3. Find the upper quartile of \(X\).
  4. Find the probability that exactly 3 out of 5 random observations of \(X\) have values greater than the upper quartile.
CAIE S2 2009 November Q6
6 Photographers often need to take many photographs of families until they find a photograph which everyone in the family likes. The number of photographs taken until obtaining one which everybody likes has mean 15.2. A new photographer claims that she can obtain a photograph which everybody likes with fewer photographs taken. To test at the \(10 \%\) level of significance whether this claim is justified, the numbers of photographs, \(x\), taken by the new photographer with a random sample of 60 families are recorded. The results are summarised by \(\Sigma x = 890\) and \(\Sigma x ^ { 2 } = 13780\).
  1. Calculate unbiased estimates of the population mean and variance of the number of photographs taken by the new photographer.
  2. State null and alternative hypotheses for the test, and state also the probability that the test results in a Type I error. Say what a Type I error means in the context of the question.
  3. Carry out the test.
CAIE S2 2009 November Q7
7 The volume of liquid in cans of cola is normally distributed with mean 330 millilitres and standard deviation 5.2 millilitres. The volume of liquid in bottles of tonic water is normally distributed with mean 500 millilitres and standard deviation 7.1 millilitres.
  1. Find the probability that 3 randomly chosen cans of cola contain less liquid than 2 randomly chosen bottles of tonic water.
  2. A new drink is made by mixing the contents of 2 cans of cola with half a bottle of tonic water. Find the probability that the volume of the new drink is more than 900 millilitres.
CAIE S2 2009 November Q1
1 There are 18 people in Millie's class. To choose a person at random she numbers the people in the class from 1 to 18 and presses the random number button on her calculator to obtain a 3-digit decimal. Millie then multiplies the first digit in this decimal by two and chooses the person corresponding to this new number. Decimals in which the first digit is zero are ignored.
  1. Give a reason why this is not a satisfactory method of choosing a person. Millie obtained a random sample of 5 people of her own age by a satisfactory sampling method and found that their heights in metres were \(1.66,1.68,1.54,1.65\) and 1.57 . Heights are known to be normally distributed with variance \(0.0052 \mathrm {~m} ^ { 2 }\).
  2. Find a \(98 \%\) confidence interval for the mean height of people of Millie's age.
CAIE S2 2009 November Q2
2 A computer user finds that unwanted emails arrive randomly at a uniform average rate of 1.27 per hour.
  1. Find the probability that more than 1 unwanted email arrives in a period of 5 hours.
  2. Find the probability that more than 850 unwanted emails arrive in a period of 700 hours.
CAIE S2 2009 November Q3
3 An airline knows that some people who have bought tickets may not arrive for the flight. The airline therefore sells more tickets than the number of seats that are available. For one flight there are 210 seats available and 213 people have bought tickets. The probability of any person who has bought a ticket not arriving for the flight is \(\frac { 1 } { 50 }\).
  1. By considering the number of people who do not arrive for the flight, use a suitable approximation to calculate the probability that more people will arrive than there are seats available. Independently, on another flight for which 135 people have bought tickets, the probability of any person not arriving is \(\frac { 1 } { 75 }\).
  2. Calculate the probability that, for both these flights, the total number of people who do not arrive is 5 .
CAIE S2 2009 November Q4
4 It is not known whether a certain coin is fair or biased. In order to perform a hypothesis test, Raj tosses the coin 10 times and counts the number of heads obtained. The probability of obtaining a head on any throw is denoted by \(p\).
  1. The null hypothesis is \(p = 0.5\). Find the acceptance region for the test, given that the probability of a Type I error is to be at most 0.1 .
  2. Calculate the probability of a Type II error in this test if the actual value of \(p\) is 0.7 .
CAIE S2 2009 November Q5
5 The masses of packets of cornflakes are normally distributed with standard deviation 11 g . A random sample of 20 packets was weighed and found to have a mean mass of 746 g .
  1. Test at the \(4 \%\) significance level whether there is enough evidence to conclude that the population mean mass is less than 750 g .
  2. Given that the population mean mass actually is 750 g , find the smallest possible sample size, \(n\), for which it is at least \(97 \%\) certain that the mean mass of the sample exceeds 745 g .
CAIE S2 2009 November Q6
6 The continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { 1 } { 3 } x ( k - x ) & 1 \leqslant x \leqslant 2
0 & \text { otherwise } \end{cases}$$
  1. Show that the value of \(k\) is \(\frac { 32 } { 9 }\).
  2. Find \(\mathrm { E } ( X )\).
  3. Is the median less than or greater than 1.5? Justify your answer numerically.
CAIE S2 2009 November Q7
7
  1. Random variables \(Y\) and \(X\) are related by \(Y = a + b X\), where \(a\) and \(b\) are constants and \(b > 0\). The standard deviation of \(Y\) is twice the standard deviation of \(X\). The mean of \(Y\) is 7.92 and is 0.8 more than the mean of \(X\). Find the values of \(a\) and \(b\).
  2. Random variables \(R\) and \(S\) are such that \(R \sim \mathrm {~N} \left( \mu , 2 ^ { 2 } \right)\) and \(S \sim \mathrm {~N} \left( 2 \mu , 3 ^ { 2 } \right)\). It is given that \(\mathrm { P } ( R + S > 1 ) = 0.9\).
    1. Find \(\mu\).
    2. Hence find \(\mathrm { P } ( S > R )\).
CAIE S2 2010 November Q1
1 In a survey of 1000 randomly chosen adults, 605 said that they used email. Calculate a \(90 \%\) confidence interval for the proportion of adults in the whole population who use email.
CAIE S2 2010 November Q2
2 People arrive randomly and independently at a supermarket checkout at an average rate of 2 people every 3 minutes.
  1. Find the probability that exactly 4 people arrive in a 5 -minute period. At another checkout in the same supermarket, people arrive randomly and independently at an average rate of 1 person each minute.
  2. Find the probability that a total of fewer than 3 people arrive at the two checkouts in a 3 -minute period.
CAIE S2 2010 November Q3
3 A book contains 40000 words. For each word, the probability that it is printed wrongly is 0.0001 and these errors occur independently. The number of words printed wrongly in the book is represented by the random variable \(X\).
  1. State the exact distribution of \(X\), including the values of any parameters.
  2. State an approximate distribution for \(X\), including the values of any parameters, and explain why this approximate distribution is appropriate.
  3. Use this approximate distribution to find the probability that there are more than 3 words printed wrongly in the book.
CAIE S2 2010 November Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{c7cbd61b-9a62-494a-b595-f624ec5c0bea-2_351_561_1562_794} The diagram shows the graph of the probability density function, f , of a random variable \(X\) which takes values between 0 and 2 only.
  1. Find \(\mathrm { P } ( 1 < X < 1.5 )\).
  2. Find the median of \(X\).
  3. Find \(\mathrm { E } ( X )\).
CAIE S2 2010 November Q5
5 The marks of candidates in Mathematics and English in 2009 were represented by the independent random variables \(X\) and \(Y\) with distributions \(\mathrm { N } \left( 28,5.6 ^ { 2 } \right)\) and \(\mathrm { N } \left( 52,12.4 ^ { 2 } \right)\) respectively. Each candidate's marks were combined to give a final mark \(F\), where \(F = X + \frac { 1 } { 2 } Y\).
  1. Find \(\mathrm { E } ( F )\) and \(\operatorname { Var } ( F )\).
  2. The final marks of a random sample of 10 candidates from Grinford in 2009 had a mean of 49. Test at the 5\% significance level whether this result suggests that the mean final mark of all candidates from Grinford in 2009 was lower than elsewhere.
CAIE S2 2010 November Q6
6 It is claimed that a certain 6-sided die is biased so that it is more likely to show a six than if it was fair. In order to test this claim at the \(10 \%\) significance level, the die is thrown 10 times and the number of sixes is noted.
  1. Given that the die shows a six on 3 of the 10 throws, carry out the test. On another occasion the same test is carried out again.
  2. Find the probability of a Type I error.
  3. Explain what is meant by a Type II error in this context.
CAIE S2 2010 November Q7
7
  1. Give a reason why sampling would be required in order to reach a conclusion about
    1. the mean height of adult males in England,
    2. the mean weight that can be supported by a single cable of a certain type without the cable breaking.
  2. The weights, in kg , of sacks of potatoes are represented by the random variable \(X\) with mean \(\mu\) and standard deviation \(\sigma\). The weights of a random sample of 500 sacks of potatoes are found and the results are summarised below. $$n = 500 , \quad \Sigma x = 9850 , \quad \Sigma x ^ { 2 } = 194125 .$$
    1. Calculate unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\).
    2. A further random sample of 60 sacks of potatoes is taken. Using your values from part (b) (i), find the probability that the mean weight of this sample exceeds 19.73 kg .
    3. Explain whether it was necessary to use the Central Limit Theorem in your calculation in part (b) (ii).
CAIE S2 2010 November Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{0784d885-5710-4eb4-8cf8-2582122bf7ed-2_351_554_1562_794} The diagram shows the graph of the probability density function, f , of a random variable \(X\) which takes values between 0 and 2 only.
  1. Find \(\mathrm { P } ( 1 < X < 1.5 )\).
  2. Find the median of \(X\).
  3. Find \(\mathrm { E } ( X )\).
CAIE S2 2010 November Q1
1 A random variable has the distribution \(\mathrm { Po } ( 31 )\). Name an appropriate approximating distribution and state the mean and standard deviation of this approximating distribution.
CAIE S2 2010 November Q2
2 The editor of a magazine wishes to obtain the views of a random sample of readers about the future of the magazine.
  1. A sub-editor proposes that they include in one issue of the magazine a questionnaire for readers to complete and return. Give two reasons why the readers who return the questionnaire would not form a random sample. The editor decides to use a table of random numbers to select a random sample of 50 readers from the 7302 regular readers. These regular readers are numbered from 1 to 7302 . The first few random numbers which the editor obtains from the table are as follows. $$49757 \quad 80239 \quad 52038 \quad 60882$$
  2. Use these random numbers to select the first three members in the sample.
CAIE S2 2010 November Q3
3 The masses of sweets produced by a machine are normally distributed with mean \(\mu\) grams and standard deviation 1.0 grams. A random sample of 65 sweets produced by the machine has a mean mass of 29.6 grams.
  1. Find a \(99 \%\) confidence interval for \(\mu\). The manufacturer claims that the machine produces sweets with a mean mass of 30 grams.
  2. Use the confidence interval found in part (i) to draw a conclusion about this claim.
  3. Another random sample of 65 sweets produced by the machine is taken. This sample gives a \(99 \%\) confidence interval that leads to a different conclusion from that found in part (ii). Assuming that the value of \(\mu\) has not changed, explain how this can be possible.