Questions S2 (1597 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure 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 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 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 Stats 1 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
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 2019 November Q3
3 The times, in minutes, taken by competitors to complete a puzzle have mean \(\mu\) and standard deviation 3 . The times taken by a random sample of 10 competitors are noted and the results are given below.
\(\begin{array} { l l l } 25.2 & 26.8 & 18.5 \end{array}\)
25.5
30.1
\(28.9 \quad 27.0\)
\(26.1 \quad 26.0\)
24.9
  1. Stating a necessary assumption, calculate a \(97 \%\) confidence interval for \(\mu\).
  2. Two more random samples, each of 10 competitors, are taken. Their times are used to calculate two more \(97 \%\) confidence intervals for \(\mu\). Find the probability that neither of these intervals contains the true value of \(\mu\).
CAIE S2 2019 November Q4
6 marks
4 A train company claims that \(92 \%\) of trains on a particular line arrive on time. Sanjeep suspects that the true percentage is less than \(92 \%\). He chooses a random sample of 20 trains on this line and finds that exactly 16 of them arrive on time. Making an assumption that should be stated, test at the 5\% significance level whether Sanjeep's suspicion is justified.
[0pt] [6]
CAIE S2 2019 November Q5
5
  1. The random variable \(X\) has the distribution \(\mathrm { B } ( 300,0.01 )\). Use a Poisson approximation to find \(\mathrm { P } ( 2 < X < 6 )\).
  2. The random variable \(Y\) has the distribution \(\mathrm { Po } ( \lambda )\), and \(\mathrm { P } ( Y = 0 ) = \mathrm { P } ( Y = 2 )\). Find \(\lambda\).
  3. The random variable \(Z\) has the distribution \(\mathrm { Po } ( 5.2 )\) and it is given that \(\mathrm { P } ( Z = n ) < \mathrm { P } ( Z = n + 1 )\).
    (a) Write down an inequality in \(n\).
    (b) Hence or otherwise find the largest possible value of \(n\).
CAIE S2 2019 November Q6
6 A random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} k \left( 3 x - x ^ { 2 } \right) & 0 \leqslant x \leqslant 3
0 & \text { otherwise } \end{cases}$$
  1. Show that \(k = \frac { 2 } { 9 }\).
  2. Find \(\mathrm { P } ( 1 \leqslant X \leqslant 2 )\).
  3. Find \(\operatorname { Var } ( X )\).
CAIE S2 2019 November Q7
7 Bob is a self-employed builder. In the past his weekly income had mean \(
) 546\( and standard deviation \)\\( 120\). Following a change in Bob's working pattern, his mean weekly income for 40 randomly chosen weeks was \(
) 581\(. You should assume that the standard deviation remains unchanged at \)\\( 120\).
  1. Test at the \(2.5 \%\) significance level whether Bob's mean weekly income has increased.
    Bob finds his mean weekly income for another random sample of 40 weeks and carries out a similar test at the \(2.5 \%\) significance level.
  2. Given that Bob's mean weekly income is now in fact \(
    ) 595$, 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 2020 November Q3
3 The masses, in kilograms, of female and male animals of a certain species have the distributions \(\mathrm { N } \left( 102,27 ^ { 2 } \right)\) and \(\mathrm { N } \left( 170,55 ^ { 2 } \right)\) respectively. Find the probability that a randomly chosen female has a mass that is less than half the mass of a randomly chosen male.
\includegraphics[max width=\textwidth, alt={}, center]{fb305858-2d96-4a5d-b1a9-a965c248fb8d-06_76_1659_484_244}
CAIE S2 2020 November Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{fb305858-2d96-4a5d-b1a9-a965c248fb8d-07_316_984_260_577} The diagram shows the probability density function, \(\mathrm { f } ( x )\), of a random variable \(X\). For \(0 \leqslant x \leqslant a\), \(\mathrm { f } ( x ) = k\); elsewhere \(\mathrm { f } ( x ) = 0\).
  1. Express \(k\) in terms of \(a\).
  2. Given that \(\operatorname { Var } ( X ) = 3\), find \(a\).
CAIE S2 2020 November Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{4a5f9f7e-b045-4c6f-8bda-6c4067668da2-04_332_1100_260_520} A random variable \(X\) takes values between 0 and 3 only and has probability density function as shown in the diagram, where \(c\) is a constant.
  1. Show that \(c = \frac { 2 } { 3 }\).
  2. Find \(\mathrm { P } ( X > 2 )\).
  3. Calculate \(\mathrm { E } ( X )\).
CAIE S2 2020 November Q3
3 The masses, in kilograms, of female and male animals of a certain species have the distributions \(\mathrm { N } \left( 102,27 ^ { 2 } \right)\) and \(\mathrm { N } \left( 170,55 ^ { 2 } \right)\) respectively. Find the probability that a randomly chosen female has a mass that is less than half the mass of a randomly chosen male.
\includegraphics[max width=\textwidth, alt={}, center]{937c15d2-fb12-4af8-96d3-c54c81d771ba-06_76_1659_484_244}
CAIE S2 2020 November Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{937c15d2-fb12-4af8-96d3-c54c81d771ba-07_316_984_260_577} The diagram shows the probability density function, \(\mathrm { f } ( x )\), of a random variable \(X\). For \(0 \leqslant x \leqslant a\), \(\mathrm { f } ( x ) = k\); elsewhere \(\mathrm { f } ( x ) = 0\).
  1. Express \(k\) in terms of \(a\).
  2. Given that \(\operatorname { Var } ( X ) = 3\), find \(a\).
CAIE S2 2020 Specimen Q3
3 Th m brg calls receid d at a small call cen re \(\mathbf { h }\) s a Pósso d strib in with mean
CAIE S2 2020 Specimen Q4
4 marks
4 Tb lifetimes, in b s, 6 L ie lig \(\mathbf { b }\) b ad Ee rlw lig \(\mathbf { b }\) b \(\mathbf { b }\) \& tb id \(\mathbf { P } \mathbf { d }\) n id strib in \(\mathrm { N } \left( \mathrm { LS } ^ { 2 } \right)\) adN ( \(\mathrm { L } ^ { 2 }\) ) resp ctie ly.
  1. Fid th pb b lity th t to to al 6 th lifetimes 6 fie rach ly cb en \(L \mathbf { b }\) ie \(\mathbf { b }\) b is less th \(\mathrm { HB } \quad \mathrm { Ch } \quad \mathrm { S }\).
    [0pt] [4]
  2. Fid th pb b lity th tth lifetime 6 a rach lycb en En rlw b b is at least th ee times th t 6 a rach lyc b erL b ir b b
CAIE S2 2020 Specimen Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{ffc7febd-0df7-4cb6-ac6c-63779e032617-08_270_648_251_712} Th diag am sh s th g a to th pb ab lityd nsityf n tiff, to a rach \& riab e \(X\),w b re $$f ( x ) = \begin{cases} \frac { 2 } { 9 } \left( 3 x - x ^ { 2 } \right) & 0 \leqslant x \leqslant 3
0 & \text { otherwise } \end{cases}$$
  1. State th \& le \(6 \mathrm { E } ( X )\) aff id \(\operatorname { Var } ( X )\).
  2. State th le \(6 \mathrm { P } (
    ) \leqslant X \leqslant 4\(.
  3. Giv it h \)\mathrm { P } \left( 1 \leqslant X \leqslant \mathcal { P } = \frac { 13 } { 27 } \right.\(, f idP \)( X > \mathcal { P }$.
CAIE S2 2020 Specimen Q7
7 Th mean weit 6 bg 6 carrb s is \(\mu \mathrm { k }\) lg ams. An in p cto wish s to test wh th \(\mathrm { r } \mu = 20\) He weits a rand sampe 6 tb g ach s resh ts are sm marised s fb low s. $$\Sigma x = 430 \quad \Sigma x ^ { 2 } = \theta$$ Carryd th test at the \%o sig fican e lee 1. If B e th follw ig lin dpg to cm p ete th an wer(s) to ay q stin (s), th q stin \(\mathrm { m } \quad \mathbf { b } \quad \mathrm { r } ( \mathrm { s } )\) ms tb clearlys n n
CAIE S2 Specimen Q1
1 Failures of two computers occur at random and independently. On average the first computer fails 1.2 times per year and the second computer fails 2.3 times per year. Find the probability that the total number of failures by the two computers in a 6-month period is more than 1 and less than 4 .
CAIE S2 Specimen Q2
2 The mean and standard deviation of the time spent by people in a certain library are 29 minutes and 6 minutes respectively.
  1. Find the probability that the mean time spent in the library by a random sample of 120 people is more than 30 minutes.
  2. Explain whether it was necessary to assume that the time spent by people in the library is normally distributed in the solution to part (i).
CAIE S2 Specimen Q3
3 Jagdeesh measured the lengths, \(x\) minutes, of 60 randomly chosen lectures. His results are summarised below.
  1. Calculate unbiased estimates of the population mean and variance.
  2. Calculate a \(98 \%\) confidence interval for the population mean.
CAIE S2 Specimen Q4
4 A random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} k ( 3 - x ) & 1 \leqslant x \leqslant 2
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = \frac { 2 } { 3 }\).
  2. Find the median of \(X\).
CAIE S2 Specimen Q5
5 On average, 1 in 2500 adults has a certain medical condition.
  1. Use a suitable approximation to find the probability that, in a random sample of 4000 people, more than 3 have this condition.
  2. In a random sample of \(n\) people, where \(n\) is large, the probability that none has the condition is less than 0.05 . Find the smallest possible value of \(n\).
CAIE S2 Specimen Q6
4 marks
6 The weights, in kilograms, of men and women have the distributions \(\mathrm { N } \left( 78,7 ^ { 2 } \right)\) and \(\mathrm { N } \left( 66,5 ^ { 2 } \right)\) respectively.
  1. The maximum load that a certain cable car can carry safely is 1200 kg . If 9 randomly chosen men and 7 randomly chosen women enter the cable car, find the probability that the cable car can operate safely.
  2. Find the probability that a randomly chosen woman weighs more than a randomly chosen man. [4]
CAIE S2 Specimen Q7
7 At a certain hospital it was found that the probability that a patient did not arrive for an appointment was 0.2 . The hospital carries out some publicity in the hope that this probability will be reduced. They wish to test whether the publicity has worked.
  1. It is suggested that the first 30 appointments on a Monday should be used for the test. Give a reason why this is not an appropriate sample.
    A suitable sample of 30 appointments is selected and the number of patients that do not arrive is noted. This figure is used to carry out a test at the 5\% significance level.
  2. Explain why the test is one-tail and state suitable null and alternative hypotheses.
  3. State what is meant by a Type I error in this context.
  4. Use the binomial distribution to find the critical region, and find the probability of a Type I error.
  5. In fact 3 patients out of the 30 do not arrive. State the conclusion of the test, explaining your answer.
CAIE S2 2019 March Q1
1 The masses of a certain variety of plums are known to have standard deviation 13.2 g . A random sample of 200 of these plums is taken and the mean mass of the plums in the sample is found to be 62.3 g .
  1. Calculate a \(98 \%\) confidence interval for the population mean mass.
  2. State with a reason whether it was necessary to use the Central Limit theorem in the calculation in part (i).
CAIE S2 2019 March Q2
2 The independent random variables \(X\) and \(Y\) have the distributions \(\mathrm { N } ( 9.2,12.1 )\) and \(\mathrm { N } ( 3.0,8.6 )\) respectively. Find \(\mathrm { P } ( X > 3 Y )\).
CAIE S2 2019 March Q3
3 At factory \(A\) the mean number of accidents per year is 32 . At factory \(B\) the records of numbers of accidents before 2018 have been lost, but the number of accidents during 2018 was 21. It is known that the number of accidents per year can be well modelled by a Poisson distribution. Use an approximating distribution to test at the \(2 \%\) significance level whether the mean number of accidents at factory \(B\) is less than at factory \(A\).
CAIE S2 2019 March Q4
4 marks
4 The lifetimes, \(X\) hours, of a random sample of 50 batteries of a certain kind were found. The results are summarised by \(\Sigma x = 420\) and \(\Sigma x ^ { 2 } = 27530\).
  1. Calculate an unbiased estimate of the population mean of \(X\) and show that an unbiased estimate of the population variance is 490 , correct to 3 significant figures.
  2. The lifetimes of a further large sample of \(n\) batteries of this kind were noted, and the sample mean, \(\bar { X }\), was found. Use your estimates from part (i) to find the value of \(n\) such that \(\mathrm { P } ( \bar { X } > 5 ) = 0.9377\).
    [0pt] [4]