Questions — CAIE S2 (717 questions)

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CAIE S2 2024 November Q6
6 The numbers of customers arriving at service desks \(A\) and \(B\) during a 10 -minute period have the independent distributions \(\operatorname { Po } ( 1.8 )\) and \(\operatorname { Po } ( 2.1 )\) respectively.
  1. Find the probability that during a randomly chosen 15 -minute period more than 2 customers will arrive at \(\operatorname { desk } A\).
  2. Find the probability that during a randomly chosen 5-minute period the total number of customers arriving at both desks is less than 4 .
    \includegraphics[max width=\textwidth, alt={}, center]{4f215475-30fd-47fb-aa77-0b53e339f50c-08_2720_35_109_2012}
    \includegraphics[max width=\textwidth, alt={}, center]{4f215475-30fd-47fb-aa77-0b53e339f50c-09_2716_29_107_22}
  3. An inspector waits at desk \(B\). She wants to wait long enough to be \(90 \%\) certain of seeing at least one customer arrive at the desk. Find the minimum time for which she should wait, giving your answer correct to the nearest minute.
CAIE S2 2024 November Q7
7 The number of accidents per year on a certain road has the distribution \(\operatorname { Po } ( \lambda )\). In the past the value of \(\lambda\) was 3.3 . Recently, a new speed limit was imposed and the council wishes to test whether the value of \(\lambda\) has decreased. The council notes the total number, \(X\), of accidents during two randomly chosen years after the speed limit was introduced and it carries out a test at the \(5 \%\) significance level.
  1. Calculate the probability of a Type I error.
  2. Given that \(X = 2\), carry out the test.
    \includegraphics[max width=\textwidth, alt={}, center]{4f215475-30fd-47fb-aa77-0b53e339f50c-10_2718_35_107_2012}
    \includegraphics[max width=\textwidth, alt={}, center]{4f215475-30fd-47fb-aa77-0b53e339f50c-11_2716_29_107_22}
  3. The council decides to carry out another similar test at the \(5 \%\) significance level using the same hypotheses and two different randomly chosen years. Given that the true value of \(\lambda\) is 0.6 , calculate the probability of a Type II error.
  4. Using \(\lambda = 0.6\) and a suitable approximating distribution, find the probability that there will be more than 10 accidents in 30 years.
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE S2 2020 Specimen Q1
1 Leat s frm a certain tro tree \(\mathbf { h }\) leg \(\mathbf { b }\) th t are \(\dot { \mathbf { d } }\) strib ed with stad \(\operatorname { rd } \mathbf { d } \dot { \mathbf { v } }\) atio \(\mathbf { Z }\) cm. A rach sample 6 6 th se lead s is tak n ad the mean leg \(\mathrm { h } \mathbf { 6 }\) th s samp e is fo d to b © cm .
  1. Calch ate a 90 cf id \(n\) e in erd \(l\) fo th \(p p\) atim earl eg \(h\)
  2. Write d n th p b b lity th t th wh e 6 a \(9 \%\) co id n e in ery l will lie b low th p atim ean
CAIE S2 2020 Specimen Q2
2 Describ b iefly to s e rach m b rs to cb e sampe \(\mathbf { b } \boldsymbol { \mathbb { I } }\) std ns frm a \(\mathbf { g }\) ar-g \(\mathbf { p } \boldsymbol { \mathbf { 6 } } \varnothing\) ste ns.
CAIE S2 2020 Specimen Q3
3 Th m brg calls receie d at a small call cen re \(\mathbf { h }\) s a Posso id strib in with mean
CAIE S2 2020 Specimen Q4
4 marks
4 Th lifetimes, in b s, b Lg ie lig b b ad Ee rlw lig b b be tb id pd 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]{8f62635a-2998-468f-8017-0db050d612be-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 Q6
6 At a certain b \(\dot { p }\) tal it was fd th \(t\) th \(p\) b b lity th \(t\) ap tien \(\dot { d } d \mathbf { p }\) arrie fo an ap \(n\) men was Q. Th b \(\dot { p }\) tal carries t sm ep icity in th th t thspb b lity willb red ed Tby wisht \(d\) est wh th \(r\) th \(p\) icityh swo k d A rach sample 6 B ap \(n\) men s is selected ad th \(m\) brg tien sth td \(\mathbf { p }\) arriw is \(\mathbf { p }\) ed Th s fig e is s ed œ arry a test at th \% sig fican e lew l.
  1. El ain wh b test is a -tailed do tate siu tabeh lad ltera tir bes. [R
  2. Use abm ial d strib in to fid to critical rego ad fid th p b b lity \(\mathbf { 6 }\) a Tr I erro . [\\(
  3. If act \)3 \boldsymbol { p }$ tien su 6 the Oh arrie . State the co lo ind th test, e ain gy ras wer.
CAIE S2 2020 Specimen Q7
7 Th mean weig 6 bg 6 carrb s is \(\mu \mathrm { klg }\) ams. An is \(\mathbf { p }\) cto wish s to test wh th \(\mathrm { r } \mu = 20\) He weig a ranch sampe 6 tb \(g\) an \(s\) resh ts are sm marised \(s\) fb low \(s\). $$\Sigma x = 430 \quad \Sigma x ^ { 2 } = 0$$ Carryo the test at the to 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 2004 June Q1
1 Each multiple choice question in a test has 4 suggested answers, exactly one of which is correct. Rehka knows nothing about the subject of the test, but claims that she has a special method for answering the questions that is better than just guessing. There are 60 questions in the test, and Rehka gets 22 correct.
  1. State null and alternative hypotheses for a test of Rehka's claim.
  2. Using a normal approximation, test at the \(5 \%\) significance level whether Rehka's claim is justified.
CAIE S2 2004 June Q2
2 In athletics matches the triple jump event consists of a hop, followed by a step, followed by a jump. The lengths covered by Albert in each part are independent normal variables with means \(3.5 \mathrm {~m} , 2.9 \mathrm {~m}\), 3.1 m and standard deviations \(0.3 \mathrm {~m} , 0.25 \mathrm {~m} , 0.35 \mathrm {~m}\) respectively. The length of the triple jump is the sum of the three parts.
  1. Find the mean and standard deviation of the length of Albert's triple jumps.
  2. Find the probability that the mean of Albert's next four triple jumps is greater than 9 m .
CAIE S2 2004 June Q3
3 The independent random variables \(X\) and \(Y\) are such that \(X\) has mean 8 and variance 4.8 and \(Y\) has a Poisson distribution with mean 6. Find
  1. \(\mathrm { E } ( 2 X - 3 Y )\),
  2. \(\operatorname { Var } ( 2 X - 3 Y )\).
CAIE S2 2004 June Q4
4 Packets of cat food are filled by a machine.
  1. In a random sample of 10 packets, the weights, in grams, of the packets were as follows.
    \(\begin{array} { l l l l l l l l l l } 374.6 & 377.4 & 376.1 & 379.2 & 371.2 & 375.0 & 372.4 & 378.6 & 377.1 & 371.5 \end{array}\)
    Find unbiased estimates of the population mean and variance.
  2. In a random sample of 200 packets, 38 were found to be underweight. Calculate a \(96 \%\) confidence interval for the population proportion of underweight packets.
CAIE S2 2004 June Q5
5 The lectures in a mathematics department are scheduled to last 54 minutes, and the times of individual lectures may be assumed to have a normal distribution with mean \(\mu\) minutes and standard deviation 3.1 minutes. One of the students commented that, on average, the lectures seemed too short. To investigate this, the times for a random sample of 10 lectures were used to test the null hypothesis \(\mu = 54\) against the alternative hypothesis \(\mu < 54\) at the \(10 \%\) significance level.
  1. Show that the null hypothesis is rejected in favour of the alternative hypothesis if \(\bar { x } < 52.74\), where \(\bar { x }\) minutes is the sample mean.
  2. Find the probability of a Type II error given that the actual mean length of lectures is 51.5 minutes.
CAIE S2 2004 June Q6
6 At a certain airfield planes land at random times at a constant average rate of one every 10 minutes.
  1. Find the probability that exactly 5 planes will land in a period of one hour.
  2. Find the probability that at least 2 planes will land in a period of 16 minutes.
  3. Given that 5 planes landed in an hour, calculate the conditional probability that 1 plane landed in the first half hour and 4 in the second half hour.
CAIE S2 2004 June Q7
7 The queuing time, \(T\) minutes, for a person queuing at a supermarket checkout has probability density function given by $$f ( t ) = \begin{cases} c t \left( 25 - t ^ { 2 } \right) & 0 \leqslant t \leqslant 5
0 & \text { otherwise } \end{cases}$$ where \(c\) is a constant.
  1. Show that the value of \(c\) is \(\frac { 4 } { 625 }\).
  2. Find the probability that a person will have to queue for between 2 and 4 minutes.
  3. Find the mean queuing time.
CAIE S2 2005 June Q1
1 Exam marks, \(X\), have mean 70 and standard deviation 8.7. The marks need to be scaled using the formula \(Y = a X + b\) so that the scaled marks, \(Y\), have mean 55 and standard deviation 6.96. Find the values of \(a\) and \(b\).
CAIE S2 2005 June Q2
2 Jenny has to do a statistics project at school on how much pocket money, in dollars, is received by students in her year group. She plans to take a sample of 7 students from her year group, which contains 122 students.
  1. Give a suitable method of taking this sample. Her sample gives the following results. $$\begin{array} { l l l l l l l } 13.40 & 10.60 & 26.50 & 20.00 & 14.50 & 15.00 & 16.50 \end{array}$$
  2. Find unbiased estimates of the population mean and variance.
  3. Is the estimated population variance more than, less than or the same as the sample variance?
  4. Describe what you understand by 'population' in this question.
CAIE S2 2005 June Q3
3 A survey of a random sample of \(n\) people found that 61 of them read The Reporter newspaper. A symmetric confidence interval for the true population proportion, \(p\), who read The Reporter is \(0.1993 < p < 0.2887\).
  1. Find the mid-point of this confidence interval and use this to find the value of \(n\).
  2. Find the confidence level of this confidence interval.
CAIE S2 2005 June Q4
4 A study of a large sample of books by a particular author shows that the number of words per sentence can be modelled by a normal distribution with mean 21.2 and standard deviation 7.3. A researcher claims to have discovered a previously unknown book by this author. The mean length of 90 sentences chosen at random in this book is found to be 19.4 words.
  1. Assuming the population standard deviation of sentence lengths in this book is also 7.3, test at the \(5 \%\) level of significance whether the mean sentence length is the same as the author's. State your null and alternative hypotheses.
  2. State in words relating to the context of the test what is meant by a Type I error and state the probability of a Type I error in the test in part (i).
CAIE S2 2005 June Q5
5 A clock contains 4 new batteries each of which gives a voltage which is normally distributed with mean 1.54 volts and standard deviation 0.05 volts. The voltages of the batteries are independent. The clock will only work if the total voltage is greater than 5.95 volts.
  1. Find the probability that the clock will work.
  2. Find the probability that the average total voltage of the batteries of 20 clocks chosen at random exceeds 6.2 volts.
CAIE S2 2005 June Q6
6 At a petrol station cars arrive independently and at random times at constant average rates of 8 cars per hour travelling east and 5 cars per hour travelling west.
  1. Find the probability that, in a quarter-hour period,
    (a) one or more cars travelling east and one or more cars travelling west will arrive,
    (b) a total of 2 or more cars will arrive.
  2. Find the approximate probability that, in a 12 -hour period, a total of more than 175 cars will arrive.
CAIE S2 2005 June Q7
7 The random variable \(X\) denotes the number of hours of cloud cover per day at a weather forecasting centre. The probability density function of \(X\) is given by $$f ( x ) = \begin{cases} \frac { ( x - 18 ) ^ { 2 } } { k } & 0 \leqslant x \leqslant 24
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = 2016\).
  2. On how many days in a year of 365 days can the centre expect to have less than 2 hours of cloud cover?
  3. Find the mean number of hours of cloud cover per day.
CAIE S2 2006 June Q1
1 Packets of fish food have weights that are distributed with standard deviation 2.3 g . A random sample of 200 packets is taken. The mean weight of this sample is found to be 99.2 g . Calculate a \(99 \%\) confidence interval for the population mean weight.
CAIE S2 2006 June Q2
2 A mathematics module is assessed by an examination and by coursework. The examination makes up \(75 \%\) of the total assessment and the coursework makes up \(25 \%\). Examination marks, \(X\), are distributed with mean 53.2 and standard deviation 9.3. Coursework marks, \(Y\), are distributed with mean 78.0 and standard deviation 5.1. Examination marks and coursework marks are independent. Find the mean and standard deviation of the combined mark \(0.75 X + 0.25 Y\).