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CAIE S2 2024 November Q5
11 marks Challenging +1.2
5 A machine puts sweets into bags at random. The numbers of lemon and orange sweets in a bag have the independent distributions \(\operatorname { Po } ( 3.7 )\) and \(\operatorname { Po } ( 2.6 )\) respectively. A bag of sweets is chosen at random.
  1. Find the probability that the number of lemon sweets in the bag is more than 2 but not more than 5 .
  2. Find the probability that the total number of lemon and orange sweets in the bag is less than 4 . \includegraphics[max width=\textwidth, alt={}, center]{9ac74d4c-f5e0-4c5d-ab25-5692dfb06f0b-06_2725_47_107_2002} \includegraphics[max width=\textwidth, alt={}, center]{9ac74d4c-f5e0-4c5d-ab25-5692dfb06f0b-07_2716_29_107_22} 10 bags of sweets are chosen at random.
  3. Use approximating distributions to find the probability that the total number of lemon sweets in the 10 bags is less than the total number of orange sweets in the 10 bags.
CAIE S2 2024 November Q6
11 marks Standard +0.3
6 The time, \(X\) hours, taken by a large number of people to complete a challenge is modelled by the probability density function given by $$f ( x ) = \left\{ \begin{array} { c l } \frac { 1 } { x ^ { 2 } } & a \leqslant x \leqslant b \\ 0 & \text { otherwise } \end{array} \right.$$ where \(a\) and \(b\) are constants.
  1. State what the constants \(a\) and \(b\) represent in this context.
  2. Show that \(a = \frac { b } { b + 1 }\).
    It is given that \(\mathrm { E } ( X ) = \ln 3\).
  3. Show that \(b = 2\) and find the value of \(a\). \includegraphics[max width=\textwidth, alt={}, center]{9ac74d4c-f5e0-4c5d-ab25-5692dfb06f0b-09_2726_35_97_20}
  4. Find the median of \(X\).
CAIE S2 2024 November Q7
9 marks Moderate -0.3
7 The heights of one-year-old trees of a certain variety are known to have mean 2.3 m . A scientist believes that, on average, trees of this age and variety in her region are slightly taller than in other places. She plans to carry out a hypothesis test, at the \(2 \%\) significance level, in order to test her belief.
  1. State the probability that she will make a Type I error.
    She takes a random sample of 100 such trees in her region and measures their heights, \(h \mathrm {~m}\). Her results are summarised below. $$n = 100 \quad \sum h = 238 \quad \sum h ^ { 2 } = 580$$
  2. Carry out the test at the \(2 \%\) significance level. \includegraphics[max width=\textwidth, alt={}, center]{9ac74d4c-f5e0-4c5d-ab25-5692dfb06f0b-10_2717_35_109_2012}
  3. The scientist carries out the test correctly, but another scientist claims that she has made a Type II error. Comment on this claim.
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE S2 2024 November Q2
5 marks Standard +0.3
2 The masses, in kilograms, of small and large bags of wheat have the independent distributions \(\mathrm { N } ( 16.0,0.4 )\) and \(\mathrm { N } ( 51.0,0.9 )\) respectively. Find the probability that the total mass of 3 randomly chosen small bags is greater than the mass of one randomly chosen large bag. \includegraphics[max width=\textwidth, alt={}, center]{4f215475-30fd-47fb-aa77-0b53e339f50c-04_2716_38_108_2009}
CAIE S2 2024 November Q6
9 marks Standard +0.3
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
14 marks Challenging +1.2
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
4 marks Moderate -0.5
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
3 marks Easy -1.8
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
10 marks Moderate -0.5
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
10 marks Standard +0.8
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
7 marks Standard +0.3
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
9 marks Standard +0.3
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 marks Standard +0.3
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
5 marks Moderate -0.3
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
6 marks Moderate -0.3
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
6 marks Moderate -0.8
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
7 marks Moderate -0.8
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
8 marks Standard +0.3
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
8 marks Standard +0.8
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
10 marks Standard +0.3
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
4 marks Moderate -0.8
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
6 marks Easy -1.3
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
7 marks Standard +0.3
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
7 marks Standard +0.3
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
7 marks Moderate -0.3
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.