Questions — CAIE S2 (717 questions)

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CAIE S2 2011 June Q4
4

  1. \includegraphics[max width=\textwidth, alt={}, center]{7c9a87ac-69c6-4850-82aa-8235bba581e8-2_611_712_1466_358}
    \includegraphics[max width=\textwidth, alt={}, center]{7c9a87ac-69c6-4850-82aa-8235bba581e8-2_618_716_1464_1155} The diagrams show the graphs of two functions, \(g\) and \(h\). For each of the functions \(g\) and \(h\), give a reason why it cannot be a probability density function.
  2. The distance, in kilometres, travelled in a given time by a cyclist is represented by the continuous random variable \(X\) with probability density function given by $$f ( x ) = \begin{cases} \frac { 30 } { x ^ { 2 } } & 10 \leqslant x \leqslant 15
    0 & \text { otherwise } \end{cases}$$
    1. Show that \(\mathrm { E } ( X ) = 30 \ln 1.5\).
    2. Find the median of \(X\). Find also the probability that \(X\) lies between the median and the mean.
CAIE S2 2011 June Q5
5 Cans of drink are packed in boxes, each containing 4 cans. The weights of these cans are normally distributed with mean 510 g and standard deviation 14 g . The weights of the boxes, when empty, are independently normally distributed with mean 200 g and standard deviation 8 g .
  1. Find the probability that the total weight of a full box of cans is between 2200 g and 2300 g .
  2. Two cans of drink are chosen at random. Find the probability that they differ in weight by more than 20 g .
CAIE S2 2011 June Q6
6 The number of injuries per month at a certain factory has a Poisson distribution. In the past the mean was 2.1 injuries per month. New safety procedures are put in place and the management wishes to use the next 3 months to test, at the \(2 \%\) significance level, whether there are now fewer injuries than before, on average.
  1. Find the critical region for the test.
  2. Find the probability of a Type I error.
  3. During the next 3 months there are a total of 3 injuries. Carry out the test.
  4. Assuming that the mean remains 2.1 , calculate an estimate of the probability that there will be fewer than 20 injuries during the next 12 months.
CAIE S2 2011 June Q1
1 The weights of bags of fuel have mean 3.2 kg and standard deviation 0.04 kg . The total weight of a random sample of three bags is denoted by \(T \mathrm {~kg}\). Find the mean and standard deviation of \(T\).
\(2 X\) is a random variable having the distribution \(\mathrm { B } \left( 12 , \frac { 1 } { 4 } \right)\). A random sample of 60 values of \(X\) is taken. Find the probability that the sample mean is less than 2.8 .
CAIE S2 2011 June Q3
3 The number of goals scored per match by Everly Rovers is represented by the random variable \(X\) which has mean 1.8.
  1. State two conditions for \(X\) to be modelled by a Poisson distribution. Assume now that \(X \sim \operatorname { Po } ( 1.8 )\).
  2. Find \(\mathrm { P } ( 2 < X < 6 )\).
  3. The manager promises the team a bonus if they score at least 1 goal in each of the next 10 matches. Find the probability that they win the bonus.
CAIE S2 2011 June Q4
4 A doctor wishes to investigate the mean fat content in low-fat burgers. He takes a random sample of 15 burgers and sends them to a laboratory where the mass, in grams, of fat in each burger is determined. The results are as follows. $$\begin{array} { l l l l l l l l l l l l l l l } 9 & 7 & 8 & 9 & 6 & 11 & 7 & 9 & 8 & 9 & 8 & 10 & 7 & 9 & 9 \end{array}$$ Assume that the mass, in grams, of fat in low-fat burgers is normally distributed with mean \(\mu\) and that the population standard deviation is 1.3 .
  1. Calculate a \(99 \%\) confidence interval for \(\mu\).
  2. Explain whether it was necessary to use the Central Limit theorem in the calculation in part (i).
  3. The manufacturer claims that the mean mass of fat in burgers of this type is 8 g . Use your answer to part (i) to comment on this claim.
CAIE S2 2011 June Q5
5 The number of adult customers arriving in a shop during a 5-minute period is modelled by a random variable with distribution \(\operatorname { Po } ( 6 )\). The number of child customers arriving in the same shop during a 10 -minute period is modelled by an independent random variable with distribution \(\mathrm { Po } ( 4.5 )\).
  1. Find the probability that during a randomly chosen 2 -minute period, the total number of adult and child customers who arrive in the shop is less than 3 .
  2. During a sale, the manager claims that more adult customers are arriving than usual. In a randomly selected 30 -minute period during the sale, 49 adult customers arrive. Test the manager's claim at the 2.5\% significance level.
CAIE S2 2011 June Q6
6 Jeevan thinks that a six-sided die is biased in favour of six. In order to test this, Jeevan throws the die 10 times. If the die shows a six on at least 4 throws out of 10 , she will conclude that she is correct.
  1. State appropriate null and alternative hypotheses.
  2. Calculate the probability of a Type I error.
  3. Explain what is meant by a Type II error in this situation.
  4. If the die is actually biased so that the probability of throwing a six is \(\frac { 1 } { 2 }\), calculate the probability of a Type II error.
CAIE S2 2011 June Q7
7 A random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} k ( 1 - x ) & - 1 \leqslant x \leqslant 1
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = \frac { 1 } { 2 }\).
  2. Find \(\mathrm { P } \left( X > \frac { 1 } { 2 } \right)\).
  3. Find the mean of \(X\).
  4. Find \(a\) such that \(\mathrm { P } ( X < a ) = \frac { 1 } { 4 }\).
CAIE S2 2012 June Q1
1 The weights, in grams, of packets of sugar are distributed with mean \(\mu\) and standard deviation 23. A random sample of 150 packets is taken. The mean weight of this sample is found to be 494 g . Calculate a 98\% confidence interval for \(\mu\).
CAIE S2 2012 June Q2
3 marks
2 An examination consists of a written paper and a practical test. The written paper marks ( \(M\) ) have mean 54.8 and standard deviation 16.0. The practical test marks ( \(P\) ) are independent of the written paper marks and have mean 82.4 and standard deviation 4.8. The final mark is found by adding \(75 \%\) of \(M\) to \(25 \%\) of \(P\). Find the mean and standard deviation of the final marks for the examination. [3]
CAIE S2 2012 June Q3
3 When the council published a plan for a new road, only \(15 \%\) of local residents approved the plan. The council then published a revised plan and, out of a random sample of 300 local residents, 60 approved the revised plan. Is there evidence, at the \(2.5 \%\) significance level, that the proportion of local residents who approve the revised plan is greater than for the original plan?
CAIE S2 2012 June Q4
4 The random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { k } { ( x + 1 ) ^ { 2 } } & 0 \leqslant x \leqslant 1
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = 2\).
  2. Find \(a\) such that \(\mathrm { P } ( X < a ) = \frac { 1 } { 5 }\).

  3. \includegraphics[max width=\textwidth, alt={}, center]{18cef198-5ca2-4700-88e9-1a2bd55f841e-2_367_524_1548_849} The diagram shows the graph of \(y = \mathrm { f } ( x )\). The median of \(X\) is denoted by \(m\). Use the diagram to explain whether \(m < 0.5\), \(m = 0.5\) or \(m > 0.5\).
CAIE S2 2012 June Q5
5 A random variable \(X\) has the distribution \(\operatorname { Po } ( 3.2 )\).
  1. A random value of \(X\) is found.
    (a) Find \(\mathrm { P } ( X \geqslant 3 )\).
    (b) Find the probability that \(X = 3\) given that \(X \geqslant 3\).
  2. Random samples of 120 values of \(X\) are taken.
    (a) Describe fully the distribution of the sample mean.
    (b) Find the probability that the mean of a random sample of size 120 is less than 3.3.
CAIE S2 2012 June Q6
6 A survey taken last year showed that the mean number of computers per household in Branley was 1.66 . This year a random sample of 50 households in Branley answered a questionnaire with the following results.
Number of computers01234\(> 4\)
Number of households512181050
  1. Calculate unbiased estimates for the population mean and variance of the number of computers per household in Branley this year.
  2. Test at the \(5 \%\) significance level whether the mean number of computers per household has changed since last year.
  3. Explain whether it is possible that a Type I error may have been made in the test in part (ii).
  4. State what is meant by a Type II error in the context of the test in part (ii), and give the set of values of the test statistic that could lead to a Type II error being made.
CAIE S2 2012 June Q7
7 At work Jerry receives emails randomly at a constant average rate of 15 emails per hour.
  1. Find the probability that Jerry receives more than 2 emails during a 20 -minute period at work.
  2. Jerry's working day is 8 hours long. Find the probability that Jerry receives fewer than 110 emails per day on each of 2 working days.
  3. At work Jerry also receives texts randomly and independently at a constant average rate of 1 text every 10 minutes. Find the probability that the total number of emails and texts that Jerry receives during a 5 -minute period at work is more than 2 and less than 6 .
CAIE S2 2021 November Q1
1 It is known that the height \(H\), in metres, of trees of a certain kind has the distribution \(\mathrm { N } ( 12.5,10.24 )\). A scientist takes a random sample of 25 trees of this kind and finds the sample mean, \(\bar { H }\), of the heights.
  1. State the distribution of \(\bar { H }\), giving the values of any parameters.
  2. Find \(\mathrm { P } ( 12 < \bar { H } < 13 )\).
CAIE S2 2021 November Q2
2 The number of enquiries received per day at a customer service desk has a Poisson distribution with mean 45.2. If more than 60 enquiries are received in a day, the customer service desk cannot deal with them all. Use a suitable approximating distribution to find the probability that, on a randomly chosen day, the customer service desk cannot deal with all the enquiries that are received.
CAIE S2 2021 November Q3
3 A random sample of 75 students at a large college was selected for a survey. 15 of these students said that they owned a car. From this result an approximate \(\alpha \%\) confidence interval for the proportion of all students at the college who own a car was calculated. The width of this interval was found to be 0.162 . Calculate the value of \(\alpha\) correct to 2 significant figures.
CAIE S2 2021 November Q4
4 A random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { 1 } { 18 } \left( 9 - x ^ { 2 } \right) & 0 \leqslant x \leqslant 3
0 & \text { otherwise } \end{cases}$$
  1. Find \(\mathrm { P } ( X < 1.2 )\).
  2. Find \(\mathrm { E } ( X )\).
    The median of \(X\) is \(m\).
  3. Show that \(m ^ { 3 } - 27 m + 27 = 0\).
CAIE S2 2021 November Q5
5
  1. The proportion of people having a particular medical condition is 1 in 100000 . A random sample of 2500 people is obtained. The number of people in the sample having the condition is denoted by \(X\).
    1. State, with a justification, a suitable approximating distribution for \(X\), giving the values of any parameters.
    2. Use the approximating distribution to calculate \(\mathrm { P } ( X > 0 )\).
  2. The percentage of people having a different medical condition is thought to be \(30 \%\). A researcher suspects that the true percentage is less than \(30 \%\). In a medical trial a random sample of 28 people was selected and 4 people were found to have this condition. Use a binomial distribution to test the researcher's suspicion at the \(2 \%\) significance level.
CAIE S2 2021 November Q6
6 The random variable \(T\) denotes the time, in seconds, for 100 m races run by Tania. \(T\) is normally distributed with mean \(\mu\) and variance \(\sigma ^ { 2 }\). A random sample of 40 races run by Tania gave the following results. $$n = 40 \quad \Sigma t = 560 \quad \Sigma t ^ { 2 } = 7850$$
  1. Calculate unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\).
    The random variable \(S\) denotes the time, in seconds, for 100 m races run by Suki. \(S\) has the independent distribution \(\mathrm { N } ( 14.2,0.3 )\).
  2. Using your answers to part (a), find the probability that, in a randomly chosen 100 m race, Suki's time will be at least 0.1 s more than Tania's time.
CAIE S2 2021 November Q7
7 The masses, in grams, of apples from a certain farm have mean \(\mu\) and standard deviation 5.2. The farmer says that the value of \(\mu\) is 64.6. A quality control inspector claims that the value of \(\mu\) is actually less than 64.6. In order to test his claim he chooses a random sample of 100 apples from the farm.
  1. The mean mass of the 100 apples is found to be 63.5 g . Carry out the test at the \(2.5 \%\) significance level.
  2. Later another test of the same hypotheses at the \(2.5 \%\) significance level, with another random sample of 100 apples from the same farm, is carried out. Given that the value of \(\mu\) is in fact 62.7 , calculate 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 2021 November Q1
1 The mass, in kilograms, of a block of cheese sold in a supermarket is denoted by the random variable \(M\). The masses of a random sample of 40 blocks are summarised as follows. $$n = 40 \quad \Sigma m = 20.50 \quad \Sigma m ^ { 2 } = 10.7280$$
  1. Calculate unbiased estimates of the population mean and variance of \(M\).
  2. The price, \(
    ) P\(, of a block of cheese of mass \)M \mathrm {~kg}\( is found using the formula \)P = 11 M + 0.50\(. Find estimates of the population mean and variance of \)P$.
CAIE S2 2021 November Q2
2 Andy and Jessica are doing a survey about musical preferences. They plan to choose a representative sample of six students from the 256 students at their college.
  1. Andy suggests that they go to the music building during the lunch hour and choose six students at random from the students who are there. Give a reason why this method is unsatisfactory.
  2. Jessica decides to use another method. She numbers all the students in the college from 1 to 256. Then she uses her calculator and generates the following random numbers. $$\begin{array} { l l l l l } 204393 & 162007 & 204028 & 587119 & 207395 \end{array}$$ From these numbers, she obtains six student numbers. The first three of her student numbers are 204, 162 and 7. Continue Jessica's method to obtain the next three student numbers.