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CAIE S2 2013 June Q4
8 marks Moderate -0.3
4 The masses, in grams, of a certain type of plum are normally distributed with mean \(\mu\) and variance \(\sigma ^ { 2 }\). The masses, \(m\) grams, of a random sample of 150 plums of this type were found and the results are summarised by \(\Sigma m = 9750\) and \(\Sigma m ^ { 2 } = 647500\).
  1. Calculate unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\).
  2. Calculate a 98\% confidence interval for \(\mu\). Two more random samples of plums of this type are taken and a \(98 \%\) confidence interval for \(\mu\) is calculated from each sample.
  3. Find the probability that neither of these two intervals contains \(\mu\).
CAIE S2 2013 June Q5
10 marks Challenging +1.2
5 Packets of cereal are packed in boxes, each containing 6 packets. The masses of the packets are normally distributed with mean 510 g and standard deviation 12 g . The masses of the empty boxes are normally distributed with mean 70 g and standard deviation 4 g .
  1. Find the probability that the total mass of a full box containing 6 packets is between 3050 g and 3150 g .
  2. A packet and an empty box are chosen at random. Find the probability that the mass of the packet is at least 8 times the mass of the empty box.
CAIE S2 2013 June Q6
12 marks Standard +0.3
6 The number of cases of asthma per month at a clinic has a Poisson distribution. In the past the mean has been 5.3 cases per month. A new treatment is introduced. In order to test at the \(5 \%\) significance level whether the mean has decreased, the number of cases in a randomly chosen month is noted.
  1. Find the critical region for the test and, given that the number of cases is 2 , carry out the test.
  2. Explain the meaning of a Type I error in this context and state the probability of a Type I error.
  3. At another clinic the mean number of cases of asthma per month has the independent distribution \(\mathrm { Po } ( 13.1 )\). Assuming that the mean for the first clinic is still 5.3, use a suitable approximating distribution to estimate the probability that the total number of cases in the two clinics in a particular month is more than 20.
CAIE S2 2013 June Q1
5 marks Moderate -0.8
1 The mean and variance of the random variable \(X\) are 5.8 and 3.1 respectively. The random variable \(S\) is the sum of three independent values of \(X\). The independent random variable \(T\) is defined by \(T = 3 X + 2\).
  1. Find the variance of \(S\).
  2. Find the variance of \(T\).
  3. Find the mean and variance of \(S - T\).
CAIE S2 2013 June Q2
5 marks Moderate -0.3
2 A hockey player found that she scored a goal on \(82 \%\) of her penalty shots. After attending a coaching course, she scored a goal on 19 out of 20 penalty shots. Making an assumption that should be stated, test at the 10\% significance level whether she has improved.
CAIE S2 2013 June Q3
6 marks Moderate -0.8
3 Each of a random sample of 15 students was asked how long they spent revising for an exam. The results, in minutes, were as follows. $$\begin{array} { l l l l l l l l l l l l l l l } 50 & 70 & 80 & 60 & 65 & 110 & 10 & 70 & 75 & 60 & 65 & 45 & 50 & 70 & 50 \end{array}$$ Assume that the times for all students are normally distributed with mean \(\mu\) minutes and standard deviation 12 minutes.
  1. Calculate a \(92 \%\) confidence interval for \(\mu\).
  2. Explain what is meant by a \(92 \%\) confidence interval for \(\mu\).
  3. Explain what is meant by saying that a sample is 'random'.
CAIE S2 2013 June Q4
6 marks Standard +0.8
4 The independent random variables \(X\) and \(Y\) have the distributions \(\operatorname { Po } ( 2 )\) and \(\operatorname { Po } ( 3 )\) respectively.
  1. Given that \(X + Y = 5\), find the probability that \(X = 1\) and \(Y = 4\).
  2. Given that \(\mathrm { P } ( X = r ) = \frac { 2 } { 3 } \mathrm { P } ( X = 0 )\), show that \(3 \times 2 ^ { r - 1 } = r\) ! and verify that \(r = 4\) satisfies this equation.
CAIE S2 2013 June Q5
7 marks Moderate -0.8
5 A random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} \frac { k } { x ^ { 3 } } & x \geqslant 1 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = 2\).
  2. Find \(\mathrm { P } ( 1 \leqslant X \leqslant 2 )\).
  3. Find \(\mathrm { E } ( X )\).
CAIE S2 2013 June Q6
7 marks Standard +0.3
6 Calls arrive at a helpdesk randomly and at a constant average rate of 1.4 calls per hour. Calculate the probability that there will be
  1. more than 3 calls in \(2 \frac { 1 } { 2 }\) hours,
  2. fewer than 1000 calls in four weeks ( 672 hours).
CAIE S2 2013 June Q7
14 marks Standard +0.8
7 In the past the weekly profit at a store had mean \(\\) 34600\( and standard deviation \)\\( 4500\). Following a change of ownership, the mean weekly profit for 90 randomly chosen weeks was \(\\) 35400$.
  1. Stating a necessary assumption, test at the \(5 \%\) significance level whether the mean weekly profit has increased.
  2. State, with a reason, whether it was necessary to use the Central Limit theorem in part (i). The mean weekly profit for another random sample of 90 weeks is found and the same test is carried out at the 5\% significance level.
  3. State the probability of a Type I error.
  4. Given that the population mean weekly profit is now \(\\) 36500$, calculate the probability of a Type II error.
CAIE S2 2014 June Q1
4 marks Moderate -0.3
1 The masses, in grams, of apples of a certain type are normally distributed with mean 60.4 and standard deviation 8.2. The apples are packed in bags, with each bag containing 8 randomly chosen apples. The bags are checked by Quality Control and any bag containing apples with a total mass of less than 436 g is rejected. Find the proportion of bags that are rejected.
CAIE S2 2014 June Q2
5 marks Moderate -0.3
2 A die is biased. The mean and variance of a random sample of 70 scores on this die are found to be 3.61 and 2.70 respectively. Calculate a \(95 \%\) confidence interval for the population mean score.
CAIE S2 2014 June Q3
5 marks Moderate -0.8
3 The lengths, in centimetres, of rods produced in a factory have mean \(\mu\) and standard deviation 0.2. The value of \(\mu\) is supposed to be 250 , but a manager claims that one machine is producing rods that are too long on average. A random sample of 40 rods from this machine is taken and the sample mean length is found to be 250.06 cm . Test at the \(5 \%\) significance level whether the manager's claim is justified.
CAIE S2 2014 June Q4
5 marks Moderate -0.8
4 The proportion of people who have a particular gene, on average, is 1 in 1000. A random sample of 3500 people in a certain country is chosen and the number of people, \(X\), having the gene is found.
  1. State the distribution of \(X\) and state also an appropriate approximating distribution. Give the values of any parameters in each case. Justify your choice of the approximating distribution.
  2. Use the approximating distribution to find \(\mathrm { P } ( X \leqslant 3 )\).
CAIE S2 2014 June Q5
5 marks Moderate -0.3
5 The score on one throw of a 4 -sided die is denoted by the random variable \(X\) with probability distribution as shown in the table.
\(x\)0123
\(\mathrm { P } ( X = x )\)0.250.250.250.25
  1. Show that \(\operatorname { Var } ( X ) = 1.25\). The die is thrown 300 times. The score on each throw is noted and the mean, \(\bar { X }\), of the 300 scores is found.
  2. Use a normal distribution to find \(\mathrm { P } ( \bar { X } < 1.4 )\).
  3. Justify the use of the normal distribution in part (ii).
CAIE S2 2014 June Q6
6 marks Standard +0.3
6 Stephan is an athlete who competes in the high jump. In the past, Stephan has succeeded in \(90 \%\) of jumps at a certain height. He suspects that his standard has recently fallen and he decides to carry out a hypothesis test to find out whether he is right. If he succeeds in fewer than 17 of his next 20 jumps at this height, he will conclude that his standard has fallen.
  1. Find the probability of a Type I error.
  2. In fact Stephan succeeds in 18 of his next 20 jumps. Which of the errors, Type I or Type II, is possible? Explain your answer.
CAIE S2 2014 June Q7
10 marks Moderate -0.3
7 A random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { k } { x } & 1 \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) and \(a\) are positive constants.
  1. Show that \(k = \frac { 1 } { \ln a }\).
  2. Find \(\mathrm { E } ( X )\) in terms of \(a\).
  3. Find the median of \(X\) in terms of \(a\).
CAIE S2 2014 June Q8
10 marks Standard +0.3
8
  1. The following tables show the probability distributions for the random variables \(V\) and \(W\).
    \(v\)- 101\(> 1\)
    \(\mathrm { P } ( V = v )\)0.3680.3680.1840.080
    \(w\)00.51\(> 1\)
    \(\mathrm { P } ( W = w )\)0.3680.3680.1840.080
    For each of the variables \(V\) and \(W\) state how you can tell from its probability distribution that it does NOT have a Poisson distribution.
  2. The random variable \(X\) has the distribution \(\operatorname { Po } ( \lambda )\). It is given that $$\mathrm { P } ( X = 0 ) = p \quad \text { and } \quad \mathrm { P } ( X = 1 ) = 2.5 p$$ where \(p\) is a constant.
    (a) Show that \(\lambda = 2.5\).
    (b) Find \(\mathrm { P } ( X \geqslant 3 )\).
  3. The random variable \(Y\) has the distribution \(\operatorname { Po } ( \mu )\), where \(\mu > 30\). Using a suitable approximating distribution, it is found that \(\mathrm { P } ( Y > 40 ) = 0.5793\) correct to 4 decimal places. Find \(\mu\).
CAIE S2 2014 June Q1
3 marks Moderate -0.5
1 On average 1 in 25000 people have a rare blood condition. Use a suitable approximating distribution to find the probability that fewer than 2 people in a random sample of 100000 have the condition.
CAIE S2 2014 June Q2
3 marks Moderate -0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{43b2498f-73e2-4d33-adaf-fc3e460fa36a-2_358_1093_495_520} A random variable \(X\) takes values between 0 and 4 only and has probability density function as shown in the diagram. Calculate the median of \(X\).
CAIE S2 2014 June Q3
4 marks Standard +0.3
3 A die is thrown 100 times and shows an odd number on 56 throws. Calculate an approximate \(97 \%\) confidence interval for the probability that the die shows an odd number on one throw.
CAIE S2 2014 June Q4
6 marks Moderate -0.8
4 The weights, \(X\) kilograms, of rabbits in a certain area have population mean \(\mu \mathrm { kg }\). A random sample of 100 rabbits from this area was taken and the weights are summarised by $$\Sigma x = 165 , \quad \Sigma x ^ { 2 } = 276.25 .$$ Test at the \(5 \%\) significance level the null hypothesis \(\mathrm { H } _ { 0 } : \mu = 1.6\) against the alternative hypothesis \(\mathrm { H } _ { 1 } : \mu \neq 1.6\).
CAIE S2 2014 June Q5
7 marks Standard +0.3
5 The lifetime, \(X\) years, of a certain type of battery has probability density function given by $$f ( x ) = \begin{cases} \frac { k } { x ^ { 2 } } & 1 \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) and \(a\) are positive constants.
  1. State what the value of \(a\) represents in this context.
  2. Show that \(k = \frac { a } { a - 1 }\).
  3. Experience has shown that the longest that any battery of this type lasts is 2.5 years. Find the mean lifetime of batteries of this type.
CAIE S2 2014 June Q6
8 marks Standard +0.3
6 A machine is designed to generate random digits between 1 and 5 inclusive. Each digit is supposed to appear with the same probability as the others, but Max claims that the digit 5 is appearing less often than it should. In order to test this claim the manufacturer uses the machine to generate 25 digits and finds that exactly 1 of these digits is a 5 .
  1. Carry out a test of Max's claim at the \(2.5 \%\) significance level.
  2. Max carried out a similar hypothesis test by generating 1000 digits between 1 and 5 inclusive. The digit 5 appeared 180 times. Without carrying out the test, state the distribution that Max should use, including the values of any parameters.
  3. State what is meant by a Type II error in this context.
CAIE S2 2014 June Q7
9 marks Moderate -0.8
7 A Lost Property office is open 7 days a week. It may be assumed that items are handed in to the office randomly, singly and independently.
  1. State another condition for the number of items handed in to have a Poisson distribution. It is now given that the number of items handed in per week has the distribution \(\operatorname { Po } ( 4.0 )\).
  2. Find the probability that exactly 2 items are handed in on a particular day.
  3. Find the probability that at least 4 items are handed in during a 10-day period.
  4. Find the probability that, during a certain week, 5 items are handed in altogether, but no items are handed in on the first day of the week.