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CAIE S1 2012 June Q5
9 marks Easy -1.3
5 The lengths of the diagonals in metres of the 9 most popular flat screen TVs and the 9 most popular conventional TVs are shown below.
Flat screen :0.850.940.910.961.040.891.070.920.76
Conventional :0.690.650.850.770.740.670.710.860.75
  1. Represent this information on a back-to-back stem-and-leaf diagram.
  2. Find the median and the interquartile range of the lengths of the diagonals of the 9 conventional TVs.
  3. Find the mean and standard deviation of the lengths of the diagonals of the 9 flat screen TVs.
CAIE S1 2012 June Q6
9 marks Standard +0.8
6 The lengths of body feathers of a particular species of bird are modelled by a normal distribution. A researcher measures the lengths of a random sample of 600 body feathers from birds of this species and finds that 63 are less than 6 cm long and 155 are more than 12 cm long.
  1. Find estimates of the mean and standard deviation of the lengths of body feathers of birds of this species.
  2. In a random sample of 1000 body feathers from birds of this species, how many would the researcher expect to find with lengths more than 1 standard deviation from the mean?
CAIE S1 2012 June Q7
10 marks Moderate -0.8
7
  1. Seven friends together with their respective partners all meet up for a meal. To commemorate the occasion they arrange for a photograph to be taken of all 14 of them standing in a line.
    1. How many different arrangements are there if each friend is standing next to his or her partner?
    2. How many different arrangements are there if the 7 friends all stand together and the 7 partners all stand together?
  2. A group of 9 people consists of 2 boys, 3 girls and 4 adults. In how many ways can a team of 4 be chosen if
    1. both boys are in the team,
    2. the adults are either all in the team or all not in the team,
    3. at least 2 girls are in the team?
CAIE S1 2012 June Q1
4 marks Easy -1.2
1 The ages, \(x\) years, of 150 cars are summarised by \(\Sigma x = 645\) and \(\Sigma x ^ { 2 } = 8287.5\). Find \(\Sigma ( x - \bar { x } ) ^ { 2 }\), where \(\bar { x }\) denotes the mean of \(x\).
CAIE S1 2012 June Q2
5 marks Standard +0.3
2 The random variable \(X\) has the probability distribution shown in the table.
\(x\)246
\(\mathrm { P } ( X = x )\)0.50.40.1
Two independent values of \(X\) are chosen at random. The random variable \(Y\) takes the value 0 if the two values of \(X\) are the same. Otherwise the value of \(Y\) is the larger value of \(X\) minus the smaller value of \(X\).
  1. Draw up the probability distribution table for \(Y\).
  2. Find the expected value of \(Y\).
CAIE S1 2012 June Q3
6 marks Standard +0.3
3 In Restaurant Bijoux 13\% of customers rated the food as 'poor', 22\% of customers rated the food as 'satisfactory' and \(65 \%\) rated it as 'good'. A random sample of 12 customers who went for a meal at Restaurant Bijoux was taken.
  1. Find the probability that more than 2 and fewer than 12 of them rated the food as 'good'. On a separate occasion, a random sample of \(n\) customers who went for a meal at the restaurant was taken.
  2. Find the smallest value of \(n\) for which the probability that at least 1 person will rate the food as 'poor' is greater than 0.95.
CAIE S1 2012 June Q4
6 marks Moderate -0.8
4 The back-to-back stem-and-leaf diagram shows the values taken by two variables \(A\) and \(B\).
\(A\)\(B\)\multirow{3}{*}{(4)}
310151335
41162234457778
8331701333466799(11)
98865543211018247(3)
998865421915(2)
98710204(1)
Key: \(4 | 16 | 7\) means \(A = 0.164\) and \(B = 0.167\).
  1. Find the median and the interquartile range for variable \(A\).
  2. You are given that, for variable \(B\), the median is 0.171 , the upper quartile is 0.179 and the lower quartile is 0.164 . Draw box-and-whisker plots for \(A\) and \(B\) in a single diagram on graph paper.
CAIE S1 2012 June Q5
7 marks Moderate -0.8
5 An English examination consists of 8 questions in Part \(A\) and 3 questions in Part \(B\). Candidates must choose 6 questions. The order in which questions are chosen does not matter. Find the number of ways in which the 6 questions can be chosen in each of the following cases.
  1. There are no restrictions on which questions can be chosen.
  2. Candidates must choose at least 4 questions from Part \(A\).
  3. Candidates must either choose both question 1 and question 2 in Part \(A\), or choose neither of these questions.
CAIE S1 2012 June Q6
9 marks Easy -1.3
6 A box of biscuits contains 30 biscuits, some of which are wrapped in gold foil and some of which are unwrapped. Some of the biscuits are chocolate-covered. 12 biscuits are wrapped in gold foil, and of these biscuits, 7 are chocolate-covered. There are 17 chocolate-covered biscuits in total.
  1. Copy and complete the table below to show the number of biscuits in each category.
    Wrapped in gold foilUnwrappedTotal
    Chocolate-covered
    Not chocolate-covered
    Total30
    A biscuit is selected at random from the box.
  2. Find the probability that the biscuit is wrapped in gold foil. The biscuit is returned to the box. An unwrapped biscuit is then selected at random from the box.
  3. Find the probability that the biscuit is chocolate-covered. The biscuit is returned to the box. A biscuit is then selected at random from the box.
  4. Find the probability that the biscuit is unwrapped, given that it is chocolate-covered. The biscuit is returned to the box. Nasir then takes 4 biscuits without replacement from the box.
  5. Find the probability that he takes exactly 2 wrapped biscuits.
CAIE S1 2012 June Q7
13 marks Standard +0.3
7 The times taken to play Beethoven's Sixth Symphony can be assumed to have a normal distribution with mean 41.1 minutes and standard deviation 3.4 minutes. Three occasions on which this symphony is played are chosen at random.
  1. Find the probability that the symphony takes longer than 42 minutes to play on exactly 1 of these occasions. The times taken to play Beethoven's Fifth Symphony can also be assumed to have a normal distribution. The probability that the time is less than 26.5 minutes is 0.1 , and the probability that the time is more than 34.6 minutes is 0.05 .
  2. Find the mean and standard deviation of the times to play this symphony.
  3. Assuming that the times to play the two symphonies are independent of each other, find the probability that, when both symphonies are played, both of the times are less than 34.6 minutes.
CAIE S1 2012 June Q1
4 marks Easy -1.8
1 Ashfaq and Kuljit have done a school statistics project on the prices of a particular model of headphones for MP3 players. Ashfaq collected prices from 21 shops. Kuljit used the internet to collect prices from 163 websites.
  1. Name a suitable statistical diagram for Ashfaq to represent his data, together with a reason for choosing this particular diagram.
  2. Name a suitable statistical diagram for Kuljit to represent her data, together with a reason for choosing this particular diagram.
CAIE S1 2012 June Q2
5 marks Moderate -0.8
2 The heights, \(x \mathrm {~cm}\), of a group of young children are summarised by $$\Sigma ( x - 100 ) = 72 , \quad \Sigma ( x - 100 ) ^ { 2 } = 499.2 .$$ The mean height is 104.8 cm .
  1. Find the number of children in the group.
  2. Find \(\Sigma ( x - 104.8 ) ^ { 2 }\).
CAIE S1 2012 June Q3
9 marks Moderate -0.8
3
  1. In how many ways can all 9 letters of the word TELEPHONE be arranged in a line if the letters P and L must be at the ends? How many different selections of 4 letters can be made from the 9 letters of the word TELEPHONE if
  2. there are no Es,
  3. there is exactly 1 E ,
  4. there are no restrictions?
CAIE S1 2012 June Q4
10 marks Moderate -0.8
4 The six faces of a fair die are numbered \(1,1,1,2,3,3\). The score for a throw of the die, denoted by the random variable \(W\), is the number on the top face after the die has landed.
  1. Find the mean and standard deviation of \(W\).
  2. The die is thrown twice and the random variable \(X\) is the sum of the two scores. Draw up a probability distribution table for \(X\).
  3. The die is thrown \(n\) times. The random variable \(Y\) is the number of times that the score is 3 . Given that \(\mathrm { E } ( Y ) = 8\), find \(\operatorname { Var } ( Y )\).
CAIE S1 2012 June Q5
10 marks Moderate -0.8
5 Suzanne has 20 pairs of shoes, some of which have designer labels. She has 6 pairs of high-heeled shoes, of which 2 pairs have designer labels. She has 4 pairs of low-heeled shoes, of which 1 pair has designer labels. The rest of her shoes are pairs of sports shoes. Suzanne has 8 pairs of shoes with designer labels in total.
  1. Copy and complete the table below to show the number of pairs in each category.
    Designer labelsNo designer labelsTotal
    High-heeled shoes
    Low-heeled shoes
    Sports shoes
    Total20
    Suzanne chooses 1 pair of shoes at random to wear.
  2. Find the probability that she wears the pair of low-heeled shoes with designer labels.
  3. Find the probability that she wears a pair of sports shoes.
  4. Find the probability that she wears a pair of high-heeled shoes, given that she wears a pair of shoes with designer labels.
  5. State with a reason whether the events 'Suzanne wears a pair of shoes with designer labels' and 'Suzanne wears a pair of sports shoes' are independent. Suzanne chooses 1 pair of shoes at random each day.
  6. Find the probability that Suzanne wears a pair of shoes with designer labels on at most 4 days out of the next 7 days.
CAIE S1 2012 June Q6
12 marks Standard +0.3
6 The lengths, in cm, of trout in a fish farm are normally distributed. 96\% of the lengths are less than 34.1 cm and 70\% of the lengths are more than 26.7 cm .
  1. Find the mean and the standard deviation of the lengths of the trout. In another fish farm, the lengths of salmon, \(X \mathrm {~cm}\), are normally distributed with mean 32.9 cm and standard deviation 2.4 cm .
  2. Find the probability that a randomly chosen salmon is 34 cm long, correct to the nearest centimetre.
  3. Find the value of \(t\) such that \(\mathrm { P } ( 31.8 < X < t ) = 0.5\).
CAIE S1 2013 June Q1
4 marks Moderate -0.8
1 A summary of 30 values of \(x\) gave the following information: $$\Sigma ( x - c ) = 234 , \quad \Sigma ( x - c ) ^ { 2 } = 1957.5 ,$$ where \(c\) is a constant.
  1. Find the standard deviation of these values of \(x\).
  2. Given that the mean of these values is 86 , find the value of \(c\).
CAIE S1 2013 June Q2
5 marks Moderate -0.3
2 Assume that, for a randomly chosen person, their next birthday is equally likely to occur on any day of the week, independently of any other person's birthday. Find the probability that, out of 350 randomly chosen people, at least 47 will have their next birthday on a Monday.
CAIE S1 2013 June Q3
5 marks Moderate -0.8
3 The following back-to-back stem-and-leaf diagram shows the annual salaries of a group of 39 females and 39 males.
FemalesMales
(4)\multirow{7}{*}{9}5200203
(9)8876400021007
(8)\multirow{5}{*}{}8753310022004566
(6)\multirow{4}{*}{}64210023002335677
(6)754000240112556889
(4)9500253457789
(2)5026046
Key: 2 | 20 | 3 means \\(20200 for females and \\)20300 for males.
  1. Find the median and the quartiles of the females' salaries. You are given that the median salary of the males is \(\\) 24000\(, the lower quartile is \)\\( 22600\) and the upper quartile is \(\\) 25300$.
  2. Represent the data by means of a pair of box-and-whisker plots in a single diagram on graph paper.
CAIE S1 2013 June Q4
7 marks Standard +0.3
4
  1. The random variable \(Y\) is normally distributed with positive mean \(\mu\) and standard deviation \(\frac { 1 } { 2 } \mu\). Find the probability that a randomly chosen value of \(Y\) is negative.
  2. The weights of bags of rice are normally distributed with mean 2.04 kg and standard deviation \(\sigma \mathrm { kg }\). In a random sample of 8000 such bags, 253 weighed over 2.1 kg . Find the value of \(\sigma\). [4]
CAIE S1 2013 June Q5
9 marks Moderate -0.3
5 Fiona uses her calculator to produce 12 random integers between 7 and 21 inclusive. The random variable \(X\) is the number of these 12 integers which are multiples of 5 .
  1. State the distribution of \(X\) and give its parameters.
  2. Calculate the probability that \(X\) is between 3 and 5 inclusive. Fiona now produces \(n\) random integers between 7 and 21 inclusive.
  3. Find the least possible value of \(n\) if the probability that none of these integers is a multiple of 5 is less than 0.01.
CAIE S1 2013 June Q6
9 marks Standard +0.3
6 Four families go to a theme park together. Mr and Mrs Lin take their 2 children. Mr O'Connor takes his 2 children. Mr and Mrs Ahmed take their 3 children. Mrs Burton takes her son. The 14 people all have to go through a turnstile one at a time to enter the theme park.
  1. In how many different orders can the 14 people go through the turnstile if each family stays together?
  2. In how many different orders can the 8 children and 6 adults go through the turnstile if no two adults go consecutively? Once inside the theme park, the children go on the roller-coaster. Each roller-coaster car holds 3 people.
  3. In how many different ways can the 8 children be divided into two groups of 3 and one group of 2 to go on the roller-coaster?
CAIE S1 2013 June Q7
11 marks Moderate -0.8
7 Box \(A\) contains 8 white balls and 2 yellow balls. Box \(B\) contains 5 white balls and \(x\) yellow balls. A ball is chosen at random from box \(A\) and placed in box \(B\). A ball is then chosen at random from box \(B\). The tree diagram below shows the possibilities for the colours of the balls chosen. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Box \(A\)} \includegraphics[alt={},max width=\textwidth]{60a9d5d4-0a6a-43e2-9828-03ea2a76ed8a-3_451_874_1774_639}
\end{figure}
  1. Justify the probability \(\frac { x } { x + 6 }\) on the tree diagram.
  2. Copy and complete the tree diagram.
  3. If the ball chosen from box \(A\) is white then the probability that the ball chosen from box \(B\) is also white is \(\frac { 1 } { 3 }\). Show that the value of \(x\) is 12 .
  4. Given that the ball chosen from box \(B\) is yellow, find the conditional probability that the ball chosen from box \(A\) was yellow.
CAIE S1 2013 June Q1
3 marks Standard +0.3
1 The random variable \(Y\) is normally distributed with mean equal to five times the standard deviation. It is given that \(\mathrm { P } ( Y > 20 ) = 0.0732\). Find the mean.
CAIE S1 2013 June Q2
4 marks Moderate -0.8
2 A summary of the speeds, \(x\) kilometres per hour, of 22 cars passing a certain point gave the following information: $$\Sigma ( x - 50 ) = 81.4 \quad \text { and } \quad \Sigma ( x - 50 ) ^ { 2 } = 671.0 .$$ Find the variance of the speeds and hence find the value of \(\Sigma x ^ { 2 }\).