Questions S1 (1967 questions)

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CAIE S1 2006 November Q1
1 The weights of 30 children in a class, to the nearest kilogram, were as follows.
50456153554752494651
60525447575942465153
56485051445249585545
Construct a grouped frequency table for these data such that there are five equal class intervals with the first class having a lower boundary of 41.5 kg and the fifth class having an upper boundary of 61.5 kg .
CAIE S1 2006 November Q2
2 The discrete random variable \(X\) has the following probability distribution.
\(x\)01234
\(\mathrm { P } ( X = x )\)0.26\(q\)\(3 q\)0.050.09
  1. Find the value of \(q\).
  2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
CAIE S1 2006 November Q3
3 In a survey, people were asked how long they took to travel to and from work, on average. The median time was 3 hours 36 minutes, the upper quartile was 4 hours 42 minutes and the interquartile range was 3 hours 48 minutes. The longest time taken was 5 hours 12 minutes and the shortest time was 30 minutes.
  1. Find the lower quartile.
  2. Represent the information by a box-and-whisker plot, using a scale of 2 cm to represent 60 minutes.
CAIE S1 2006 November Q4
4 Two fair dice are thrown.
  1. Event \(A\) is 'the scores differ by 3 or more'. Find the probability of event \(A\).
  2. Event \(B\) is 'the product of the scores is greater than 8 '. Find the probability of event \(B\).
  3. State with a reason whether events \(A\) and \(B\) are mutually exclusive.
CAIE S1 2006 November Q5
5
  1. Give an example of a variable in real life which could be modelled by a normal distribution.
  2. The random variable \(X\) is normally distributed with mean \(\mu\) and variance 21.0. Given that \(\mathrm { P } ( X > 10.0 ) = 0.7389\), find the value of \(\mu\).
  3. If 300 observations are taken at random from the distribution in part (ii), estimate how many of these would be greater than 22.0.
CAIE S1 2006 November Q6
6 Six men and three women are standing in a supermarket queue.
  1. How many possible arrangements are there if there are no restrictions on order?
  2. How many possible arrangements are there if no two of the women are standing next to each other?
  3. Three of the people in the queue are chosen to take part in a customer survey. How many different choices are possible if at least one woman must be included?
CAIE S1 2006 November Q7
7 A manufacturer makes two sizes of elastic bands: large and small. \(40 \%\) of the bands produced are large bands and \(60 \%\) are small bands. Assuming that each pack of these elastic bands contains a random selection, calculate the probability that, in a pack containing 20 bands, there are
  1. equal numbers of large and small bands,
  2. more than 17 small bands. An office pack contains 150 elastic bands.
  3. Using a suitable approximation, calculate the probability that the number of small bands in the office pack is between 88 and 97 inclusive.
CAIE S1 2007 November Q1
1 A summary of 24 observations of \(x\) gave the following information: $$\Sigma ( x - a ) = - 73.2 \quad \text { and } \quad \Sigma ( x - a ) ^ { 2 } = 2115 .$$ The mean of these values of \(x\) is 8.95 .
  1. Find the value of the constant \(a\).
  2. Find the standard deviation of these values of \(x\).
CAIE S1 2007 November Q2
2 The random variable \(X\) takes the values \(- 2,0\) and 4 only. It is given that \(\mathrm { P } ( X = - 2 ) = 2 p , \mathrm { P } ( X = 0 ) = p\) and \(\mathrm { P } ( X = 4 ) = 3 p\).
  1. Find \(p\).
  2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
CAIE S1 2007 November Q3
3 The six digits 4, 5, 6, 7, 7, 7 can be arranged to give many different 6-digit numbers.
  1. How many different 6-digit numbers can be made?
  2. How many of these 6-digit numbers start with an odd digit and end with an odd digit?
CAIE S1 2007 November Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{1a10471c-5810-44ca-9353-c2c76e190a2b-2_542_876_1425_632} The random variable \(X\) has a normal distribution with mean 4.5. It is given that \(\mathrm { P } ( X > 5.5 ) = 0.0465\) (see diagram).
  1. Find the standard deviation of \(X\).
  2. Find the probability that a random observation of \(X\) lies between 3.8 and 4.8.
CAIE S1 2007 November Q5
5 The arrival times of 204 trains were noted and the number of minutes, \(t\), that each train was late was recorded. The results are summarised in the table.
Number of minutes late \(( t )\)\(- 2 \leqslant t < 0\)\(0 \leqslant t < 2\)\(2 \leqslant t < 4\)\(4 \leqslant t < 6\)\(6 \leqslant t < 10\)
Number of trains4351692219
  1. Explain what \(- 2 \leqslant t < 0\) means about the arrival times of trains.
  2. Draw a cumulative frequency graph, and from it estimate the median and the interquartile range of the number of minutes late of these trains.
CAIE S1 2007 November Q6
6 On any occasion when a particular gymnast performs a certain routine, the probability that she will perform it correctly is 0.65 , independently of all other occasions.
  1. Find the probability that she will perform the routine correctly on exactly 5 occasions out 7 .
  2. On one day she performs the routine 50 times. Use a suitable approximation to estimate the probability that she will perform the routine correctly on fewer than 29 occasions.
  3. On another day she performs the routine \(n\) times. Find the smallest value of \(n\) for which the expected number of correct performances is at least 8 .
CAIE S1 2007 November Q7
7 Box \(A\) contains 5 red paper clips and 1 white paper clip. Box \(B\) contains 7 red paper clips and 2 white paper clips. One paper clip is taken at random from box \(A\) and transferred to box \(B\). One paper clip is then taken at random from box \(B\).
  1. Find the probability of taking both a white paper clip from box \(A\) and a red paper clip from box \(B\).
  2. Find the probability that the paper clip taken from box \(B\) is red.
  3. Find the probability that the paper clip taken from box \(A\) was red, given that the paper clip taken from box \(B\) is red.
  4. The random variable \(X\) denotes the number of times that a red paper clip is taken. Draw up a table to show the probability distribution of \(X\).
CAIE S1 2008 November Q1
1 Rachel measured the lengths in millimetres of some of the leaves on a tree. Her results are recorded below. $$\begin{array} { l l l l l l l l l l } 32 & 35 & 45 & 37 & 38 & 44 & 33 & 39 & 36 & 45 \end{array}$$ Find the mean and standard deviation of the lengths of these leaves.
CAIE S1 2008 November Q2
2 On a production line making toys, the probability of any toy being faulty is 0.08 . A random sample of 200 toys is checked. Use a suitable approximation to find the probability that there are at least 15 faulty toys.
CAIE S1 2008 November Q3
3
  1. The daily minimum temperature in degrees Celsius ( \({ } ^ { \circ } \mathrm { C }\) ) in January in Ottawa is a random variable with distribution \(\mathrm { N } ( - 15.1,62.0 )\). Find the probability that a randomly chosen day in January in Ottawa has a minimum temperature above \(0 ^ { \circ } \mathrm { C }\).
  2. In another city the daily minimum temperature in \({ } ^ { \circ } \mathrm { C }\) in January is a random variable with distribution \(\mathrm { N } ( \mu , 40.0 )\). In this city the probability that a randomly chosen day in January has a minimum temperature above \(0 ^ { \circ } \mathrm { C }\) is 0.8888 . Find the value of \(\mu\).
CAIE S1 2008 November Q4
4 A builder is planning to build 12 houses along one side of a road. He will build 2 houses in style \(A\), 2 houses in style \(B , 3\) houses in style \(C , 4\) houses in style \(D\) and 1 house in style \(E\).
  1. Find the number of possible arrangements of these 12 houses.
  2. Road
    \(\square \square \square \square \square \square \square \square \square\)\(\square \square \square\)
    The 12 houses will be in two groups of 6 (see diagram). Find the number of possible arrangements if all the houses in styles \(A\) and \(D\) are in the first group and all the houses in styles \(B , C\) and \(E\) are in the second group.
  3. Four of the 12 houses will be selected for a survey. Exactly one house must be in style \(B\) and exactly one house in style \(C\). Find the number of ways in which these four houses can be selected.
CAIE S1 2008 November Q5
5 The pulse rates, in beats per minute, of a random sample of 15 small animals are shown in the following table.
115120158132125
104142160145104
162117109124134
  1. Draw a stem-and-leaf diagram to represent the data.
  2. Find the median and the quartiles.
  3. On graph paper, using a scale of 2 cm to represent 10 beats per minute, draw a box-and-whisker plot of the data.
CAIE S1 2008 November Q6
6 There are three sets of traffic lights on Karinne's journey to work. The independent probabilities that Karinne has to stop at the first, second and third set of lights are \(0.4,0.8\) and 0.3 respectively.
  1. Draw a tree diagram to show this information.
  2. Find the probability that Karinne has to stop at each of the first two sets of lights but does not have to stop at the third set.
  3. Find the probability that Karinne has to stop at exactly two of the three sets of lights.
  4. Find the probability that Karinne has to stop at the first set of lights, given that she has to stop at exactly two sets of lights.
CAIE S1 2008 November Q7
7 A fair die has one face numbered 1, one face numbered 3, two faces numbered 5 and two faces numbered 6 .
  1. Find the probability of obtaining at least 7 odd numbers in 8 throws of the die. The die is thrown twice. Let \(X\) be the sum of the two scores. The following table shows the possible values of \(X\). \begin{table}[h]
    \captionsetup{labelformat=empty} \caption{Second throw}
    135566
    \cline { 2 - 8 }1246677
    3468899
    First56810101111
    throw56810101111
    67911111212
    67911111212
    \end{table}
  2. Draw up a table showing the probability distribution of \(X\).
  3. Calculate \(\mathrm { E } ( X )\).
  4. Find the probability that \(X\) is greater than \(\mathrm { E } ( X )\).
CAIE S1 2009 November Q1
1 The mean number of defective batteries in packs of 20 is 1.6 . Use a binomial distribution to calculate the probability that a randomly chosen pack of 20 will have more than 2 defective batteries.
CAIE S1 2009 November Q2
2 The probability distribution of the random variable \(X\) is shown in the following table.
\(x\)- 2- 10123
\(\mathrm { P } ( X = x )\)0.08\(p\)0.120.16\(q\)0.22
The mean of \(X\) is 1.05 .
  1. Write down two equations involving \(p\) and \(q\) and hence find the values of \(p\) and \(q\).
  2. Find the variance of \(X\).
CAIE S1 2009 November Q3
3 The times for a certain car journey have a normal distribution with mean 100 minutes and standard deviation 7 minutes. Journey times are classified as follows: \begin{displayquote} 'short' (the shortest \(33 \%\) of times),
'long' (the longest \(33 \%\) of times),
'standard' (the remaining 34\% of times).
  1. Find the probability that a randomly chosen car journey takes between 85 and 100 minutes.
  2. Find the least and greatest times for 'standard' journeys. \end{displayquote}
CAIE S1 2009 November Q4
4 A library has many identical shelves. All the shelves are full and the numbers of books on each shelf in a certain section are summarised by the following stem-and-leaf diagram.
33699
467
50122
600112344444556667889
7113335667899
80245568
9001244445567788999
Key: 3 | 6 represents 36 books
  1. Find the number of shelves in this section of the library.
  2. Draw a box-and-whisker plot to represent the data. In another section all the shelves are full and the numbers of books on each shelf are summarised by the following stem-and-leaf diagram.
    212222334566679\(( 13 )\)
    301112334456677788\(( 15 )\)
    4223357789
    Key: 3 | 6 represents 36 books
  3. There are fewer books in this section than in the previous section. State one other difference between the books in this section and the books in the previous section.