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Edexcel S1 2006 June Q6
13 marks Moderate -0.8
A group of 100 people produced the following information relating to three attributes. The attributes were wearing glasses, being left handed and having dark hair.
Glasses were worn by 36 people, 28 were left handed and 36 had dark hair. There were 17 who wore glasses and were left handed, 19 who wore glasses and had dark hair and 15 who were left handed and had dark hair. Only 10 people wore glasses, were left handed and had dark hair.
  1. Represent these data on a Venn diagram. A person was selected at random from this group.
    Find the probability that this person
  2. wore glasses but was not left handed and did not have dark hair,
  3. did not wear glasses, was not left handed and did not have dark hair,
  4. had only two of the attributes,
  5. wore glasses given that they were left handed and had dark hair.
Edexcel S1 2007 June Q1
4 marks Easy -1.2
  1. A young family were looking for a new 3 bedroom semi-detached house. A local survey recorded the price \(x\), in \(\pounds 1000\), and the distance \(y\), in miles, from the station of such houses. The following summary statistics were provided
$$S _ { x x } = 113573 , \quad S _ { y y } = 8.657 , \quad S _ { x y } = - 808.917$$
  1. Use these values to calculate the product moment correlation coefficient.
  2. Give an interpretation of your answer to part (a). Another family asked for the distances to be measured in km rather than miles.
  3. State the value of the product moment correlation coefficient in this case.
Edexcel S1 2007 June Q2
10 marks Moderate -0.8
2. The box plot in Figure 1 shows a summary of the weights of the luggage, in kg, for each musician in an orchestra on an overseas tour. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{045e10d2-1766-4399-aa0a-5619dd0cce0f-03_346_1452_324_228} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The airline's recommended weight limit for each musician's luggage was 45 kg . Given that none of the musicians' luggage weighed exactly 45 kg ,
  1. state the proportion of the musicians whose luggage was below the recommended weight limit. A quarter of the musicians had to pay a charge for taking heavy luggage.
  2. State the smallest weight for which the charge was made.
  3. Explain what you understand by the + on the box plot in Figure 1, and suggest an instrument that the owner of this luggage might play.
  4. Describe the skewness of this distribution. Give a reason for your answer. One musician of the orchestra suggests that the weights of luggage, in kg, can be modelled by a normal distribution with quartiles as given in Figure 1.
  5. Find the standard deviation of this normal distribution.
Edexcel S1 2007 June Q3
15 marks Moderate -0.3
3. A student is investigating the relationship between the price ( \(y\) pence) of 100 g of chocolate and the percentage ( \(x \%\) ) of cocoa solids in the chocolate.
The following data is obtained
Chocolate brandABC\(D\)\(E\)\(F\)G\(H\)
\(x\) (\% cocoa)1020303540506070
\(y\) (pence)3555401006090110130
(You may use: \(\sum x = 315 , \sum x ^ { 2 } = 15225 , \sum y = 620 , \sum y ^ { 2 } = 56550 , \sum x y = 28750\) )
  1. On the graph paper on page 9 draw a scatter diagram to represent these data.
  2. Show that \(S _ { x y } = 4337.5\) and find \(S _ { x x }\). The student believes that a linear relationship of the form \(y = a + b x\) could be used to describe these data.
  3. Use linear regression to find the value of \(a\) and the value of \(b\), giving your answers to 1 decimal place.
  4. Draw the regression line on your scatter diagram. The student believes that one brand of chocolate is overpriced.
  5. Use the scatter diagram to
    1. state which brand is overpriced,
    2. suggest a fair price for this brand. Give reasons for both your answers.
      \includegraphics[max width=\textwidth, alt={}]{045e10d2-1766-4399-aa0a-5619dd0cce0f-06_2454_1485_282_228}
      The data on page 8 has been repeated here to help you
      Chocolate brandA\(B\)\(C\)D\(E\)\(F\)G\(H\)
      \(x\) (\% cocoa)1020303540506070
      \(y\) (pence)3555401006090110130
      (You may use: \(\sum x = 315 , \sum x ^ { 2 } = 15225 , \sum y = 620 , \sum y ^ { 2 } = 56550 , \sum x y = 28750\) )
Edexcel S1 2007 June Q4
9 marks Moderate -0.8
A survey of the reading habits of some students revealed that, on a regular basis, \(25 \%\) read quality newspapers, 45\% read tabloid newspapers and 40\% do not read newspapers at all.
  1. Find the proportion of students who read both quality and tabloid newspapers.
  2. In the space on page 13 draw a Venn diagram to represent this information. A student is selected at random. Given that this student reads newspapers on a regular basis,
  3. find the probability that this student only reads quality newspapers.
Edexcel S1 2007 June Q5
17 marks Moderate -0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{045e10d2-1766-4399-aa0a-5619dd0cce0f-10_726_1509_255_278} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a histogram for the variable \(t\) which represents the time taken, in minutes, by a group of people to swim 500 m .
  1. Complete the frequency table for \(t\).
    \(t\)\(5 - 10\)\(10 - 14\)\(14 - 18\)\(18 - 25\)\(25 - 40\)
    Frequency101624
  2. Estimate the number of people who took longer than 20 minutes to swim 500 m .
  3. Find an estimate of the mean time taken.
  4. Find an estimate for the standard deviation of \(t\).
  5. Find the median and quartiles for \(t\). One measure of skewness is found using \(\frac { 3 ( \text { mean } - \text { median } ) } { \text { standard deviation } }\).
  6. Evaluate this measure and describe the skewness of these data.
Edexcel S1 2007 June Q6
7 marks Moderate -0.8
6. The random variable \(X\) has a normal distribution with mean 20 and standard deviation 4 .
  1. Find \(\mathrm { P } ( X > 25 )\).
  2. Find the value of \(d\) such that \(\mathrm { P } ( 20 < X < d ) = 0.4641\)
Edexcel S1 2007 June Q7
13 marks Moderate -0.3
7. The random variable \(X\) has probability distribution
\(x\)13579
\(\mathrm { P } ( X = x )\)0.2\(p\)0.2\(q\)0.15
  1. Given that \(\mathrm { E } ( X ) = 4.5\), write down two equations involving \(p\) and \(q\). Find
  2. the value of \(p\) and the value of \(q\),
  3. \(\mathrm { P } ( 4 < X \leqslant 7 )\). Given that \(\mathrm { E } \left( X ^ { 2 } \right) = 27.4\), find
  4. \(\operatorname { Var } ( X )\),
  5. \(\mathrm { E } ( 19 - 4 X )\),
  6. \(\operatorname { Var } ( 19 - 4 X )\).
Edexcel S1 2008 June Q1
9 marks Moderate -0.3
  1. A disease is known to be present in \(2 \%\) of a population. A test is developed to help determine whether or not someone has the disease.
Given that a person has the disease, the test is positive with probability 0.95
Given that a person does not have the disease, the test is positive with probability 0.03
  1. Draw a tree diagram to represent this information. A person is selected at random from the population and tested for this disease.
  2. Find the probability that the test is positive. A doctor randomly selects a person from the population and tests him for the disease. Given that the test is positive,
  3. find the probability that he does not have the disease.
  4. Comment on the usefulness of this test.
Edexcel S1 2008 June Q2
14 marks Moderate -0.8
2. The age in years of the residents of two hotels are shown in the back to back stem and leaf diagram below. Abbey Hotel \(8 | 5 | 0\) means 58 years in Abbey hotel and 50 years in Balmoral hotel Balmoral Hotel
(1)20
(4)97511
(4)983126(1)
(11)999976653323447(3)
(6)9877504005569(6)
\multirow[t]{3}{*}{(1)}85000013667(9)
6233457(6)
7015(3)
For the Balmoral Hotel,
  1. write down the mode of the age of the residents,
  2. find the values of the lower quartile, the median and the upper quartile.
    1. Find the mean, \(\bar { x }\), of the age of the residents.
    2. Given that \(\sum x ^ { 2 } = 81213\) find the standard deviation of the age of the residents. One measure of skewness is found using $$\frac { \text { mean - mode } } { \text { standard deviation } }$$
  4. Evaluate this measure for the Balmoral Hotel. For the Abbey Hotel, the mode is 39 , the mean is 33.2 , the standard deviation is 12.7 and the measure of skewness is - 0.454
  5. Compare the two age distributions of the residents of each hotel.
Edexcel S1 2008 June Q3
11 marks Moderate -0.3
3. The random variable \(X\) has probability distribution given in the table below.
\(x\)- 10123
\(\mathrm { P } ( X = x )\)\(p\)\(q\)0.20.150.15
Given that \(\mathrm { E } ( X ) = 0.55\), find
  1. the value of \(p\) and the value of \(q\),
  2. \(\operatorname { Var } ( X )\),
  3. \(\mathrm { E } ( 2 X - 4 )\).
Edexcel S1 2008 June Q4
15 marks Moderate -0.8
4. Crickets make a noise. The pitch, \(v \mathrm { kHz }\), of the noise made by a cricket was recorded at 15 different temperatures, \(t ^ { \circ } \mathrm { C }\). These data are summarised below. $$\sum t ^ { 2 } = 10922.81 , \sum v ^ { 2 } = 42.3356 , \sum t v = 677.971 , \sum t = 401.3 , \sum v = 25.08$$
  1. Find \(S _ { t t } , S _ { v v }\) and \(S _ { t v }\) for these data.
  2. Find the product moment correlation coefficient between \(t\) and \(v\).
  3. State, with a reason, which variable is the explanatory variable.
  4. Give a reason to support fitting a regression model of the form \(v = a + b t\) to these data.
  5. Find the value of \(a\) and the value of \(b\). Give your answers to 3 significant figures.
  6. Using this model, predict the pitch of the noise at \(19 ^ { \circ } \mathrm { C }\).
Edexcel S1 2008 June Q5
10 marks Easy -1.2
5. A person's blood group is determined by whether or not it contains any of 3 substances \(A , B\) and \(C\). A doctor surveyed 300 patients' blood and produced the table below.
Blood containsNo. of Patients
only \(C\)100
\(A\) and \(C\) but not \(B\)100
only A30
\(B\) and \(C\) but not \(A\)25
only \(B\)12
\(A , B\) and \(C\)10
\(A\) and \(B\) but not \(C\)3
  1. Draw a Venn diagram to represent this information.
  2. Find the probability that a randomly chosen patient's blood contains substance \(C\). Harry is one of the patients. Given that his blood contains substance \(A\),
  3. find the probability that his blood contains all 3 substances. Patients whose blood contains none of these substances are called universal blood donors.
  4. Find the probability that a randomly chosen patient is a universal blood donor.
Edexcel S1 2008 June Q6
5 marks Moderate -0.8
6. The discrete random variable \(X\) can take only the values 2,3 or 4 . For these values the cumulative distribution function is defined by $$F ( x ) = \frac { ( x + k ) ^ { 2 } } { 25 } \text { for } x = 2,3,4$$ where \(k\) is a positive integer.
  1. Find \(k\).
  2. Find the probability distribution of \(X\).
Edexcel S1 2008 June Q7
12 marks Moderate -0.3
7. A packing plant fills bags with cement. The weight \(X \mathrm {~kg}\) of a bag of cement can be modelled by a normal distribution with mean 50 kg and standard deviation 2 kg .
  1. Find \(\mathrm { P } ( X > 53 )\).
  2. Find the weight that is exceeded by \(99 \%\) of the bags. Three bags are selected at random.
  3. Find the probability that two weigh more than 53 kg and one weighs less than 53 kg .
Edexcel S1 2009 June Q1
6 marks Easy -1.2
The volume of a sample of gas is kept constant. The gas is heated and the pressure, \(p\), is measured at 10 different temperatures, \(t\). The results are summarised below. \(\sum p = 445 \quad \sum p ^ { 2 } = 38125 \quad \sum t = 240 \quad \sum t ^ { 2 } = 27520 \quad \sum p t = 26830\)
  1. Find \(\mathrm { S } _ { p p }\) and \(\mathrm { S } _ { p t }\). Given that \(\mathrm { S } _ { t t } = 21760\),
  2. calculate the product moment correlation coefficient.
  3. Give an interpretation of your answer to part (b).
Edexcel S1 2009 June Q2
11 marks Easy -1.2
2. On a randomly chosen day the probability that Bill travels to school by car, by bicycle or on foot is \(\frac { 1 } { 2 } , \frac { 1 } { 6 }\) and \(\frac { 1 } { 3 }\) respectively. The probability of being late when using these methods of travel is \(\frac { 1 } { 5 } , \frac { 2 } { 5 }\) and \(\frac { 1 } { 10 }\) respectively.
  1. Draw a tree diagram to represent this information.
  2. Find the probability that on a randomly chosen day
    1. Bill travels by foot and is late,
    2. Bill is not late.
  3. Given that Bill is late, find the probability that he did not travel on foot.
Edexcel S1 2009 June Q3
3 marks Easy -1.3
3. The variable \(x\) was measured to the nearest whole number. Forty observations are given in the table below.
\(x\)\(10 - 15\)\(16 - 18\)\(19 -\)
Frequency15916
A histogram was drawn and the bar representing the \(10 - 15\) class has a width of 2 cm and a height of 5 cm . For the \(16 - 18\) class find
  1. the width,
  2. the height
    of the bar representing this class.
Edexcel S1 2009 June Q4
13 marks Moderate -0.3
4. A researcher measured the foot lengths of a random sample of 120 ten-year-old children. The lengths are summarised in the table below.
Foot length, \(l\), (cm)Number of children
\(10 \leqslant l < 12\)5
\(12 \leqslant l < 17\)53
\(17 \leqslant l < 19\)29
\(19 \leqslant l < 21\)15
\(21 \leqslant l < 23\)11
\(23 \leqslant l < 25\)7
  1. Use interpolation to estimate the median of this distribution.
  2. Calculate estimates for the mean and the standard deviation of these data. One measure of skewness is given by $$\text { Coefficient of skewness } = \frac { 3 ( \text { mean } - \text { median } ) } { \text { standard deviation } }$$
  3. Evaluate this coefficient and comment on the skewness of these data. Greg suggests that a normal distribution is a suitable model for the foot lengths of ten-year-old children.
  4. Using the value found in part (c), comment on Greg's suggestion, giving a reason for your answer.
Edexcel S1 2009 June Q5
9 marks Moderate -0.8
5. The weight, \(w\) grams, and the length, \(l \mathrm {~mm}\), of 10 randomly selected newborn turtles are given in the table below.
\(l\)49.052.053.054.554.153.450.051.649.551.2
\(w\)29323439383530312930
$$\text { (You may use } \mathrm { S } _ { l l } = 33.381 \quad \mathrm {~S} _ { w l } = 59.99 \quad \mathrm {~S} _ { w w } = 120.1 \text { ) }$$
  1. Find the equation of the regression line of \(w\) on \(l\) in the form \(w = a + b l\).
  2. Use your regression line to estimate the weight of a newborn turtle of length 60 mm .
  3. Comment on the reliability of your estimate giving a reason for your answer.
Edexcel S1 2009 June Q6
15 marks Standard +0.3
6. The discrete random variable \(X\) has probability function $$\mathrm { P } ( X = x ) = \left\{ \begin{array} { c l } a ( 3 - x ) & x = 0,1,2 \\ b & x = 3 \end{array} \right.$$
  1. Find \(\mathrm { P } ( X = 2 )\) and complete the table below.
    \(x\)0123
    \(\mathrm { P } ( X = x )\)\(3 a\)\(2 a\)\(b\)
    Given that \(\mathrm { E } ( X ) = 1.6\)
  2. Find the value of \(a\) and the value of \(b\). Find
  3. \(\mathrm { P } ( 0.5 < X < 3 )\),
  4. \(\mathrm { E } ( 3 X - 2 )\).
  5. Show that the \(\operatorname { Var } ( X ) = 1.64\)
  6. Calculate \(\operatorname { Var } ( 3 X - 2 )\).
Edexcel S1 2009 June Q7
7 marks Moderate -0.5
7. (a) Given that \(\mathrm { P } ( A ) = a\) and \(\mathrm { P } ( B ) = b\) express \(\mathrm { P } ( A \cup B )\) in terms of \(a\) and \(b\) when
  1. \(A\) and \(B\) are mutually exclusive,
  2. \(A\) and \(B\) are independent. Two events \(R\) and \(Q\) are such that \(\mathrm { P } \left( R \cap Q ^ { \prime } \right) = 0.15 , \quad \mathrm { P } ( Q ) = 0.35\) and \(\mathrm { P } ( R \mid Q ) = 0.1\) Find the value of
    (b) \(\mathrm { P } ( R \cup Q )\),
    (c) \(\mathrm { P } ( R \cap Q )\),
    (d) \(\mathrm { P } ( R )\).
Edexcel S1 2009 June Q8
11 marks Moderate -0.3
8. The lifetimes of bulbs used in a lamp are normally distributed. A company \(X\) sells bulbs with a mean lifetime of 850 hours and a standard deviation of 50 hours.
  1. Find the probability of a bulb, from company \(X\), having a lifetime of less than 830 hours.
  2. In a box of 500 bulbs, from company \(X\), find the expected number having a lifetime of less than 830 hours. A rival company \(Y\) sells bulbs with a mean lifetime of 860 hours and \(20 \%\) of these bulbs have a lifetime of less than 818 hours.
  3. Find the standard deviation of the lifetimes of bulbs from company \(Y\). Both companies sell the bulbs for the same price.
  4. State which company you would recommend. Give reasons for your answer.
Edexcel S1 2010 June Q2
10 marks Easy -1.2
2. An experiment consists of selecting a ball from a bag and spinning a coin. The bag contains 5 red balls and 7 blue balls. A ball is selected at random from the bag, its colour is noted and then the ball is returned to the bag. When a red ball is selected, a biased coin with probability \(\frac { 2 } { 3 }\) of landing heads is spun.
When a blue ball is selected a fair coin is spun.
  1. Complete the tree diagram below to show the possible outcomes and associated probabilities. \includegraphics[max width=\textwidth, alt={}, center]{039e6fcf-3222-40cc-95ea-37b8dc4a4ddb-03_787_395_734_548} \section*{Coin}
    \includegraphics[max width=\textwidth, alt={}]{039e6fcf-3222-40cc-95ea-37b8dc4a4ddb-03_1007_488_808_950}
    Shivani selects a ball and spins the appropriate coin.
  2. Find the probability that she obtains a head. Given that Tom selected a ball at random and obtained a head when he spun the appropriate coin,
  3. find the probability that Tom selected a red ball. Shivani and Tom each repeat this experiment.
  4. Find the probability that the colour of the ball Shivani selects is the same as the colour of the ball Tom selects.
Edexcel S1 2010 June Q3
11 marks Easy -1.3
3. The discrete random variable \(X\) has probability distribution given by
\(x\)- 10123
\(\mathrm { P } ( X = x )\)\(\frac { 1 } { 5 }\)\(a\)\(\frac { 1 } { 10 }\)\(a\)\(\frac { 1 } { 5 }\)
where \(a\) is a constant.
  1. Find the value of \(a\).
  2. Write down \(\mathrm { E } ( X )\).
  3. Find \(\operatorname { Var } ( X )\). The random variable \(Y = 6 - 2 X\)
  4. Find \(\operatorname { Var } ( Y )\).
  5. Calculate \(\mathrm { P } ( X \geqslant Y )\).