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Edexcel S1 2005 June Q4
10 marks Easy -1.2
4. Aeroplanes fly from City \(A\) to City \(B\). Over a long period of time the number of minutes delay in take-off from City \(A\) was recorded. The minimum delay was 5 minutes and the maximum delay was 63 minutes. A quarter of all delays were at most 12 minutes, half were at most 17 minutes and \(75 \%\) were at most 28 minutes. Only one of the delays was longer than 45 minutes. An outlier is an observation that falls either \(1.5 \times\) (interquartile range) above the upper quartile or \(1.5 \times\) (interquartile range) below the lower quartile.
  1. On the graph paper opposite draw a box plot to represent these data.
  2. Comment on the distribution of delays. Justify your answer.
  3. Suggest how the distribution might be interpreted by a passenger who frequently flies from City \(A\) to City \(B\). \includegraphics[max width=\textwidth, alt={}, center]{9698650f-ef85-468d-a703-1b40df7f9d02-07_1190_1487_278_223}
Edexcel S1 2005 June Q5
10 marks Moderate -0.8
5. The random variable \(X\) has probability function $$P ( X = x ) = \begin{cases} k x , & x = 1,2,3 \\ k ( x + 1 ) , & x = 4,5 \end{cases}$$ where \(k\) is a constant.
  1. Find the value of \(k\).
  2. Find the exact value of \(\mathrm { E } ( X )\).
  3. Show that, to 3 significant figures, \(\operatorname { Var } ( X ) = 1.47\).
  4. Find, to 1 decimal place, \(\operatorname { Var } ( 4 - 3 X )\).
Edexcel S1 2005 June Q6
11 marks Moderate -0.8
  1. A scientist found that the time taken, \(M\) minutes, to carry out an experiment can be modelled by a normal random variable with mean 155 minutes and standard deviation 3.5 minutes.
Find
  1. \(\mathrm { P } ( M > 160 )\).
  2. \(\mathrm { P } ( 150 \leqslant M \leqslant 157 )\).
  3. the value of \(m\), to 1 decimal place, such that \(\mathrm { P } ( M \leqslant m ) = 0.30\).
Edexcel S1 2005 June Q7
12 marks Moderate -0.8
7. In a school there are 148 students in Years 12 and 13 studying Science, Humanities or Arts subjects. Of these students, 89 wear glasses and the others do not. There are 30 Science students of whom 18 wear glasses. The corresponding figures for the Humanities students are 68 and 44 respectively. A student is chosen at random. Find the probability that this student
  1. is studying Arts subjects,
  2. does not wear glasses, given that the student is studying Arts subjects. Amongst the Science students, \(80 \%\) are right-handed. Corresponding percentages for Humanities and Arts students are 75\% and 70\% respectively. A student is again chosen at random.
  3. Find the probability that this student is right-handed.
  4. Given that this student is right-handed, find the probability that the student is studying Science subjects.
Edexcel S1 2006 June Q1
15 marks Easy -1.8
  1. (a) Describe the main features and uses of a box plot.
Children from schools \(A\) and \(B\) took part in a fun run for charity. The times, to the nearest minute, taken by the children from school \(A\) are summarised in Figure 1. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{c8bade79-a39a-4055-bfae-928f5338fdfc-02_398_1045_946_461}
\end{figure} (b) (i) Write down the time by which \(75 \%\) of the children in school \(A\) had completed the run.
(ii) State the name given to this value.
(c) Explain what you understand by the two crosses ( X ) on Figure 1.
For school \(B\) the least time taken by any of the children was 25 minutes and the longest time was 55 minutes. The three quartiles were 30,37 and 50 respectively.
(d) Draw a box plot to represent the data from school \(B\). \includegraphics[max width=\textwidth, alt={}, center]{c8bade79-a39a-4055-bfae-928f5338fdfc-03_798_1196_580_372}
(e) Compare and contrast these two box plots.
Edexcel S1 2006 June Q2
10 marks Moderate -0.8
2. Sunita and Shelley talk to one another once a week on the telephone. Over many weeks they recorded, to the nearest minute, the number of minutes spent in conversation on each occasion. The following table summarises their results.
Time
(to the nearest minute)
Number of
Conversations
\(5 - 9\)2
\(10 - 14\)9
\(15 - 19\)20
\(20 - 24\)13
\(25 - 29\)8
\(30 - 34\)3
Two of the conversations were chosen at random.
  1. Find the probability that both of them were longer than 24.5 minutes. The mid-point of each class was represented by \(x\) and its corresponding frequency by \(f\), giving \(\Sigma f x = 1060\).
  2. Calculate an estimate of the mean time spent on their conversations. During the following 25 weeks they monitored their weekly conversations and found that at the end of the 80 weeks their overall mean length of conversation was 21 minutes.
  3. Find the mean time spent in conversation during these 25 weeks.
  4. Comment on these two mean values.
Edexcel S1 2006 June Q3
18 marks Moderate -0.3
  1. A metallurgist measured the length, \(l \mathrm {~mm}\), of a copper rod at various temperatures, \(t ^ { \circ } \mathrm { C }\), and recorded the following results.
\(t\)\(l\)
20.42461.12
27.32461.41
32.12461.73
39.02461.88
42.92462.03
49.72462.37
58.32462.69
67.42463.05
The results were then coded such that \(x = t\) and \(y = l - 2460.00\).
  1. Calculate \(S _ { x y }\) and \(S _ { x x }\).
    (You may use \(\Sigma x ^ { 2 } = 15965.01\) and \(\Sigma x y = 757.467\) )
  2. Find the equation of the regression line of \(y\) on \(x\) in the form \(y = a + b x\).
  3. Estimate the length of the rod at \(40 ^ { \circ } \mathrm { C }\).
  4. Find the equation of the regression line of \(l\) on \(t\).
  5. Estimate the length of the rod at \(90 ^ { \circ } \mathrm { C }\).
  6. Comment on the reliability of your estimate in part (e).
Edexcel S1 2006 June Q4
7 marks Easy -1.2
  1. The random variable \(X\) has the discrete uniform distribution
$$\mathrm { P } ( X = x ) = \frac { 1 } { 5 } , \quad x = 1,2,3,4,5$$
  1. Write down the value of \(\mathrm { E } ( X )\) and show that \(\operatorname { Var } ( X ) = 2\). Find
  2. \(\mathrm { E } ( 3 X - 2 )\),
  3. \(\operatorname { Var } ( 4 - 3 X )\).
Edexcel S1 2006 June Q5
12 marks Standard +0.3
5. From experience a high-jumper knows that he can clear a height of at least 1.78 m once in 5 attempts. He also knows that he can clear a height of at least 1.65 m on 7 out of 10 attempts. Assuming that the heights the high-jumper can reach follow a Normal distribution,
  1. draw a sketch to illustrate the above information,
  2. find, to 3 decimal places, the mean and the standard deviation of the heights the high-jumper can reach,
  3. calculate the probability that he can jump at least 1.74 m .
Edexcel S1 2006 June Q6
13 marks Moderate -0.8
  1. 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
  1. 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,
  2. 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
  1. 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).