Questions S1 (1967 questions)

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Edexcel S1 2004 June Q1
  1. A fair die has six faces numbered \(1,2,2,3,3\) and 3 . The die is rolled twice and the number showing on the uppermost face is recorded each time.
Find the probability that the sum of the two numbers recorded is at least 5 .
(5)
Edexcel S1 2004 June Q2
2. A researcher thinks there is a link between a person's height and level of confidence. She measured the height \(h\), to the nearest cm , of a random sample of 9 people. She also devised a test to measure the level of confidence \(c\) of each person. The data are shown in the table below.
\(h\)179169187166162193161177168
\(c\)569561579561540598542565573
[You may use \(\Sigma h ^ { 2 } = 272094 , \Sigma c ^ { 2 } = 2878966 , \Sigma h c = 884484\) ]
  1. Draw a scatter diagram to illustrate these data.
  2. Find exact values of \(S _ { h c } S _ { h h }\) and \(S _ { c c }\).
  3. Calculate the value of the product moment correlation coefficient for these data.
  4. Give an interpretation of your correlation coefficient.
  5. Calculate the equation of the regression line of \(c\) on \(h\) in the form \(c = a + b h\).
  6. Estimate the level of confidence of a person of height 180 cm .
  7. State the range of values of \(h\) for which estimates of \(c\) are reliable.
Edexcel S1 2004 June Q3
3. A discrete random variable \(X\) has a probability function as shown in the table below, where \(a\) and \(b\) are constants.
\(x\)0123
\(\mathrm { P } ( X = x )\)0.20.3\(b\)\(a\)
Given that \(\mathrm { E } ( X ) = 1.7\),
  1. find the value of \(a\) and the value of \(b\). Find
  2. \(\mathrm { P } ( 0 < X < 1.5 )\),
  3. \(\mathrm { E } ( 2 X - 3 )\).
  4. Show that \(\operatorname { Var } ( X ) = 1.41\).
  5. Evaluate \(\operatorname { Var } ( 2 X - 3 )\).
Edexcel S1 2004 June Q4
4. The attendance at college of a group of 18 students was recorded for a 4-week period. The number of students actually attending each of 16 classes are shown below.
18181717
16171618
18141718
15171816
    1. Calculate the mean and the standard deviation of the number of students attending these classes.
    2. Express the mean as a percentage of the 18 students in the group. In the same 4-week period, the attendance of a different group of 20, students is shown below.
      20161819
      15141415
      18151617
      16181514
  2. Construct a back-to-back stem and leaf diagram to represent the attendance in both groups.
  3. Find the mode, median and inter-quartile range for each group of students. The mean percentage attendance and standard deviation for the second group of students are 81.25 and 1.82 respectively.
  4. Compare and contrast the attendance of these 2 groups of students.
Edexcel S1 2004 June Q5
5. A health club lets members use, on each visit, its facilities for as long as they wish. The club's records suggest that the length of a visit can be modelled by a normal distribution with mean 90 minutes. Only \(20 \%\) of members stay for more than 125 minutes.
  1. Find the standard deviation of the normal distribution.
  2. Find the probability that a visit lasts less than 25 minutes. The club introduce a closing time of 10:00 pm. Tara arrives at the club at 8:00 pm.
  3. Explain whether or not this normal distribution is still a suitable model for the length of her visit.
Edexcel S1 2004 June Q6
6. Three events \(A , B\) and \(C\) are defined in the sample space \(S\). The events \(A\) and \(B\) are mutually exclusive and \(A\) and \(C\) are independent.
  1. Draw a Venn diagram to illustrate the relationships between the 3 events and the sample space. Given that \(\mathrm { P } ( A ) = 0.2 , \mathrm { P } ( B ) = 0.4\) and \(\mathrm { P } ( A \cup C ) = 0.7\), find
  2. \(\mathrm { P } ( A C )\),
  3. \(\mathrm { P } ( A \cup B )\),
  4. \(\mathrm { P } ( C )\). END
Edexcel S1 2005 June Q1
  1. The scatter diagrams below were drawn by a student.
$$\begin{aligned} & y \underset { x } { \begin{array} { l l l l } & &
+ & & &
+ & + & + &
+ & + & + \end{array} } \end{aligned}$$ The student calculated the value of the product moment correlation coefficient for each of the sets of data. The values were $$\begin{array} { l l l } 0.68 & - 0.79 & 0.08 \end{array}$$ Write down, with a reason, which value corresponds to which scatter diagram.
(6)
Edexcel S1 2005 June Q2
2. The following table summarises the distances, to the nearest km , that 134 examiners travelled to attend a meeting in London.
Distance (km)Number of examiners
41-454
46-5019
51-6053
61-7037
71-9015
91-1506
  1. Give a reason to justify the use of a histogram to represent these data.
  2. Calculate the frequency densities needed to draw a histogram for these data.
    (DO NOT DRAW THE HISTOGRAM)
  3. Use interpolation to estimate the median \(Q _ { 2 }\), the lower quartile \(Q _ { 1 }\), and the upper quartile \(Q _ { 3 }\) of these data. The mid-point of each class is represented by \(x\) and the corresponding frequency by \(f\). Calculations then give the following values $$\Sigma f _ { x } = 8379.5 \quad \text { and } \quad \Sigma f _ { x ^ { 2 } } = 557489.75$$
  4. Calculate an estimate of the mean and an estimate of the standard deviation for these data. One coefficient of skewness is given by $$\frac { Q _ { 3 } - 2 Q _ { 2 } + Q _ { 1 } } { Q _ { 3 } - Q _ { 1 } }$$
  5. Evaluate this coefficient and comment on the skewness of these data.
  6. Give another justification of your comment in part (e).
Edexcel S1 2005 June Q3
  1. A long distance lorry driver recorded the distance travelled, \(m\) miles, and the amount of fuel used, \(f\) litres, each day. Summarised below are data from the driver's records for a random sample of 8 days.
The data are coded such that \(x = m - 250\) and \(y = f - 100\). $$\Sigma x = 130 \quad \Sigma y = 48 \quad \Sigma x y = 8880 \quad \mathrm {~S} _ { x x } = 20487.5$$
  1. Find the equation of the regression line of \(y\) on \(x\) in the form \(y = a + b x\).
  2. Hence find the equation of the regression line of \(f\) on \(m\).
  3. Predict the amount of fuel used on a journey of 235 miles.
Edexcel S1 2005 June Q4
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
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
  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
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
  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
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
  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
  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
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
  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
  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
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
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
  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
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
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 \(\mathrm { P } ( 20 < X < d ) = 0.4641\)