Questions — Edexcel (10514 questions)

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Edexcel S1 2007 January Q4
14 marks Moderate -0.8
  1. Summarised below are the distances, to the nearest mile, travelled to work by a random sample of 120 commuters.
Distance (to the nearest mile)Number of commuters
0-910
10-1919
20-2943
30-3925
40-498
50-596
60-695
70-793
80-891
For this distribution,
  1. describe its shape,
  2. use linear interpolation to estimate its median. The mid-point of each class was represented by \(x\) and its corresponding frequency by \(f\) giving $$\Sigma f x = 3550 \text { and } \Sigma f x ^ { 2 } = 138020$$
  3. Estimate the mean and the standard deviation of this distribution. One coefficient of skewness is given by $$\frac { 3 ( \text { mean - median } ) } { \text { standard deviation } } .$$
  4. Evaluate this coefficient for this distribution.
  5. State whether or not the value of your coefficient is consistent with your description in part (a). Justify your answer.
  6. State, with a reason, whether you should use the mean or the median to represent the data in this distribution.
  7. State the circumstance under which it would not matter whether you used the mean or the median to represent a set of data.
Edexcel S1 2007 January Q5
7 marks Easy -1.2
  1. A teacher recorded, to the nearest hour, the time spent watching television during a particular week by each child in a random sample. The times were summarised in a grouped frequency table and represented by a histogram.
One of the classes in the grouped frequency distribution was 20-29 and its associated frequency was 9. On the histogram the height of the rectangle representing that class was 3.6 cm and the width was 2 cm .
  1. Give a reason to support the use of a histogram to represent these data.
  2. Write down the underlying feature associated with each of the bars in a histogram.
  3. Show that on this histogram each child was represented by \(0.8 \mathrm {~cm} ^ { 2 }\). The total area under the histogram was \(24 \mathrm {~cm} ^ { 2 }\).
  4. Find the total number of children in the group.
Edexcel S1 2007 January Q6
5 marks Easy -2.0
  1. Give two reasons to justify the use of statistical models. It has been suggested that there are 7 stages involved in creating a statistical model. They are summarised below, with stages 3 , 4 and 7 missing. Stage 1. The recognition of a real-world problem. Stage 2. A statistical model is devised. Stage 3. Stage 4. Stage 5. Comparisons are made against the devised model. Stage 6. Statistical concepts are used to test how well the model describes the real-world problem. Stage 7.
  2. Write down the missing stages.
Edexcel S1 2007 January Q7
10 marks Moderate -0.8
The measure of intelligence, IQ, of a group of students is assumed to be Normally distributed with mean 100 and standard deviation 15.
  1. Find the probability that a student selected at random has an IQ less than 91. The probability that a randomly selected student has an IQ of at least \(100 + k\) is 0.2090 .
  2. Find, to the nearest integer, the value of \(k\).
Edexcel S1 2008 January Q1
7 marks Moderate -0.3
  1. A personnel manager wants to find out if a test carried out during an employee's interview and a skills assessment at the end of basic training is a guide to performance after working for the company for one year.
The table below shows the results of the interview test of 10 employees and their performance after one year.
EmployeeA\(B\)CD\(E\)\(F\)G\(H\)IJ
Interview test, \(x\) \%.65717977857885908162
Performance after one year, \(y \%\).65748264877861657969
$$\text { [You may use } \sum x ^ { 2 } = 60475 , \sum y ^ { 2 } = 53122 , \sum x y = 56076 \text { ] }$$
  1. Showing your working clearly, calculate the product moment correlation coefficient between the interview test and the performance after one year. The product moment correlation coefficient between the skills assessment and the performance after one year is - 0.156 to 3 significant figures.
  2. Use your answer to part (a) to comment on whether or not the interview test and skills assessment are a guide to the performance after one year. Give clear reasons for your answers.
Edexcel S1 2008 January Q2
14 marks Easy -1.3
2. Cotinine is a chemical that is made by the body from nicotine which is found in cigarette smoke. A doctor tested the blood of 12 patients, who claimed to smoke a packet of cigarettes a day, for cotinine. The results, in appropriate units, are shown below.
Patient\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)\(I\)\(J\)\(K\)\(L\)
Cotinine
level, \(X\)
160390169175125420171250210258186243
$$\text { [You may use } \sum x ^ { 2 } = 724 \text { 961] }$$
  1. Find the mean and standard deviation of the level of cotinine in a patient's blood.
  2. Find the median, upper and lower quartiles of these data. A doctor suspects that some of his patients have been smoking more than a packet of cigarettes per day. He decides to use \(\mathrm { Q } _ { 3 } + 1.5 \left( \mathrm { Q } _ { 3 } - \mathrm { Q } _ { 1 } \right)\) to determine if any of the cotinine results are far enough away from the upper quartile to be outliers.
  3. Identify which patient(s) may have been smoking more than a packet of cigarettes a day. Show your working clearly. Research suggests that cotinine levels in the blood form a skewed distribution.
    One measure of skewness is found using \(\frac { \left( Q _ { 1 } - 2 Q _ { 2 } + Q _ { 3 } \right) } { \left( Q _ { 3 } - Q _ { 1 } \right) }\).
  4. Evaluate this measure and describe the skewness of these data.
Edexcel S1 2008 January Q3
5 marks Easy -1.2
3. The histogram in Figure 1 shows the time taken, to the nearest minute, for 140 runners to complete a fun run. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{af84d17b-5308-4b1e-99b5-40c5df5bf01e-06_1027_1509_367_258} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Use the histogram to calculate the number of runners who took between 78.5 and 90.5 minutes to complete the fun run.
Edexcel S1 2008 January Q4
10 marks Moderate -0.8
4. A second hand car dealer has 10 cars for sale. She decides to investigate the link between the age of the cars, \(x\) years, and the mileage, \(y\) thousand miles. The data collected from the cars are shown in the table below.
Age, \(x\)
(years)
22.5344.54.55366.5
Mileage, \(y\)
(thousands)
22343337404549305858
[You may assume that \(\sum x = 41 , \sum y = 406 , \sum x ^ { 2 } = 188 , \sum x y = 1818.5\) ]
  1. Find \(S _ { x x }\) and \(S _ { x y }\).
  2. Find the equation of the least squares regression line in the form \(y = a + b x\). Give the values of \(a\) and \(b\) to 2 decimal places.
  3. Give a practical interpretation of the slope \(b\).
  4. Using your answer to part (b), find the mileage predicted by the regression line for a 5 year old car. \(\_\_\_\_\)
Edexcel S1 2008 January Q5
16 marks Easy -1.2
5. The following shows the results of a wine tasting survey of 100 people. \begin{displayquote} 96 like wine \(A\),
93 like wine \(B\),
96 like wine \(C\),
92 like \(A\) and \(B\),
91 like \(B\) and \(C\),
93 like \(A\) and \(C\),
90 like all three wines.
  1. Draw a Venn Diagram to represent these data. \end{displayquote} Find the probability that a randomly selected person from the survey likes
  2. none of the three wines,
  3. wine \(A\) but not wine \(B\),
  4. any wine in the survey except wine \(C\),
  5. exactly two of the three kinds of wine. Given that a person from the survey likes wine \(A\),
  6. find the probability that the person likes wine \(C\).
Edexcel S1 2008 January Q6
9 marks Moderate -0.8
6. The weights of bags of popcorn are normally distributed with mean of 200 g and \(60 \%\) of all bags weighing between 190 g and 210 g .
  1. Write down the median weight of the bags of popcorn.
  2. Find the standard deviation of the weights of the bags of popcorn. A shopkeeper finds that customers will complain if their bag of popcorn weighs less than 180 g .
  3. Find the probability that a customer will complain.
Edexcel S1 2008 January Q7
14 marks Moderate -0.8
7. Tetrahedral dice have four faces. Two fair tetrahedral dice, one red and one blue, have faces numbered \(0,1,2\), and 3 respectively. The dice are rolled and the numbers face down on the two dice are recorded. The random variable \(R\) is the score on the red die and the random variable \(B\) is the score on the blue die.
  1. Find \(\mathrm { P } ( R = 3\) and \(B = 0 )\). The random variable \(T\) is \(R\) multiplied by \(B\).
  2. Complete the diagram below to represent the sample space that shows all the possible values of \(T\). \includegraphics[max width=\textwidth, alt={}, center]{af84d17b-5308-4b1e-99b5-40c5df5bf01e-13_732_771_834_621} \section*{Sample space diagram of \(T\)}
  3. The table below represents the probability distribution of the random variable \(T\).
    \(t\)0123469
    \(\mathrm { P } ( T = t )\)\(a\)\(b\)\(1 / 8\)\(1 / 8\)\(c\)\(1 / 8\)\(d\)
    Find the values of \(a , b , c\) and \(d\). Find the values of
  4. \(\mathrm { E } ( T )\),
  5. \(\operatorname { Var } ( T )\).
Edexcel S1 2009 January Q1
11 marks Moderate -0.8
  1. A teacher is monitoring the progress of students using a computer based revision course. The improvement in performance, \(y\) marks, is recorded for each student along with the time, \(x\) hours, that the student spent using the revision course. The results for a random sample of 10 students are recorded below.
\(x\)
hours
1.03.54.01.51.30.51.82.52.33.0
\(y\)
marks
5302710- 3- 5715- 1020
$$\text { [You may use } \sum x = 21.4 , \quad \sum y = 96 , \quad \sum x ^ { 2 } = 57.22 , \quad \sum x y = 313.7 \text { ] }$$
  1. Calculate \(S _ { x x }\) and \(S _ { x y }\).
  2. Find the equation of the least squares regression line of \(y\) on \(x\) in the form \(y = a + b x\).
  3. Give an interpretation of the gradient of your regression line. Rosemary spends 3.3 hours using the revision course.
  4. Predict her improvement in marks. Lee spends 8 hours using the revision course claiming that this should give him an improvement in performance of over 60 marks.
  5. Comment on Lee's claim.
Edexcel S1 2009 January Q2
8 marks Moderate -0.8
2. A group of office workers were questioned for a health magazine and \(\frac { 2 } { 5 }\) were found to take regular exercise. When questioned about their eating habits \(\frac { 2 } { 3 }\) said they always eat breakfast and, of those who always eat breakfast \(\frac { 9 } { 25 }\) also took regular exercise. Find the probability that a randomly selected member of the group
  1. always eats breakfast and takes regular exercise,
  2. does not always eat breakfast and does not take regular exercise.
  3. Determine, giving your reason, whether or not always eating breakfast and taking regular exercise are statistically independent.
Edexcel S1 2009 January Q3
16 marks Moderate -0.3
3. When Rohit plays a game, the number of points he receives is given by the discrete random variable \(X\) with the following probability distribution.
\(x\)0123
\(\mathrm { P } ( X = x )\)0.40.30.20.1
  1. Find \(\mathrm { E } ( X )\).
  2. Find \(\mathrm { F } ( 1.5 )\).
  3. Show that \(\operatorname { Var } ( X ) = 1\)
  4. Find \(\operatorname { Var } ( 5 - 3 X )\). Rohit can win a prize if the total number of points he has scored after 5 games is at least 10. After 3 games he has a total of 6 points. You may assume that games are independent.
  5. Find the probability that Rohit wins the prize.
Edexcel S1 2009 January Q4
14 marks Moderate -0.8
4. In a study of how students use their mobile telephones, the phone usage of a random sample of 11 students was examined for a particular week. The total length of calls, \(y\) minutes, for the 11 students were $$17,23,35,36,51,53,54,55,60,77,110$$
  1. Find the median and quartiles for these data. A value that is greater than \(Q _ { 3 } + 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)\) or smaller than \(Q _ { 1 } - 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)\) is defined as an outlier.
  2. Show that 110 is the only outlier.
  3. Using the graph paper on page 15 draw a box plot for these data indicating clearly the position of the outlier. The value of 110 is omitted.
  4. Show that \(S _ { y y }\) for the remaining 10 students is 2966.9 These 10 students were each asked how many text messages, \(x\), they sent in the same week. The values of \(S _ { x x }\) and \(S _ { x y }\) for these 10 students are \(S _ { x x } = 3463.6\) and \(S _ { x y } = - 18.3\).
  5. Calculate the product moment correlation coefficient between the number of text messages sent and the total length of calls for these 10 students. A parent believes that a student who sends a large number of text messages will spend fewer minutes on calls.
  6. Comment on this belief in the light of your calculation in part (e). \includegraphics[max width=\textwidth, alt={}, center]{d5d000c7-de42-461a-ba05-6c8b2c333780-09_611_1593_297_178}
Edexcel S1 2009 January Q5
16 marks Standard +0.3
5. In a shopping survey a random sample of 104 teenagers were asked how many hours, to the nearest hour, they spent shopping in the last month. The results are summarised in the table below.
Number of hoursMid-pointFrequency
0-52.7520
6-76.516
8-10918
11-151325
16-2520.515
26-503810
A histogram was drawn and the group ( \(8 - 10\) ) hours was represented by a rectangle that was 1.5 cm wide and 3 cm high.
  1. Calculate the width and height of the rectangle representing the group (16-25) hours.
  2. Use linear interpolation to estimate the median and interquartile range.
  3. Estimate the mean and standard deviation of the number of hours spent shopping.
  4. State, giving a reason, the skewness of these data.
  5. State, giving a reason, which average and measure of dispersion you would recommend to use to summarise these data.
Edexcel S1 2009 January Q6
10 marks Moderate -0.8
6. The random variable \(X\) has a normal distribution with mean 30 and standard deviation 5 .
  1. Find \(\mathrm { P } ( X < 39 )\).
  2. Find the value of \(d\) such that \(\mathrm { P } ( X < d ) = 0.1151\)
  3. Find the value of \(e\) such that \(\mathrm { P } ( X > e ) = 0.1151\)
  4. Find \(\mathrm { P } ( d < X < e )\).
Edexcel S1 2011 January Q1
6 marks Easy -1.2
  1. A random sample of 50 salmon was caught by a scientist. He recorded the length \(l \mathrm {~cm}\) and weight \(w \mathrm {~kg}\) of each salmon.
The following summary statistics were calculated from these data. \(\sum l = 4027 \quad \sum l ^ { 2 } = 327754.5 \quad \sum w = 357.1 \quad \sum l w = 29330.5 \quad S _ { w w } = 289.6\)
  1. Find \(S _ { l l }\) and \(S _ { l w }\)
  2. Calculate, to 3 significant figures, the product moment correlation coefficient between \(l\) and \(w\).
  3. Give an interpretation of your coefficient.
Edexcel S1 2011 January Q2
4 marks Easy -1.8
  1. Keith records the amount of rainfall, in mm , at his school, each day for a week. The results are given below.
    0.0
    0.5
    1.8
    2.8
    2.3
    5.6
    9.4
Jenny then records the amount of rainfall, \(x \mathrm {~mm}\), at the school each day for the following 21 days. The results for the 21 days are summarised below. $$\sum x = 84.6$$
  1. Calculate the mean amount of rainfall during the whole 28 days. Keith realises that he has transposed two of his figures. The number 9.4 should have been 4.9 and the number 0.5 should have been 5.0 Keith corrects these figures.
  2. State, giving your reason, the effect this will have on the mean.
Edexcel S1 2011 January Q3
9 marks Easy -1.2
3. Over a long period of time a small company recorded the amount it received in sales per month. The results are summarised below.
Amount received in sales (£1000s)
Two lowest values3,4
Lower quartile7
Median12
Upper quartile14
Two highest values20,25
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 below, draw a box plot to represent these data, indicating clearly any outliers.
    (5) \includegraphics[max width=\textwidth, alt={}, center]{c78ec7b6-dd06-4de1-94c2-052a5577dd10-05_933_1226_1283_367}
  2. State the skewness of the distribution of the amount of sales received. Justify your answer.
  3. The company claims that for \(75 \%\) of the months, the amount received per month is greater than \(\pounds 10000\). Comment on this claim, giving a reason for your answer.
    (2)
Edexcel S1 2011 January Q4
6 marks Moderate -0.8
  1. A farmer collected data on the annual rainfall, \(x \mathrm {~cm}\), and the annual yield of peas, \(p\) tonnes per acre.
The data for annual rainfall was coded using \(v = \frac { x - 5 } { 10 }\) and the following statistics were found. $$S _ { v v } = 5.753 \quad S _ { p v } = 1.688 \quad S _ { p p } = 1.168 \quad \bar { p } = 3.22 \quad \bar { v } = 4.42$$
  1. Find the equation of the regression line of \(p\) on \(v\) in the form \(p = a + b v\).
  2. Using your regression line estimate the annual yield of peas per acre when the annual rainfall is 85 cm .
Edexcel S1 2011 January Q5
7 marks Moderate -0.8
5. On a randomly chosen day, each of the 32 students in a class recorded the time, \(t\) minutes to the nearest minute, they spent on their homework. The data for the class is summarised in the following table.
Time, \(t\)Number of students
10-192
20-294
30-398
40-4911
50-695
70-792
  1. Use interpolation to estimate the value of the median. Given that $$\sum t = 1414 \quad \text { and } \quad \sum t ^ { 2 } = 69378$$
  2. find the mean and the standard deviation of the times spent by the students on their homework.
  3. Comment on the skewness of the distribution of the times spent by the students on their homework. Give a reason for your answer.
Edexcel S1 2011 January Q6
14 marks Moderate -0.8
  1. The discrete random variable \(X\) has the probability distribution
\(x\)1234
\(\mathrm { P } ( X = x )\)\(k\)\(2 k\)\(3 k\)\(4 k\)
  1. Show that \(k = 0.1\) Find
  2. \(\mathrm { E } ( X )\)
  3. \(\mathrm { E } \left( X ^ { 2 } \right)\)
  4. \(\operatorname { Var } ( 2 - 5 X )\) Two independent observations \(X _ { 1 }\) and \(X _ { 2 }\) are made of \(X\).
  5. Show that \(\mathrm { P } \left( X _ { 1 } + X _ { 2 } = 4 \right) = 0.1\)
  6. Complete the probability distribution table for \(X _ { 1 } + X _ { 2 }\)
    \(y\)2345678
    \(\mathrm { P } \left( X _ { 1 } + X _ { 2 } = y \right)\)0.010.040.100.250.24
  7. Find \(\mathrm { P } \left( 1.5 < X _ { 1 } + X _ { 2 } \leqslant 3.5 \right)\)
Edexcel S1 2011 January Q7
17 marks Standard +0.3
  1. The bag \(P\) contains 6 balls of which 3 are red and 3 are yellow.
The bag \(Q\) contains 7 balls of which 4 are red and 3 are yellow.
A ball is drawn at random from bag \(P\) and placed in bag \(Q\). A second ball is drawn at random from bag \(P\) and placed in bag \(Q\).
A third ball is then drawn at random from the 9 balls in bag \(Q\). The event \(A\) occurs when the 2 balls drawn from bag \(P\) are of the same colour. The event \(B\) occurs when the ball drawn from bag \(Q\) is red.
  1. Complete the tree diagram shown below.
    (4) \includegraphics[max width=\textwidth, alt={}, center]{c78ec7b6-dd06-4de1-94c2-052a5577dd10-12_1201_1390_753_269}
  2. Find \(\mathrm { P } ( A )\)
  3. Show that \(\mathrm { P } ( B ) = \frac { 5 } { 9 }\)
  4. Show that \(\mathrm { P } ( A \cap B ) = \frac { 2 } { 9 }\)
  5. Hence find \(\mathrm { P } ( A \cup B )\)
  6. Given that all three balls drawn are the same colour, find the probability that they are all red.
    (3)
Edexcel S1 2011 January Q8
12 marks Standard +0.3
The weight, \(X\) grams, of soup put in a tin by machine \(A\) is normally distributed with a mean of 160 g and a standard deviation of 5 g .
A tin is selected at random.
  1. Find the probability that this tin contains more than 168 g . The weight stated on the tin is \(w\) grams.
  2. Find \(w\) such that \(\mathrm { P } ( X < w ) = 0.01\) The weight, \(Y\) grams, of soup put into a carton by machine \(B\) is normally distributed with mean \(\mu\) grams and standard deviation \(\sigma\) grams.
  3. Given that \(\mathrm { P } ( Y < 160 ) = 0.99\) and \(\mathrm { P } ( Y > 152 ) = 0.90\) find the value of \(\mu\) and the value of \(\sigma\).