Questions — Edexcel (10514 questions)

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Edexcel S1 Q5
16 marks Easy -1.3
5. Each child in class 3A was given a packet of seeds to plant. The stem and leaf diagram below shows how many seedlings were visible in each child's tray one week after planting.
Number of seedlings(2 | 1 means 21)Totals
002(2)
0(0)
11(1)
157(2)
201334(5)
25777899(7)
30001224(7)
35688(4)
4134(3)
  1. Find the median and interquartile range for these data.
  2. Use the quartiles to describe the skewness of the data. Show your method clearly. The mean and standard deviation for these data were 27.2 and 10.3 respectively.
  3. Explaining your answer, state whether you would recommend using these values or your answers to part (a) to summarise these data. Outliers are defined to be values outside of the limits \(\mathrm { Q } _ { 1 } - 2 s\) and \(\mathrm { Q } _ { 3 } + 2 s\) where \(s\) is the standard deviation given above.
  4. Represent these data with a boxplot identifying clearly any outliers.
Edexcel S1 Q6
17 marks Moderate -0.8
6. A school introduced a new programme of support lessons in 1994 with a view to improving grades in GCSE English. The table below shows the number of years since 1994, n, and the corresponding percentage of students achieving A to C grades in GCSE English, \(p\), for each year.
\(n\)123456
\(p ( \% )\)35.237.140.639.043.444.8
  1. Represent these data on a scatter diagram. You may use the following values. $$\Sigma n = 21 , \quad \Sigma p = 240.1 , \quad \Sigma n ^ { 2 } = 91 , \quad \Sigma p ^ { 2 } = 9675.41 , \quad \Sigma n p = 873 .$$
  2. Find an equation of the regression line of \(p\) on \(n\) and draw it on your graph.
  3. Calculate the product moment correlation coefficient for these data and comment on the suitability of a linear model for the relationship between \(n\) and \(p\) during this period.
Edexcel S1 Q1
7 marks Easy -1.3
  1. The discrete random variable \(Y\) has the following probability distribution.
\(y\)\({ } ^ { - } 2\)\({ } ^ { - } 1\)012
\(\mathrm { P } ( Y = y )\)0.10.150.20.30.25
Find
  1. \(\mathrm { F } ( 0.5 )\),
  2. \(\mathrm { P } \left( { } ^ { - } 1 < Y < 1.9 \right)\),
  3. \(\mathrm { E } ( Y )\),
  4. \(\mathrm { E } ( 3 Y - 1 )\).
Edexcel S1 Q2
8 marks Easy -1.2
2. A supermarket manager believes that those of her staff on lower rates of pay tend to work more hours of overtime.
  1. Suggest why this might be the case. To investigate her theory the manager recorded the number of hours of overtime, \(h\), worked by each of the store's 18 full-time staff during one week. She also recorded each employee's hourly rate of pay, \(\pounds p\), and summarised her results as follows: $$\Sigma p = 86 , \quad \Sigma h = 104.5 , \quad \Sigma p ^ { 2 } = 420.58 , \quad \Sigma h ^ { 2 } = 830.25 , \quad \Sigma p h = 487.3$$
  2. Calculate the product moment correlation coefficient for these data.
  3. Comment on the manager's hypothesis.
Edexcel S1 Q3
9 marks Moderate -0.8
3. A magazine collected data on the total cost of the reception at each of a random sample of 80 weddings. The data is grouped and coded using \(y = \frac { C - 3250 } { 250 }\), where \(C\) is the mid-point in pounds of each class, giving \(\sum f y = 37\) and \(\sum f y ^ { 2 } = 2317\).
  1. Using these values, calculate estimates of the mean and standard deviation of the cost of the receptions in the sample.
  2. Explain why your answers to part (a) are only estimates. The median of the data was \(\pounds 3050\).
  3. Comment on the skewness of the data and suggest a reason for it.
Edexcel S1 Q4
11 marks Moderate -0.3
4. The random variable \(A\) is normally distributed with a mean of 32.5 and a variance of 18.6 Find
  1. \(\mathrm { P } ( A < 38.2 )\),
  2. \(\mathrm { P } ( 31 \leq A \leq 35 )\), The random variable \(B\) is normally distributed with a standard deviation of 7.2
    Given also that \(\mathrm { P } ( B > 110 ) = 0.138\),
  3. find the mean of \(B\).
Edexcel S1 Q5
11 marks Standard +0.3
5. A group of children were each asked to try and complete a task to test hand-eye coordination. Each child repeated the task until he or she had been successful or had made four attempts. The number of attempts made by the children in the group are summarised in the table below.
Number of attempts1234
Number of children4326133
  1. Calculate the mean and standard deviation of the number of attempts made by each child. It is suggested that the number of attempts made by each child could be modelled by a discrete random variable \(X\) with the probability function $$P ( X = x ) = \left\{ \begin{array} { c c } k \left( 20 - x ^ { 2 } \right) , & x = 1,2,3,4 \\ 0 , & \text { otherwise } \end{array} \right.$$
  2. Show that \(k = \frac { 1 } { 50 }\).
  3. Find \(\mathrm { E } ( X )\).
  4. Comment on the suitability of this model.
Edexcel S1 Q6
14 marks Moderate -0.8
6. Serving against his regular opponent, a tennis player has a \(65 \%\) chance of getting his first serve in. If his first serve is in he then has a \(70 \%\) chance of winning the point but if his first serve is not in, he only has a \(45 \%\) chance of winning the point.
  1. Represent this information on a tree diagram. For a point on which this player served to his regular opponent, find the probability that
  2. he won the point,
  3. his first serve went in given that he won the point,
  4. his first serve didn't go in given that he lost the point.
Edexcel S1 Q7
15 marks Moderate -0.8
7. Pipes-R-us manufacture a special lightweight aluminium tubing. The price \(\pounds P\), for each length, \(l\) metres, that the company sells is shown in the table.
\(l\) (metres)0.50.81.01.5246
\(P ( \pounds )\)2.503.404.005.206.0010.5015.00
  1. Represent these data on a scatter diagram. You may use $$\Sigma l = 15.8 , \quad \Sigma P = 46.6 , \quad \Sigma l ^ { 2 } = 60.14 , \quad \Sigma l P = 159.77$$
  2. Find the equation of the regression line of \(P\) on \(l\) in the form \(P = a + b l\).
  3. Give a practical interpretation of the constant b. In response to customer demand Pipes- \(R\)-us decide to start selling tubes cut to specific lengths. Initially the company decides to use the regression line found in part (b) as a pricing formula for this new service.
  4. Calculate the price that Pipes- \(R\)-us should charge for 5.2 metres of the tubing.
  5. Suggest a reason why Pipes- \(R\)-us might not offer prices based on the regression line for any length of tubing.
Edexcel S1 Q1
5 marks Easy -1.2
  1. The discrete random variable \(X\) has the following probability distribution.
\(x\)\(k\)\(k + 4\)\(2 k\)
\(\mathrm { P } ( X = x )\)\(\frac { 1 } { 8 }\)\(\frac { 3 } { 8 }\)\(\frac { 1 } { 2 }\)
  1. Find and simplify an expression in terms of \(k\) for \(\mathrm { E } ( X )\). Given that \(\mathrm { E } ( X ) = 9\),
  2. find the value of \(k\).
Edexcel S1 Q2
8 marks Easy -1.2
2.
  1. Explain briefly what is meant by a statistical model.
  2. State, with a reason, whether or not the normal distribution might be suitable for modelling each of the following:
    1. The number of children in a family;
    2. The time taken for a particular employee to cycle to work each day using the same route;
    3. The quarterly electricity bills for a particular house.
Edexcel S1 Q3
9 marks Moderate -0.3
3. The probability that Ajita gets up before 6.30 am in the morning is 0.7 The probability that she goes for a run in the morning is 0.35
The probability that Ajita gets up after 6.30 am and does not go for a run is 0.22
Let \(A\) represent the event that Ajita gets up before 6.30 am and \(B\) represent the event that she goes for a run in the morning. Find
  1. \(\mathrm { P } ( A \cup B )\),
  2. \(\mathrm { P } \left( A \cap B ^ { \prime } \right)\),
  3. \(\mathrm { P } ( B \mid A )\).
  4. State, with a reason, whether or not events \(A\) and \(B\) are independent.
Edexcel S1 Q4
10 marks Standard +0.3
4. A company produces jars of English Honey. The weight of the glass jars used is normally distributed with a mean of 122.3 g and a standard deviation of 2.6 g . Calculate the probability that a randomly chosen jar will weigh
  1. less than 127 g ,
  2. less than 121.5 g . The weight of honey put into each jar by a machine is normally distributed with a standard deviation of 1.6 g . The machine operator can adjust the mean weight of the honey put into each jar without changing the standard deviation.
  3. Find, correct to 4 significant figures, the minimum that the mean weight can be set to such that at most 1 in 20 of the jars will contain less than 454 g .
    (4 marks)
Edexcel S1 Q5
13 marks Standard +0.3
5. The letters of the word DISTRIBUTION are written on separate cards. The cards are then shuffled and the top three are turned over. Let the random variable \(V\) be the number of vowels that are turned over.
  1. Show that \(\mathrm { P } ( V = 1 ) = \frac { 21 } { 44 }\).
  2. Find the probability distribution of \(V\).
  3. Find \(\mathrm { E } ( V )\) and \(\operatorname { Var } ( V )\).
Edexcel S1 Q6
13 marks Moderate -0.3
6. A cinema recorded the number of people at each showing of each film during a one-week period. The results are summarised in the table below.
Number of peopleNumber of showings
1-4036
41-6020
61-8033
81-10024
101-15036
151-20039
201-30052
  1. Draw a histogram on graph paper to illustrate these data.
  2. Calculate estimates of the median and quartiles of these data.
  3. Use your answers to part (b) to show that the data is positively skewed.
Edexcel S1 Q7
17 marks Standard +0.3
7. A new vaccine is tested over a six-month period in one health authority. The table shows the number of new cases of the disease, \(d\), reported in the \(m\) th month after the trials began.
\(m\)123456
\(d\)1026961585248
A doctor suggests that a relationship of the form \(d = a + b x\) where \(x = \frac { 1 } { m }\) can be used to model the situation.
  1. Tabulate the values of \(x\) corresponding to the given values of \(d\) and plot a scatter diagram of \(d\) against \(x\).
  2. Explain how your scatter diagram supports the suggested model. You may use $$\Sigma x = 2.45 , \quad \Sigma d = 390 , \quad \Sigma x ^ { 2 } = 1.491 , \quad \Sigma x d = 189.733$$
  3. Find an equation of the regression line \(d\) on \(x\) in the form \(d = a + b x\).
  4. Use your regression line to estimate how many new cases of the disease there will be in the 13th month after the trial began.
  5. Comment on the reliability of your answer to part (d).
Edexcel S1 Q1
11 marks Moderate -0.8
  1. A net was used to catch swallows so that they could be ringed and examined. The weights of 55 adult birds were recorded and the results are summarised in the table below.
Weight (g)\(14 - 19\)\(20 - 21\)\(22 - 23\)\(24 - 25\)\(26 - 29\)\(30 - 35\)
Frequency36152092
  1. For these data calculate estimates of
    1. the median,
    2. the \(33 ^ { \text {rd } }\) percentile. These data are represented by a histogram and the bar representing the 24-25 group is 1 cm wide and 20 cm high.
  2. Calculate the dimensions of the bars representing the groups
    1. 20-21
    2. 26-29
Edexcel S1 Q2
11 marks Moderate -0.8
2. The discrete random variable \(X\) has the probability function shown below. $$P ( X = x ) = \left\{ \begin{array} { l c } \frac { k } { x } , & x = 1,2,3,4 \\ 0 , & \text { otherwise } . \end{array} \right.$$
  1. Show that \(k = \frac { 12 } { 25 }\) Find
  2. \(\mathrm { F } ( 2 )\),
  3. \(\mathrm { E } ( X )\),
  4. \(\mathrm { E } \left( X ^ { 2 } + 2 \right)\).
Edexcel S1 Q3
12 marks Standard +0.3
3. A study was made of the heights of boys of different ages in Lancashire. The study concluded that the heights of 13 year-old boys are normally distributed with a mean of 156 cm and a variance of \(73 \mathrm {~cm} ^ { 2 }\). Find the probability that a 13 year-old boy chosen at random will be
  1. more than 165 cm tall,
  2. between 156 and 165 cm tall. The study also concluded that the heights of 14 year-old boys are normally distributed with a mean of 160 cm and a variance of \(79 \mathrm {~cm} ^ { 2 }\). One 13 year-old and one 14 year-old boy are chosen at random.
  3. Find the probability that both boys are more than 165 cm tall.
  4. State, with a reason, whether the probability that the combined height of the two boys is more than 330 cm is more or less than your answer to part (c).
    (2 marks)
Edexcel S1 Q4
13 marks Moderate -0.8
4. A company offering a bicycle courier service within London collected data on the delivery times for a sample of jobs completed by staff at each of its two offices. The times, \(t\) minutes, for 20 deliveries handled by the company's Hammersmith office were summarised by $$\Sigma t = 427 , \text { and } \Sigma t ^ { 2 } = 11077$$
  1. Find the mean and variance of the delivery times in this sample. The company's Holborn office handles more business, so the delivery times for a sample of 30 jobs handled by this office was taken. The mean and standard deviation of this sample were 18.5 minutes and 8.2 minutes respectively.
  2. Find the mean and variance of the delivery times of the combined sample of 50 deliveries.
Edexcel S1 Q5
14 marks Moderate -0.8
5. A College employs 75 teachers, of whom 47 are full-time and the rest are part-time. Of the 39 male teachers at the College, 26 are full-time.
  1. Represent this information on a Venn diagram.
  2. One teacher is selected at random to be interviewed by an inspector. Find the probability that the teacher chosen
    1. works full-time and is female,
    2. works part-time, given that he is male.
  3. Three teachers are selected at random to be observed by an inspector during one day. Find correct to 3 significant figures the probability that
    1. all three teachers chosen work full-time,
    2. at least one of the three teachers chosen is female.
Edexcel S1 Q6
14 marks Moderate -0.8
6. A physics student recorded the length, \(l \mathrm {~cm}\), of a spring when different masses, \(m\) grams, were suspended from it giving the following results.
\(m ( \mathrm {~g} )\)50100200300400500600700
\(l ( \mathrm {~cm} )\)7.810.716.522.128.033.935.235.6
  1. Represent these data on a scatter diagram with \(l\) on the vertical axis. The student decides to find the equation of a regression line of the form \(l = a + b m\) using only the data for \(m \leq 500 \mathrm {~g}\).
  2. Give a reason to support the fitting of such a regression line and explain why the student is excluding two of his values.
    (2 marks)
    You may use $$\Sigma m = 1550 , \quad \Sigma l = 119 , \quad \Sigma m ^ { 2 } = 552500 , \quad \Sigma l ^ { 2 } = 2869.2 , \quad \Sigma m l = 39540 .$$
  3. Find the values of \(a\) and \(b\).
  4. Explain the significance of the values of \(a\) and \(b\) in this situation.
Edexcel S1 Q1
10 marks Moderate -0.8
  1. There are 16 competitors in a table-tennis competition, 5 of which come from Racknor Comprehensive School. Prizes are awarded to the competitors finishing in each of first, second and third place.
Assuming that all the competitors have an equal chance of success, find the probability that the students from Racknor Comprehensive
  1. win no prizes,
  2. win the \(1 ^ { \text {st } }\) and \(3 ^ { \text {rd } }\) place prizes but not the \(2 ^ { \text {nd } }\) place prize,
  3. win exactly one of the prizes.
Edexcel S1 Q2
10 marks Moderate -0.3
2. A statistics student gave a questionnaire to a random sample of 50 pupils at his school. The sample included pupils aged from 11 to 18 years old. The student summarised the data on age in completed years, \(A\), and the number of hours spent doing homework in the previous week, \(H\), giving the following: $$\Sigma A = 703 , \quad \Sigma H = 217 , \quad \Sigma A ^ { 2 } = 10131 , \quad \Sigma H ^ { 2 } = 1338.5 , \quad \Sigma A H = 3253.5$$
  1. Calculate the product moment correlation coefficient for these data and explain what is shown by your result.
    (6 marks)
    The student also asked each pupil how many hours of paid work they had done in the previous week. He then calculated the product moment correlation coefficient for the data on hours doing homework and hours doing paid work, giving a value of \(r = 0.5213\) The student concluded that paid work did not interfere with homework as pupils doing more paid work also tended to do more homework.
  2. Explain why this conclusion may not be valid.
  3. Explain briefly how the student could more effectively investigate the effect of paid work on homework.
    (2 marks)
Edexcel S1 Q3
11 marks Moderate -0.3
3. A soccer fan collected data on the number of minutes of league football, \(m\), played by each team in the four main divisions before first scoring a goal at the start of a new season. Her results are shown in the table below.
\(m\) (minutes)Number of teams
\(0 \leq m < 40\)36
\(40 \leq m < 80\)28
\(80 \leq m < 120\)10
\(120 \leq m < 160\)4
\(160 \leq m < 200\)5
\(200 \leq m < 300\)4
\(300 \leq m < 400\)2
\(400 \leq m < 600\)3
  1. Calculate estimates of the mean and standard deviation of these data.
  2. Explain why the mean and standard deviation might not be the best summary statistics to use with these data.
  3. Suggest alternative summary statistics that would better represent these data.