Questions — Edexcel S1 (574 questions)

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Edexcel S1 Q1
  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
2. (a) Explain briefly what is meant by a statistical model.
(b) 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
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
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
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
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
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
  1. (a) (i) Name a suitable distribution for modelling the volume of liquid in bottles of wine sold as containing 75 cl .
    (ii) Explain why the mean in such a model would probably be greater than 75 cl .
    (b) (i) Name a suitable distribution for modelling the score on a single throw of a fair four-sided die with the numbers \(1,2,3\) and 4 on its faces.
    (ii) Use your suggested model to find the mean and variance of the score on a single throw of the die.
    (6 marks)
  2. The events \(A\) and \(B\) are independent and such that
$$\mathrm { P } ( A ) = 2 \mathrm { P } ( B ) \text { and } \mathrm { P } ( A \cap B ) = \frac { 1 } { 8 }$$ (a) Show that \(\mathrm { P } ( B ) = \frac { 1 } { 4 }\).
(b) Find \(\mathrm { P } ( A \cup B )\).
(c) Find \(\mathrm { P } \left( A \mid B ^ { \prime } \right)\).
Edexcel S1 Q3
3. A call-centre dealing with complaints collected data on how long customers had to wait before an operator was free to take their call. The lower quartile of the data was 12.7 minutes and the interquartile range was 5.8 minutes.
  1. Find the value of the upper quartile of the data. It is suggested that a normal distribution could be used to model the waiting time.
  2. Calculate correct to 3 significant figures the mean and variance of this normal distribution based on the values of the quartiles.
    (8 marks)
    The actual mean and variance of the data were 15.3 minutes and 20.1 minutes \(^ { 2 }\) respectively.
  3. Comment on the suitability of the model.
    (2 marks)
Edexcel S1 Q4
4. A College offers evening classes in GCSE Mathematics and English. In order to assess which age groups were reluctant to use the classes, the College collected data on the age in completed years of those currently attending each course. The results are shown in this back-to-back stem and leaf diagram.
TotalsMathematicsAgeEnglishTotals
(6)999887199(2)
(8)853111002013558(6)
(7)766422132379(4)
(4)9754402689(5)
(3)86050377(4)
(2)5262448(4)
(0)71(1)
Key: 1|3|2 means age 31 doing Mathematics and age 32 doing English
  1. Find the median and quartiles of the age in completed years of those attending the Mathematics classes.
    (4 marks)
  2. On graph paper, draw a box plot representing the data for the Mathematics class. The median and quartiles of the age in completed years of those attending the English classes are 25,41 and 57 years respectively.
  3. Draw a box plot representing the data for the English class using the same scale as for the data from the Mathematics class.
    (3 marks)
  4. Using your box plots, compare and contrast the ages of those taking each class.
Edexcel S1 Q5
5. A netball team are in a league with three other teams from which one team will progress to the next stage of the competition. The team's coach estimates their chances of winning each of their three matches in the league to be \(0.6,0.5\) and 0.3 respectively, and believes these probabilities to be independent of each other.
  1. Show that the probability of the team winning exactly two of their three matches is 0.36
    (4 marks)
    Let the random variable \(W\) be the number of matches that the team win in the league.
  2. Find the probability distribution of \(W\).
  3. Find \(\mathrm { E } ( W )\) and \(\operatorname { Var } ( W )\).
  4. Comment on the coach's assumption that the probabilities of success in each of the three matches are independent.
    (2 marks)
Edexcel S1 Q6
6. The Principal of a school believes that more students are absent on days when the temperature is lower. Over a two-week period in December she records the percentage of students who are absent, \(A \%\), and the temperature, \(T ^ { \circ } \mathrm { C }\), at 9 am each morning giving these results.
\(T \left( { } ^ { \circ } \mathrm { C } \right)\)4\({ } ^ { - } 3\)\({ } ^ { - } 2\)\({ } ^ { - } 6\)037\({ } ^ { - } 1\)32
\(A ( \% )\)8.514.117.020.317.915.512.412.813.711.6
  1. Represent these data on a scatter diagram. You may use $$\Sigma T = 7 , \quad \Sigma A = 143.8 , \quad \Sigma T ^ { 2 } = 137 , \quad \Sigma A ^ { 2 } = 2172.66 , \quad \Sigma T A = 20.7$$
  2. Calculate the product moment correlation coefficient for these data and comment on the Principal’s hypothesis.
  3. Find an equation of the regression line of \(A\) on \(T\) in the form \(A = p + q T\).
  4. Draw the regression line on your scatter diagram.
Edexcel S1 Q1
  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
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
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
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
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
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
  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
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
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.
Edexcel S1 Q4
4. Alan runs on a treadmill each day for as long as he can at 7 miles per hour. The length of time for which he runs is normally distributed with a mean of 21.6 minutes and a standard deviation of 1.8 minutes.
  1. Calculate the probability that on any one day Alan will run for less than 20 minutes.
  2. Estimate the number of times in a ninety-day period that Alan will run for more than 24 minutes.
  3. On a particular day Alan is still running after 22 minutes. Find the probability that he will stop running in the next 2 minutes.
Edexcel S1 Q5
5. In a survey unemployed people were asked how many months it had been, to the nearest month, since they were last employed on a full-time basis. The data collected is summarised in this stem and leaf diagram.
Number of months(2 | 1 means 21 months)Totals
011224446779(11)
102355689( )
21568( )
3079( )
45( )
527(2)
63(1)
70(1)
  1. Write down the values needed to complete the totals column on the stem and leaf diagram.
  2. State the mode of these data.
  3. Find the median and quartiles of these data. Given that any values outside of the limits \(\mathrm { Q } _ { 1 } - 1.5 \left( \mathrm { Q } _ { 3 } - \mathrm { Q } _ { 1 } \right)\) and \(\mathrm { Q } _ { 3 } + 1.5 \left( \mathrm { Q } _ { 3 } - \mathrm { Q } _ { 1 } \right)\) are to be regarded as outliers,
  4. determine if there are any outliers in these data,
  5. draw a box plot representing these data on graph paper,
  6. describe the skewness of these data and suggest a reason for it.
Edexcel S1 Q6
6. In a game two spinners are used. The score on the first spinner is given by the random variable \(A\), which has the following probability distribution:
\(a\)123
\(\mathrm { P } ( A = a )\)\(\frac { 1 } { 3 }\)\(\frac { 1 } { 3 }\)\(\frac { 1 } { 3 }\)
  1. State the name of this distribution.
  2. Write down \(\mathrm { E } ( A )\). The score on the second spinner is given by the random variable \(B\), which has the following probability distribution:
    \(b\)123
    \(\mathrm { P } ( B = b )\)\(\frac { 1 } { 2 }\)\(\frac { 1 } { 4 }\)\(\frac { 1 } { 4 }\)
  3. Find \(\mathrm { E } ( B )\). On each player's turn in the game, both spinners are used and the scores on the two spinners are added together. The total score on the two spinners is given by the random variable \(C\).
  4. Show that \(\mathrm { P } ( C = 2 ) = \frac { 1 } { 6 }\).
  5. Find the probability distribution of \(C\).
  6. Show that \(\mathrm { E } ( C ) = \mathrm { E } ( A ) + \mathrm { E } ( B )\).
Edexcel S1 Q1
  1. A shop recorded the number of pairs of gloves, \(n\), that it sold and the average daytime temperature, \(T ^ { \circ } \mathrm { C }\), for each month over a 12-month period.
The data was then summarised as follows: $$\Sigma T = 124 , \quad \Sigma n = 384 , \quad \Sigma T ^ { 2 } = 1802 , \quad \Sigma n ^ { 2 } = 18518 , \quad \Sigma T n = 2583 .$$
  1. Calculate the product moment correlation coefficient for these data.
  2. Comment on what your value shows and suggest a reason for this.