Questions — Edexcel S1 (574 questions)

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Edexcel S1 2023 January Q5
  1. The lengths, \(L \mathrm {~mm}\), of housefly wings are normally distributed with \(L \sim \mathrm {~N} \left( 4.5,0.4 ^ { 2 } \right)\)
    1. Find the probability that a randomly selected housefly has a wing length of less than 3.86 mm .
    2. Find
      1. the upper quartile ( \(Q _ { 3 }\) ) of \(L\)
      2. the lower quartile ( \(Q _ { 1 }\) ) of \(L\)
    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. Find these two outlier limits. A housefly is selected at random.
  3. Using standardisation, show that the probability that this housefly is not an outlier is 0.993 to 3 decimal places. Given that this housefly is not an outlier,
  4. showing your working, find the probability that the wing length of this housefly is greater than 5 mm .
Edexcel S1 2023 January Q6
  1. A research student is investigating the maximum weight, \(y\) grams, of sugar that will dissolve in 100 grams of water at various temperatures, \(x ^ { \circ } \mathrm { C }\), where \(10 \leqslant x \leqslant 80\)
The research student calculated the regression line of \(y\) on \(x\) and found it to be $$y = 151.2 + 2.72 x$$
  1. Give an interpretation of the gradient of the regression line.
  2. Use the regression line to estimate the maximum weight of sugar that will dissolve in 100 grams of water when the temperature is \(90 ^ { \circ } \mathrm { C }\).
  3. Comment on the reliability of your estimate, giving a reason for your answer. Using the regression line of \(y\) on \(x\) and the following summary statistics $$\sum y = 3119 \quad \sum y ^ { 2 } = 851093 \quad \sum x ^ { 2 } = 24500 \quad n = 12$$
  4. show that the product moment correlation coefficient for these data is 0.988 to 3 decimal places. The research student's supervisor plotted the original data on a scatter diagram, shown on page 23 With reference to both the scatter diagram and the correlation coefficient,
  5. discuss the suitability of a linear regression model to describe the relationship between \(x\) and \(y\).
    \includegraphics[max width=\textwidth, alt={}]{c316fa29-dedc-4890-bd82-31eb0bb819f9-23_990_1138_205_356}
Edexcel S1 2024 January Q1
  1. The histogram below shows the distribution of the heights, to the nearest cm , of 408 plants.
    \includegraphics[max width=\textwidth, alt={}, center]{86446ce3-496a-4f02-9566-9b207bac9efa-02_1001_1473_340_296}
    1. Use the histogram to complete the following table.
    Height \(( h\) cm)\(5 \leqslant h < 9\)\(9 \leqslant h < 13\)\(13 \leqslant h < 15\)\(15 \leqslant h < 17\)\(17 \leqslant h < 25\)
    Frequency32152120
  2. Use interpolation to estimate the median. The mean height of these plants is 13.2 cm correct to one decimal place.
  3. Describe the skew of these data. Give a reason for your answer. Two of these plants are chosen at random.
  4. Estimate the probability that both of their heights are between 8 cm and 14 cm
Edexcel S1 2024 January Q2
  1. The average minimum monthly temperature, \(x\) degrees Fahrenheit ( \({ } ^ { \circ } \mathrm { F }\) ), and the average maximum monthly temperature, \(y\) degrees Fahrenheit ( \({ } ^ { \circ } \mathrm { F }\) ), in Kolkata were recorded for 12 months.
Some of the summary statistics are given below. $$\sum x = 862 \quad \sum x ^ { 2 } = 62802 \quad \mathrm {~S} _ { y y } = 413.67 \quad S _ { x y } = 512.67 \quad n = 12$$
    1. Calculate the mean of the 12 values of the average minimum
      monthly temperature.
    2. Show that the standard deviation of the 12 values of the average minimum monthly temperature is \(8.57 ^ { \circ } \mathrm { F }\) to 3 significant figures.
  2. Calculate the product moment correlation coefficient between \(x\) and \(y\) For comparative purposes with a UK city, it was necessary to convert the temperatures from degrees Fahrenheit ( \({ } ^ { \circ } \mathrm { F }\) ) to degrees Celsius ( \({ } ^ { \circ } \mathrm { C }\) ). The formula used was $$c = \frac { 5 } { 9 } ( f - 32 )$$ where \(f\) is the temperature in \({ } ^ { \circ } \mathrm { F }\) and \(c\) is the temperature in \({ } ^ { \circ } \mathrm { C }\)
  3. Use this formula and the values from part (a) to calculate, in \({ } ^ { \circ } \mathrm { C }\), the mean and the standard deviation of the 12 values of the average minimum monthly temperature in Kolkata.
    Give your answers to 3 significant figures. Given that
    • \(u\) is the equivalent temperature in \({ } ^ { \circ } \mathrm { C }\) of \(x\)
    • \(\quad v\) is the equivalent temperature in \({ } ^ { \circ } \mathrm { C }\) of \(y\)
    • state, giving a reason, the product moment correlation coefficient between \(u\) and \(v\)
Edexcel S1 2024 January Q3
  1. In a sixth form college each student in Year 12 and Year 13 is either left-handed (L) or right-handed (R).
The partially completed tree diagram, where \(p\) is a probability, gives information about these students.
\includegraphics[max width=\textwidth, alt={}, center]{86446ce3-496a-4f02-9566-9b207bac9efa-10_960_981_477_543}
  1. Complete the tree diagram, in terms of \(p\) where necessary. The probability that a student is left-handed is 0.11
  2. Find the value of \(p\)
  3. Find the probability that a student selected at random is in Year 12 and left-handed. Given that a student is right-handed,
  4. find the probability that the student is in Year 12
Edexcel S1 2024 January Q4
  1. A French test and a Spanish test were sat by 11 students.
The table below shows their marks.
StudentABCDEFGHIJK
French mark ( f )2430323236364044506068
Spanish mark ( \(\boldsymbol { s }\) )1690242832363844484868
Greg says that if these points were plotted on a scatter diagram, then the point \(( 30,90 )\) would be an outlier because 90 is an outlier for the Spanish marks. An outlier is defined as a value that is $$\text { greater than } Q _ { 3 } + 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right) \text { or smaller than } Q _ { 1 } - 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)$$
  1. Show that 90 is an outlier for the Spanish marks. Ignoring the point (30, 90), Greg calculated the following summary statistics. $$\sum f = 422 \quad \sum s = 382 \quad S _ { f f } = 1667.6 \quad S _ { f s } = 1735.6$$
  2. Use these summary statistics to show that the equation of the least squares regression line of \(s\) on \(f\) for the remaining 10 students is $$s = - 5.72 + 1.04 f$$ where the values of the intercept and gradient are given to 3 significant figures. You must show your working.
  3. Give an interpretation of the gradient of the regression line. Two further students sat the French test but missed the Spanish test.
  4. Using the equation given in part (b), estimate
    1. a Spanish mark for the student who scored 55 marks in their French test,
    2. a Spanish mark for the student who scored 18 marks in their French test.
  5. State, giving a reason, which of the two estimates found in part (d) would be the more reliable estimate.
Edexcel S1 2024 January Q5
  1. The distance an athlete can throw a discus is normally distributed with mean 40 m and standard deviation 4 m
    1. Using standardisation, show that the probability that this athlete throws the discus less than 38.8 m is 0.3821
    This athlete enters a discus competition.
    To qualify for the final, they have 3 attempts to throw the discus a distance of more than 38.8 m
    Once they qualify, they do not use any of their remaining attempts.
    Given that they qualified for the final and that throws are independent,
  2. find the probability that this athlete qualified for the final on their second throw with a distance of more than 44 m
Edexcel S1 2024 January Q6
  1. The events \(A\) and \(B\) satisfy
$$\mathrm { P } ( A ) = x \quad \mathrm { P } ( B ) = y \quad \mathrm { P } ( A \cup B ) = 0.65 \quad \mathrm { P } ( B \mid A ) = 0.3$$
  1. Show that $$14 x + 20 y = 13$$ The events \(B\) and \(C\) are mutually exclusive such that $$\mathrm { P } ( B \cup C ) = 0.85 \quad \mathrm { P } ( C ) = \frac { 1 } { 2 } x + y$$
    1. Find a second equation in \(x\) and \(y\)
    2. Hence find the value of \(x\) and the value of \(y\)
  3. Determine whether or not \(A\) and \(B\) are statistically independent. You must show your working clearly.
Edexcel S1 2024 January Q7
  1. The cumulative distribution of a discrete random variable \(X\) is given by
\(x\)1234
\(\mathrm {~F} ( x )\)\(\frac { 1 } { 13 }\)\(\frac { 2 k - 1 } { 26 }\)\(\frac { 3 ( k + 1 ) } { 26 }\)\(\frac { k + 4 } { 8 }\)
where \(k\) is a positive constant.
  1. Show that \(k = 4\)
  2. Find the probability distribution of the discrete random variable \(X\)
  3. Using your answer to part (b), write down the mode of \(X\)
  4. Calculate \(\operatorname { Var } ( 13 X - 6 )\)
Edexcel S1 2024 January Q8
  1. The random variable \(X\) is normally distributed with mean \(\mu\) and variance 36
Given that $$\mathrm { P } ( \mu - 2 k < X < \mu + 2 k ) = 0.6$$
  1. find the value of \(k\) The random variable \(Y\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma\) Given that $$2 \mu = 3 \sigma ^ { 2 } \quad \text { and } \quad \mathrm { P } \left( \mathrm { Y } > \frac { 3 } { 2 } \mu \right) = 0.0668$$
  2. find the value of \(\mu\) and the value of \(\sigma\)
Edexcel S1 2014 June Q1
  1. A medical researcher is studying the relationship between age ( \(x\) years) and volume of blood ( \(y \mathrm { ml }\) ) pumped by each contraction of the heart. The researcher obtained the following data from a random sample of 8 patients.
Age (x)2025304555606570
Volume (y)7476777268676462
[You may use \(\sum x = 370 , \mathrm {~S} _ { x x } = 2587.5 , \sum y = 560 , \sum y ^ { 2 } = 39418 , \mathrm {~S} _ { x y } = - 710\) ]
  1. Calculate \(\mathrm { S } _ { y y }\)
  2. Calculate the product moment correlation coefficient for these data.
  3. Interpret your value of the correlation coefficient. The researcher believes that a linear regression model may be appropriate to describe these data.
  4. State, giving a reason, whether or not your value of the correlation coefficient supports the researcher's belief.
  5. Find the equation of the regression line of \(y\) on \(x\), giving your answer in the form \(y = a + b x\) Jack is a 40-year-old patient.
    1. Use your regression line to estimate the volume of blood pumped by each contraction of Jack's heart.
    2. Comment, giving a reason, on the reliability of your estimate.
Edexcel S1 2014 June Q2
  1. The table below shows the distances (to the nearest km ) travelled to work by the 50 employees in an office.
Distance (km)Frequency (f)Distance midpoint (x)
0-2161.25
3-5124
6-10108
11-20815.5
21-40430.5
$$\text { [You may use } \left. \sum \mathrm { f } x = 394 , \quad \sum \mathrm { f } x ^ { 2 } = 6500 \right]$$ A histogram has been drawn to represent these data.
The bar representing the distance of \(3 - 5\) has a width of 1.5 cm and a height of 6 cm .
  1. Calculate the width and height of the bar representing the distance of 6-10
  2. Use linear interpolation to estimate the median distance travelled to work.
    1. Show that an estimate of the mean distance travelled to work is 7.88 km .
    2. Estimate the standard deviation of the distances travelled to work.
  4. Describe, giving a reason, the skewness of these data. Peng starts to work in this office as the \(51 ^ { \text {st } }\) employee.
    She travels a distance of 7.88 km to work.
  5. Without carrying out any further calculations, state, giving a reason, what effect Peng's addition to the workforce would have on your estimates of the
    1. mean,
    2. median,
    3. standard deviation
      of the distances travelled to work.
Edexcel S1 2014 June Q3
  1. A biased four-sided die has faces marked \(1,3,5\) and 7 . The random variable \(X\) represents the score on the die when it is rolled. The cumulative distribution function of \(X , \mathrm {~F} ( x )\), is given in the table below.
\(x\)1357
\(\mathrm {~F} ( x )\)0.20.50.91
  1. Find the probability distribution of \(X\)
  2. Find \(\mathrm { P } ( 2 < X \leqslant 6 )\)
  3. Write down the value of \(\mathrm { F } ( 4 )\)
Edexcel S1 2014 June Q4
4. The random variable \(Y \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\) Given that \(\mathrm { P } ( Y < 17 ) = 0.6\) find
  1. \(\mathrm { P } ( Y > 17 )\)
  2. \(\mathrm { P } ( \mu < Y < 17 )\)
  3. \(\mathrm { P } ( Y < \mu \mid Y < 17 )\)
Edexcel S1 2014 June Q5
5. The discrete random variable \(X\) has the following probability distribution
\(x\)- 2024
\(\mathrm { P } ( X = x )\)\(a\)\(b\)\(a\)\(c\)
where \(a\), \(b\) and \(c\) are probabilities.
Given that \(\mathrm { E } ( X ) = 0.8\)
  1. find the value of \(c\). Given also that \(\mathrm { E } \left( X ^ { 2 } \right) = 5\) find
  2. the value of \(a\) and the value of \(b\),
  3. \(\operatorname { Var } ( X )\) The random variable \(Y = 5 - 3 X\)
    Find
  4. \(\mathrm { E } ( Y )\)
  5. \(\operatorname { Var } ( Y )\)
  6. \(\mathrm { P } ( Y \geqslant 0 )\)
Edexcel S1 2014 June Q6
6. The Venn diagram below shows the probabilities of customers having various combinations of a starter, main course or dessert at Polly’s restaurant.
\(S =\) the event a customer has a starter.
\(M =\) the event a customer has a main course.
\(D =\) the event a customer has a dessert.
\includegraphics[max width=\textwidth, alt={}, center]{fa0dbe16-ace8-4c44-8404-2bc4e1879d57-10_602_1125_607_413} Given that the events \(S\) and \(D\) are statistically independent
  1. find the value of \(p\).
  2. Hence find the value of \(q\).
  3. Find
    1. \(\quad\) P( \(D \mid M \cap S\) )
    2. \(\operatorname { P } \left( D \mid M \cap S ^ { \prime } \right)\) One evening 63 customers are booked into Polly's restaurant for an office party. Polly has asked for their starter and main course orders before they arrive. Of these 63 customers 27 ordered a main course and a starter, 36 ordered a main course without a starter.
  4. Estimate the number of desserts that these 63 customers will have.
Edexcel S1 2014 June Q7
7. One event at Pentor sports day is throwing a tennis ball. The distance a child throws a tennis ball is modelled by a normal distribution with mean 32 m and standard deviation 12 m . Any child who throws the tennis ball more than 50 m is awarded a gold certificate.
  1. Show that, to 3 significant figures, 6.68\% of children are awarded a gold certificate. A silver certificate is awarded to any child who throws the tennis ball more than \(d\) metres but less than 50 m . Given that 19.1\% of the children are awarded a silver certificate,
  2. find the value of \(d\). Three children are selected at random from those who take part in the throwing a tennis ball event.
  3. Find the probability that 1 is awarded a gold certificate and 2 are awarded silver certificates. Give your answer to 2 significant figures.
Edexcel S1 2015 June Q1
  1. The discrete random variable \(X\) can only take the values \(1,2,3\) and 4 For these values the cumulative distribution function is defined by
$$\mathrm { F } ( x ) = k x ^ { 2 } \text { for } x = 1,2,3,4$$ where \(k\) is a constant.
  1. Find the value of \(k\).
  2. Find the probability distribution of \(X\).
Edexcel S1 2015 June Q2
2. Paul believes there is a relationship between the value and the floor size of a house. He takes a random sample of 20 houses and records the value, \(\pounds v\), and the floor size, \(s \mathrm {~m} ^ { 2 }\) The data were coded using \(x = \frac { s - 50 } { 10 }\) and \(y = \frac { v } { 100000 }\) and the following statistics obtained. $$\sum x = 441.5 , \quad \sum y = 59.8 , \quad \sum x ^ { 2 } = 11261.25 , \quad \sum y ^ { 2 } = 196.66 , \quad \sum x y = 1474.1$$
  1. Find the value of \(S _ { x y }\) and the value of \(S _ { x x }\)
  2. Find the equation of the least squares regression line of \(y\) on \(x\) in the form \(y = a + b x\) The least squares regression line of \(v\) on \(s\) is \(v = c + d s\)
  3. Show that \(d = 1020\) to 3 significant figures and find the value of \(c\)
  4. Estimate the value of a house of floor size \(130 \mathrm {~m} ^ { 2 }\)
  5. Interpret the value \(d\) Paul wants to increase the value of his house. He decides to add an extension to increase the floor size by \(31 \mathrm {~m} ^ { 2 }\)
  6. Estimate the increase in the value of Paul's house after adding the extension.
Edexcel S1 2015 June Q3
  1. A company employs 90 administrators. The length of time that they have been employed by the company and their gender are summarised in the table below.
Length of time employed, \(x\) yearsFemaleMale
\(x < 4\)916
\(4 \leqslant x < 10\)1420
\(10 \leqslant x\)724
One of the 90 administrators is selected at random.
  1. Find the probability that the administrator is female.
  2. Given that the administrator has been employed by the company for less than 4 years, find the probability that this administrator is male.
  3. Given that the administrator has been employed by the company for less than 10 years, find the probability that this administrator is male.
  4. State, with a reason, whether or not the event 'selecting a male' is independent of the event 'selecting an administrator who has been employed by the company for less than 4 years'.
Edexcel S1 2015 June Q4
  1. A bag contains 19 red beads and 1 blue bead only.
Linda selects a bead at random from the bag. She notes its colour and replaces the bead in the bag. She then selects a second bead at random from the bag and notes its colour. Find the probability that
  1. both beads selected are blue,
  2. exactly one bead selected is red. In another bag there are 9 beads, 4 of which are green and the rest are yellow.
    Linda selects 3 beads from this bag at random without replacement.
  3. Find the probability that 2 of these beads are yellow and 1 is green. Linda replaces the 3 beads and then selects another 4 at random without replacement.
  4. Find the probability that at least 1 of the beads is green.
Edexcel S1 2015 June Q5
  1. Police measure the speed of cars passing a particular point on a motorway. The random variable \(X\) is the speed of a car.
    \(X\) is modelled by a normal distribution with mean 55 mph (miles per hour).
    1. Draw a sketch to illustrate the distribution of \(X\). Label the mean on your sketch.
    The speed limit on the motorway is 70 mph . Car drivers can choose to travel faster than the speed limit but risk being caught by the police. The distribution of \(X\) has a standard deviation of 20 mph .
  2. Find the percentage of cars that are travelling faster than the speed limit. The fastest \(1 \%\) of car drivers will be banned from driving.
  3. Show that the lowest speed, correct to 3 significant figures, for a car driver to be banned is 102 mph . Show your working clearly. Car drivers will just be given a caution if they are travelling at a speed \(m\) such that $$\mathrm { P } ( 70 < X < m ) = 0.1315$$
  4. Find the value of \(m\). Show your working clearly.
Edexcel S1 2015 June Q6
  1. The random variable \(X\) has a discrete uniform distribution and takes the values \(1,2,3,4\) Find
    1. \(\mathrm { F } ( 3 )\), where \(\mathrm { F } ( x )\) is the cumulative distribution function of \(X\),
    2. \(\mathrm { E } ( X )\).
    3. Show that \(\operatorname { Var } ( X ) = \frac { 5 } { 4 }\)
    The random variable \(Y\) has a discrete uniform distribution and takes the values $$3,3 + k , 3 + 2 k , 3 + 3 k$$ where \(k\) is a constant.
  2. Write down \(\mathrm { P } ( Y = y )\) for \(y = 3,3 + k , 3 + 2 k , 3 + 3 k\) The relationship between \(X\) and \(Y\) may be written in the form \(Y = k X + c\) where \(c\) is a constant.
  3. Find \(\operatorname { Var } ( Y )\) in terms of \(k\).
  4. Express \(c\) in terms of \(k\).
Edexcel S1 2015 June Q7
7. A doctor is investigating the correlation between blood protein, \(p\), and body mass index, \(b\). He takes a random sample of 8 patients and the data are shown in the table below.
Patient\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)
\(b\)3236404442212737
\(p\)1821313921121970
  1. Draw a scatter diagram of these data on the axes provided.
    \includegraphics[max width=\textwidth, alt={}, center]{36cf6341-1957-45b9-9f7d-0914506f5919-13_938_673_785_614} The doctor decides to leave out patient \(H\) from his calculations.
  2. Give a reason for the doctor's decision. For the 7 patients \(A , B , C , D , E , F\) and \(G\), $$S _ { b p } = 369 , \quad S _ { p p } = 490 \text { and } S _ { b b } = 423 \frac { 5 } { 7 }$$
  3. Find the product moment correlation coefficient, \(r\), for these 7 patients.
  4. Without any further calculations, state how \(r\) would differ from your answer in part (c) if it was calculated for all 8 patients. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{36cf6341-1957-45b9-9f7d-0914506f5919-15_1322_1593_207_173} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} The histogram in Figure 1 summarises the times, in minutes, that 200 people spent shopping in a supermarket.
  5. Give a reason to justify the use of a histogram to represent these data. Given that 40 people spent between 11 and 21 minutes shopping in the supermarket, estimate
  6. the number of people that spent between 18 and 25 minutes shopping in the supermarket,
  7. the median time spent shopping in the supermarket by these 200 people. The mid-point of each bar is represented by \(x\) and the corresponding frequency by f .
  8. Show that \(\sum \mathrm { f } x = 6390\) Given that \(\sum \mathrm { f } x ^ { 2 } = 238430\)
  9. for the data shown in the histogram, calculate estimates of
    1. the mean,
    2. the standard deviation. A coefficient of skewness is given by \(\frac { 3 ( \text { mean } - \text { median } ) } { \text { standard deviation } }\)
  10. Calculate this coefficient of skewness for these data. The manager of the supermarket decides to model these data with a normal distribution.
  11. Comment on the manager's decision. Give a justification for your answer.
Edexcel S1 2004 January Q1
  1. An office has the heating switched on at 7.00 a.m. each morning. On a particular day, the temperature of the office, \(t { } ^ { \circ } \mathrm { C }\), was recorded \(m\) minutes after 7.00 a.m. The results are shown in the table below.
\(m\)01020304050
\(t\)6.08.911.813.515.316.1
  1. Calculate the exact values of \(S _ { m t }\) and \(S _ { m m }\).
  2. Calculate the equation of the regression line of \(t\) on \(m\) in the form \(t = a + b m\).
  3. Use your equation to estimate the value of \(t\) at 7.35 a.m.
  4. State, giving a reason, whether or not you would use the regression equation in (b) to estimate the temperature
    1. at 9.00 a.m. that day,
    2. at 7.15 a.m. one month later.