Questions — Edexcel FS2 AS (30 questions)

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Edexcel FS2 AS 2018 June Q1
  1. The scores achieved on a maths test, \(m\), and the scores achieved on a physics test, \(p\), by 16 students are summarised below.
$$\sum m = 392 \quad \sum p = 254 \quad \sum p ^ { 2 } = 4748 \quad \mathrm {~S} _ { m m } = 1846 \quad \mathrm {~S} _ { m p } = 1115$$
  1. Find the product moment correlation coefficient between \(m\) and \(p\)
  2. Find the equation of the linear regression line of \(p\) on \(m\) Figure 1 shows a plot of the residuals. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{0fcb4d83-9763-4edd-8006-93f75a44c596-02_808_1222_997_429} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure}
  3. Calculate the residual sum of squares (RSS). For the person who scored 30 marks on the maths test,
  4. find the score on the physics test. The data for the person who scored 20 on the maths test is removed from the data set.
  5. Suggest a reason why. The product moment correlation coefficient between \(m\) and \(p\) is now recalculated for the remaining 15 students.
  6. Without carrying out any further calculations, suggest how you would expect this recalculated value to compare with your answer to part (a).
    Give a reason for your answer.
    V349 SIHI NI IMIMM ION OCVJYV SIHIL NI LIIIM ION OOVJYV SIHIL NI JIIYM ION OC
Edexcel FS2 AS 2018 June Q2
  1. The continuous random variable X has probability density function
$$f ( x ) = \begin{cases} \frac { 1 } { 8 } & 1 \leqslant x \leqslant 9
0 & \text { otherwise } \end{cases}$$
  1. Write down the name given to this distribution. The continuous random variable \(Y = 5 - 2 X\)
  2. Find \(\mathrm { P } ( Y > 0 )\)
  3. Find \(\mathrm { E } ( Y )\)
  4. Find \(\mathrm { P } ( Y < 0 \mid X < 7.5 )\)
    VILU SIHI NI IIIUM ION OCVGHV SIHILNI IMAM ION OOVJYV SIHI NI JIIYM ION OC
Edexcel FS2 AS 2018 June Q3
  1. The table below shows the heights cleared, in metres, for each of 6 competitors in a high jump competition.
CompetitorABCDEF
Height (m)2.051.932.021.961.812.02
These 6 competitors also took part in a long jump competition and finished in the following order, with C jumping the furthest.
C
A
F
D
B
E
  1. Calculate Spearman's rank correlation coefficient for these data.
  2. Stating your hypotheses clearly, test at the \(5 \%\) level of significance whether or not there is a positive correlation between results in the high jump and results in the long jump. The product moment correlation coefficient between the height of the high jump and the length of the long jump for each competitor is found to be 0.678
  3. Use this value to test, at the \(5 \%\) level of significance, for evidence of positive correlation between results in the high jump and results in the long jump.
  4. State the condition required for the test in part (c) to be valid.
  5. Explain what your conclusions in part (b) and part (c) suggest about the relationship between results in the high jump and results in the long jump.
    V349 SIHI NI IMIMM ION OCVJYV SIHIL NI LIIIM ION OOVJYV SIHIL NI JIIYM ION OC
Edexcel FS2 AS 2018 June Q4
  1. The continuous random variable \(X\) has cumulative distribution function
$$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 & x < 3
c - 4.5 x ^ { n } & 3 \leqslant x \leqslant 9
1 & x > 9 \end{array} \right.$$ where \(c\) is a positive constant and \(n\) is an integer.
  1. Showing all stages of your working, find the value of \(c\) and the value of \(n\)
  2. Find the lower quartile of \(X\)
Edexcel FS2 AS 2019 June Q1
  1. Bara is investigating whether or not the two judges of a skating competition are in agreement. The two judges gave a score to each of the 8 skaters in the competition as shown in the table below.
\cline { 2 - 9 } \multicolumn{1}{c|}{}Skater
\cline { 2 - 9 }\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)
Judge 17170726263615753
Judge 27371676462565253
Bara decided to calculate Spearman's rank correlation coefficient for these data.
  1. Calculate Spearman's rank correlation coefficient between the ranks of the two judges.
  2. Test, at the \(1 \%\) level of significance, whether or not the two judges are in agreement. Judge 1 accidentally swapped the scores for skaters \(D\) and \(E\). The score for skater \(D\) should be 63 and the score for skater \(E\) should be 62
  3. Without carrying out any further calculations, explain how Spearman's rank correlation coefficient will change. Give a reason for your answer.
Edexcel FS2 AS 2019 June Q2
  1. Lloyd regularly takes a break from work to go to the local cafe. The amount of time Lloyd waits to be served, in minutes, is modelled by the continuous random variable \(T\), having probability density function
$$f ( t ) = \left\{ \begin{array} { c c } \frac { t } { 120 } & 4 \leqslant t \leqslant 16
0 & \text { otherwise } \end{array} \right.$$
  1. Show that the cumulative distribution function is given by $$\mathrm { F } ( t ) = \left\{ \begin{array} { c r } 0 & t < 4
    \frac { t ^ { 2 } } { 240 } - c & 4 \leqslant t \leqslant 16
    1 & t > 16 \end{array} \right.$$ where the value of \(c\) is to be found.
  2. Find the exact probability that the amount of time Lloyd waits to be served is between 5 and 10 minutes.
  3. Find the median of \(T\).
  4. Find the value of \(k\) such that $$\mathrm { P } ( T < k ) = \frac { 2 } { 3 } \mathrm { P } ( T > k )$$ giving your answer to 3 significant figures.
Edexcel FS2 AS 2019 June Q3
  1. Two students, Jim and Dora, collected data on the mean annual rainfall, \(w \mathrm {~cm}\), and the annual yield of leeks, \(l\) tonnes per hectare, for 10 years.
Jim summarised the data as follows $$\mathrm { S } _ { w l } = 42.786 \quad \mathrm {~S} _ { w w } = 9936.9 \quad \sum l ^ { 2 } = 26.2326 \quad \sum l = 16.06$$
  1. Find the product moment correlation coefficient between \(l\) and \(w\) Dora decided to code the data first using \(s = w - 6\) and \(t = l - 20\)
  2. Write down the value of the product moment correlation coefficient between \(s\) and \(t\). Give a justification for your answer. Dora calculates the equation of the regression line of \(t\) on \(s\) to be \(t = 0.00431 s - 18.87\)
  3. Find the equation of the regression line of \(l\) on \(w\) in the form \(l = a + b w\), giving the values of \(a\) and \(b\) to 3 significant figures.
  4. Use your equation to estimate the yield of leeks when \(w\) is 100 cm .
  5. Calculate the residual sum of squares. The graph shows the residual for each value of \(l\)
    \includegraphics[max width=\textwidth, alt={}, center]{7e46e14a-0f5a-4d02-8f00-a92bc4def6d7-08_716_1594_1594_239}
    1. State whether this graph suggests that the use of a linear regression model is suitable for these data. Give a reason for your answer.
    2. Other than collecting more data, suggest how to improve the fit of the model in part (c) to the data.
Edexcel FS2 AS 2019 June Q4
  1. The random variable \(X\) has a continuous uniform distribution over the interval [5,a], where \(a\) is a constant.
    Given that \(\operatorname { Var } ( X ) = \frac { 27 } { 4 }\)
    1. show that \(a = 14\)
    The continuous random variable \(Y\) has probability density function $$f ( y ) = \left\{ \begin{array} { c c } \frac { 1 } { 20 } ( 2 y - 3 ) & 2 \leqslant y \leqslant 6
    0 & \text { otherwise } \end{array} \right.$$ The random variable \(T = 3 \left( X ^ { 2 } + X \right) + 2 Y\)
  2. Show that \(\mathrm { E } ( T ) = \frac { 9857 } { 30 }\)
Edexcel FS2 AS 2020 June Q1
  1. An estate agent in Tornep believes that houses further from the railway station are more expensive than those that are closer. She took a random sample of 22 three-bedroom houses in Tornep and calculated the product moment correlation coefficient between the house price and the distance from the station to be 0.3892
Stating your hypotheses clearly, use a \(5 \%\) level of significance to test the estate agent's belief. State the critical region used in your test.
Edexcel FS2 AS 2020 June Q2
  1. Mary, Jahil and Dawn are judging the cakes in a village show. They have 5 features to consider and each feature is awarded up to 5 points. The total score the judges gave each cake are given in the table below.
CakeA\(B\)C\(D\)\(E\)\(F\)\(G\)\(H\)I
Mary19172310211512814
Jahil221821102420161215
Dawn911618915132013
  1. Calculate Spearman's rank correlation coefficient between Mary's scores and Jahil’s scores.
  2. Calculate Spearman's rank correlation coefficient between Jahil's scores and Dawn's scores. The judges discussed their interpretation of the points system and agreed that the first prize should go to cake \(C\).
  3. Explain how different interpretations of the points system could give rise to the results in part (a) and part (b).
Edexcel FS2 AS 2020 June Q3
  1. The continuous random variable \(X\) has cumulative distribution function
$$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < 4
p x - k \sqrt { x } & 4 \leqslant x \leqslant 9
1 & x > 9 \end{array} \right.$$ where \(p\) and \(k\) are constants.
  1. Find the value of \(p\) and the value of \(k\). Given that \(\mathrm { E } ( X ) = \frac { 119 } { 18 }\)
  2. show that \(\operatorname { Var } ( X ) = 2.05\) to 3 significant figures.
  3. Write down the mode of \(X\).
  4. Find the exact value of the constant \(a\) such that \(\mathrm { P } ( X \leqslant a ) = \frac { 7 } { 27 }\)
Edexcel FS2 AS 2020 June Q4
  1. Some students are investigating the strength of wire by suspending a weight at the end of the wire. They measure the diameter of the wire, \(d \mathrm {~mm}\), and the weight, \(w\) grams, when the wire fails. Their results are given in the following table.
\cline { 2 - 13 } \multicolumn{1}{l|}{}These 14 points are plotted on page 13Not yet plotted
\(d\)0.50.60.70.80.91.11.31.622.42.83.33.53.9\(\mathbf { 4 . 5 }\)\(\mathbf { 4 . 6 }\)\(\mathbf { 4 . 8 }\)\(\mathbf { 5 . 4 }\)
\(w\)1.21.72.33.03.85.67.711.61825.934.947.452.763.9\(\mathbf { 8 1 }\)\(\mathbf { 8 3 . 6 }\)\(\mathbf { 8 9 . 9 }\)\(\mathbf { 1 0 9 . 4 }\)
The first 14 points are plotted on the axes on page 13.
  1. On the axes on page 13, complete the scatter diagram for these data.
  2. Use your calculator to write down the equation of the regression line of \(w\) on \(d\).
  3. With reference to the scatter diagram, comment on the appropriateness of using this linear regression model to make predictions for \(w\) for different values of \(d\) between 0.5 and 5.4 The product moment correlation coefficient for these data is \(r = 0.987\) (to 3 significant figures).
  4. Calculate the residual sum of squares (RSS) for this model. Robert, one of the students, suggests that the model could be improved and intends to find the equation of the line of regression of \(w\) on \(u\), where \(u = d ^ { 2 }\)
    He finds the following statistics $$\mathrm { S } _ { w u } = 5721.625 \quad \mathrm {~S} _ { u u } = 1482.619 \quad \sum u = 157.57$$
  5. By considering the physical nature of the problem, give a reason to support Robert's suggestion.
  6. Find the equation of the regression line of \(w\) on \(u\).
  7. Find the residual sum of squares (RSS) for Robert's model.
  8. State, giving a reason based on these calculations, which of these models better describes these data.
    1. Hence estimate the weight at which a piece of wire with diameter 3 mm will fail. \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Question 4 continued} \includegraphics[alt={},max width=\textwidth]{fbd7b196-5372-4956-8d38-92f05c92a5f7-13_2315_1363_301_358}
      \end{figure}
Edexcel FS2 AS 2022 June Q1
  1. Abena and Meghan are both given the same list of 10 films.
Each of them ranks the 10 films from most favourite to least favourite.
For the differences, \(d\), between their ranks for these 10 films, \(\sum d ^ { 2 } = 84\)
  1. Calculate Spearman's rank correlation coefficient between Abena's ranks and Meghan's ranks. A test is carried out at the 5\% level of significance to see if there is agreement between their ranks for the films. The hypotheses for the test are $$\mathrm { H } _ { 0 } : \rho _ { \mathrm { S } } = 0 \quad \mathrm { H } _ { 1 } : \rho _ { \mathrm { S } } > 0$$
    1. Find the critical region for the test.
    2. State the conclusion of the test. An 11th film is added to the list. Abena and Meghan both agree that this film is their least favourite. A new test is carried out at the \(5 \%\) level of significance using the same hypotheses.
  3. Determine the conclusion of this test. You should state the test statistic and the critical value used.
Edexcel FS2 AS 2022 June Q2
  1. The graph shows the probability density function \(\mathrm { f } ( x )\) of the continuous random variable \(X\)
    \includegraphics[max width=\textwidth, alt={}, center]{128c408d-3e08-4f74-8f19-d33ecd5c882f-04_951_1365_322_331}
    1. Find \(\mathrm { P } ( X < 4 )\)
    2. Specify the cumulative distribution function of \(X\) for \(7 \leqslant x \leqslant 11\)
Edexcel FS2 AS 2022 June Q3
  1. Gabriela is investigating a particular type of fish, called bream. She wants to create a model to predict the weight, \(w\) grams, of bream based on their length, \(x \mathrm {~cm}\).
For a sample of 27 bream, some summary statistics are given below. $$\begin{gathered} \bar { x } = 31.07 \quad \bar { w } = 628.59 \quad \sum w ^ { 2 } = 11386134
\mathrm {~S} _ { x w } = 13082.3 \quad \mathrm {~S} _ { x x } = 260.8 \end{gathered}$$
  1. Find the value of the product moment correlation coefficient between \(x\) and \(w\)
  2. Explain whether the answer to part (a) is consistent with a linear model for these data.
  3. Find the equation of the regression line of \(w\) on \(x\) in the form \(w = a + b x\) A residual plot for these data is shown below.
    \includegraphics[max width=\textwidth, alt={}, center]{128c408d-3e08-4f74-8f19-d33ecd5c882f-06_931_1790_1107_139} One of the bream in the sample has a length of 32 cm .
  4. Find its weight.
  5. With reference to the residual plot, comment on the model for bream with lengths above 33 cm .
Edexcel FS2 AS 2022 June Q4
  1. A random variable \(X\) has probability density function given by
$$f ( x ) = \left\{ \begin{array} { c c } 0.8 - 6.4 x ^ { - 3 } & 2 \leqslant x \leqslant 4
0 & \text { otherwise } \end{array} \right.$$ The median of \(X\) is \(m\)
  1. Show that \(m ^ { 3 } - 3.625 m ^ { 2 } + 4 = 0\)
    1. Find \(\mathrm { f } ^ { \prime } ( x )\)
    2. Explain why the mode of \(X\) is 4 Given that \(\mathrm { E } \left( X ^ { 2 } \right) = 10.5\) to 3 significant figures,
  3. find \(\operatorname { Var } ( X )\), showing your working clearly.
Edexcel FS2 AS 2022 June Q5
  1. The random variable \(X\) has the continuous uniform distribution over the interval [0.5, 2.5]
Talia selects a number, \(T\), at random from the distribution of \(X\)
  1. Find \(\mathrm { P } ( T < 1 )\) Malik takes Talia's number, \(T\), and calculates his number, \(M\), where \(M = \frac { 1 } { T ^ { 2 } }\)
  2. Find the probability that both \(T\) and \(M\) are less than 2.25 Raja and Greta play a game many times.
    Each time they play they use a number, \(R\), randomly selected from the distribution of \(X\)
    Raja's score is \(R\)
    Greta's score is \(G\), where \(G = \frac { 2 } { R ^ { 2 } }\)
  3. Determine, giving a reason, who you would expect to have the higher total score.
Edexcel FS2 AS 2023 June Q1
  1. Every applicant for a job at Donala is given three different tasks, \(P , Q\) and \(R\).
For each task the applicant is awarded a score.
The scores awarded to 9 of the applicants, for the tasks \(P\) and \(Q\), are given below.
Applicant\(A\)\(B\)C\(D\)E\(F\)GHI
Task \(\boldsymbol { P }\)1916161281712125
Task \(Q\)1711147618151110
  1. Calculate Spearman's rank correlation coefficient for the scores awarded for the tasks \(P\) and \(Q\).
  2. Test, at the \(1 \%\) level of significance, whether or not there is evidence for a positive correlation between the ranks of scores for tasks \(P\) and \(Q\). You should state your hypotheses and critical value clearly. The Spearman's rank correlation coefficient for \(P\) and \(R\) is 0.290 and for \(Q\) and \(R\) is 0.795 The manager of Donala wishes to reduce the number of tasks given to job applicants from three to two.
  3. Giving a reason for your answer, state which 2 tasks you would recommend the manager uses.
Edexcel FS2 AS 2023 June Q2
  1. A continuous random variable \(X\) has probability density function
$$f ( x ) = \left\{ \begin{array} { c c } \frac { x } { 16 } \left( 9 - x ^ { 2 } \right) & 1 \leqslant x \leqslant 3
0 & \text { otherwise } \end{array} \right.$$
  1. Find the cumulative distribution function of \(X\)
  2. Calculate \(\mathrm { P } ( X > 1.8 )\)
  3. Use calculus to find \(\mathrm { E } \left( \frac { 3 } { X } + 2 \right)\)
  4. Show that the mode of \(X\) is \(\sqrt { 3 }\)
Edexcel FS2 AS 2023 June Q3
  1. Pat is investigating the relationship between the height of professional tennis players and the speed of their serve. Data from 9 randomly selected professional male tennis players were collected. The variables recorded were the height of each player, \(h\) metres, and the maximum speed of their serve, \(v \mathrm {~km} / \mathrm { h }\).
Pat summarised these data as follows $$\sum h = 17.63 \quad \sum v = 2174.9 \quad \sum v ^ { 2 } = 526407.8 \quad S _ { h h } = 0.0487 \quad S _ { h v } = 5.1376$$
  1. Calculate the product moment correlation coefficient between \(h\) and \(v\)
  2. Explain whether the answer to part (a) is consistent with a linear model for these data.
  3. Find the equation of the regression line of \(v\) on \(h\) in the form \(v = a + b h\) where \(a\) and \(b\) are to be given to one decimal place. Pat calculated the sum of the residuals for the 9 tennis players as 1.04
  4. Without doing a calculation, explain how you know Pat has made a mistake. Pat made one mistake in the calculation. For the tennis player of height 1.96 m Pat misread the residual as 2.27
  5. Find the maximum speed of serve, in km/h, for the tennis player of height 1.96 m
Edexcel FS2 AS 2023 June Q4
  1. The random variable \(X\) has a continuous uniform distribution over the interval \([ - 3 , k ]\) Given that \(\mathrm { P } ( - 4 < X < 2 ) = \frac { 1 } { 3 }\)
    1. find the value of \(k\)
    A computer generates a random number, \(Y\), where
    • \(\quad Y\) has a continuous uniform distribution over the interval \([ a , b ]\)
    • \(\mathrm { E } ( Y ) = 6\)
    • \(\operatorname { Var } ( Y ) = 192\)
    The computer generates 5 random numbers.
  2. Calculate the probability that at least 2 of the 5 numbers generated are greater than 7.5
Edexcel FS2 AS 2024 June Q1
  1. A continuous random variable \(X\) has cumulative distribution function \(\mathrm { F } ( x )\) given by
$$\mathrm { F } ( x ) = \left\{ \begin{array} { c r } 0 & x < - 1
\frac { 1 } { 5 } ( x + 1 ) ^ { 2 } & - 1 \leqslant x \leqslant 0
1 - \frac { 1 } { 20 } ( 4 - x ) ^ { 2 } & 0 < x \leqslant 4
1 & x > 4 \end{array} \right.$$
  1. Find the probability density function, \(\mathrm { f } ( x )\)
    1. Sketch \(\mathrm { f } ( x )\)
    2. Hence describe the skewness of the distribution.
  3. Find, to 3 significant figures, the value of \(c\) such that $$\mathrm { P } ( 1 < X < c ) = \mathrm { P } ( c < X < 2 )$$
Edexcel FS2 AS 2024 June Q2
  1. A random sample of size \(n = 8\) of paired data is taken from a population. The data are plotted below.
    \includegraphics[max width=\textwidth, alt={}, center]{ba41c616-0805-4466-81b8-b985b0bdd94b-06_572_983_335_541}
Test, at the \(1 \%\) level of significance, whether or not there is evidence of a negative rank correlation between the two variables. You should state your hypotheses and critical value and show your working clearly.
Edexcel FS2 AS 2024 June Q3
  1. The continuous random variable \(Y\) has probability density function
$$f ( y ) = \left\{ \begin{array} { c c } \frac { 1 } { 24 } ( y + 2 ) ( 4 - y ) & 0 \leqslant y \leqslant 3
0 & \text { otherwise } \end{array} \right.$$
  1. Show that the mode of \(Y\) is 1 , justifying your reasoning. Given that \(\mathrm { P } ( Y < 1 ) = \frac { 13 } { 36 }\)
  2. determine whether the median of \(Y\) is less than, equal to, or greater than 2 Give a reason for your answer. Given that \(\mathrm { E } \left( Y ^ { 2 } \right) = \frac { 213 } { 80 }\)
  3. find, using algebraic integration, \(\operatorname { Var } ( 2 Y )\)
Edexcel FS2 AS 2024 June Q4
  1. The continuous random variable \(X\) is uniformly distributed over the interval [2, 7]
    1. Write down the value of \(\mathrm { E } ( X )\)
    2. Find \(\mathrm { P } ( 1 < X < 4 )\)
    3. Find \(\mathrm { P } \left( 2 X ^ { 2 } - 15 X + 27 > 0 \right)\)
    4. Find \(\mathrm { E } \left( \frac { 3 } { X ^ { 2 } } \right)\)