Questions (30179 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 Stats 1 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Sort by: Default | Easiest first | Hardest first
AQA Further Paper 3 Statistics 2023 June Q3
3 marks Moderate -0.8
3 The masses of tins of a particular brand of spaghetti are normally distributed with mean \(\mu\) grams and standard deviation 4.1 grams. A random sample of 11 tins of spaghetti has a mean mass of 401.8 grams.
Construct a \(98 \%\) confidence interval for \(\mu\), giving your values to one decimal place.
AQA Further Paper 3 Statistics 2023 June Q4
5 marks Standard +0.3
4 The random variable \(X\) has a normal distribution with unknown mean \(\mu\) and unknown variance \(\sigma ^ { 2 }\) A random sample of 8 observations of \(X\) has mean \(\bar { x } = 101.5\) and gives the unbiased estimate of the variance as \(s ^ { 2 } = 4.8\) The random sample is used to conduct a hypothesis test at the \(10 \%\) level of significance with the hypotheses $$\begin{aligned} & \mathrm { H } _ { 0 } : \mu = 100 \\ & \mathrm { H } _ { 1 } : \mu \neq 100 \end{aligned}$$ Carry out the hypothesis test.
AQA Further Paper 3 Statistics 2023 June Q5
8 marks Standard +0.3
5 A school management team oversees 11 different schools.
The school management team allows each student in the schools to choose one enrichment activity from 11 possible activities. The school management team count the number of students in each school choosing each of the possible activities. They then conduct a \(\chi ^ { 2 }\)-test for association with the data they have gathered. 5
  1. Exactly one of the calculated expected frequencies for the \(\chi ^ { 2 }\)-test is less than 5
    Explain why the number of degrees of freedom for the test is 90
    5
  2. The school management team claims that there is an association between the school a student attends and the activity they choose. The test statistic is 124.8 Test the claim using the \(1 \%\) level of significance.
    5
  3. During the hypothesis test, the value of \(\frac { ( O - E ) ^ { 2 } } { E }\), where \(O\) is the observed frequency and \(E\) is the expected frequency, was calculated for each group of students. The values for four groups of students are shown in the table below.
    Group\(\frac { ( O - E ) ^ { 2 } } { E }\)
    Attends school 3 and chose activity 10.01
    Attends school 8 and chose activity 318.5
    Attends school 8 and chose activity 724.2
    Attends school 11 and chose activity 749.0
    State, with a reason, which of the four groups of students represents the strongest source of association.
AQA Further Paper 3 Statistics 2023 June Q6
7 marks Easy -1.2
6 A game consists of two rounds. The first round of the game uses a random number generator to output the score \(X\), a real number between 0 and 10 6
  1. Find \(\mathrm { P } ( X > 4 )\) 6
  2. The second round of the game uses an unbiased dice, with faces numbered 1 to 6 , to give the score \(Y\) The variables \(X\) and \(Y\) are independent.
    6
    1. Find the mean total score of the game.
      6
  4. (ii) Find the variance of the total score of the game.
AQA Further Paper 3 Statistics 2023 June Q7
11 marks Standard +0.3
7 Company \(A\) uses a machine to produce toys. The number of toys in a week that do not pass Company \(A\) 's quality checks is modelled by a Poisson distribution \(X\) with standard deviation 5 The machine producing the toys breaks down.
After it is repaired, 16 toys in the next week do not pass the quality checks.
7
  1. Investigate whether the average number of toys that do not pass the quality checks in a week has changed, using the \(5 \%\) level of significance.
    7
  2. For the test carried out in part (a), state in context the meaning of a Type II error. 7
  3. Company \(B\) uses a different machine to produce toys.
    The number of toys in a week that do not pass Company B's quality checks is modelled by a Poisson distribution \(Y\) with mean 18 The variables \(X\) and \(Y\) are independent.
    Find the distribution of the total number of toys in a week produced by companies \(A\) and \(B\) that do not pass their quality checks. 7
  4. State two reasons why a Poisson distribution may not be a valid model for the number of toys that do not pass the quality checks in a week. Reason 1 \(\_\_\_\_\) Reason 2 \(\_\_\_\_\)
AQA Further Paper 3 Statistics 2023 June Q8
14 marks Standard +0.3
8 The continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} k \sin 2 x & 0 \leq x \leq \frac { \pi } { 6 } \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant. 8
  1. Show that \(k = 4\) 8
  2. Find the cumulative distribution function \(\mathrm { F } ( x )\) 8
  3. Find the median of \(X\), giving your answer to three significant figures. 8
  4. Find the mean of \(X\) giving your answer in the form \(\frac { 1 } { a } ( b \sqrt { 3 } - \pi )\) where \(a\) and \(b\) are integers. \includegraphics[max width=\textwidth, alt={}, center]{1e2fdd33-afa4-486f-a9e2-1d425ed14eee-14_2492_1721_217_150}
AQA Further Paper 3 Statistics 2024 June Q1
1 marks Easy -1.8
1 The random variable \(X\) has a Poisson distribution with mean 16 Find the standard deviation of \(X\) Circle your answer.
4
8
16
256
AQA Further Paper 3 Statistics 2024 June Q4
6 marks Moderate -0.3
4
8
16
256 2 The random variable \(T\) has an exponential distribution with mean 2 Find \(\mathrm { P } ( T \leq 1.4 )\) Circle your answer. \(\mathrm { e } ^ { - 2.8 }\) \(\mathrm { e } ^ { - 0.7 }\) \(1 - e ^ { - 0.7 }\) \(1 - \mathrm { e } ^ { - 2.8 }\) The continuous random variable \(Y\) has cumulative distribution function $$\mathrm { F } ( y ) = \left\{ \begin{array} { l r } 0 & y < 2 \\ - \frac { 1 } { 9 } y ^ { 2 } + \frac { 10 } { 9 } y - \frac { 16 } { 9 } & 2 \leq y < 5 \\ 1 & y \geq 5 \end{array} \right.$$ Find the median of \(Y\) Circle your answer. 2 \(\frac { 10 - 3 \sqrt { 2 } } { 2 }\) \(\frac { 7 } { 2 }\) \(\frac { 10 + 3 \sqrt { 2 } } { 2 }\) Turn over for the next question 4 Research has shown that the mean number of volcanic eruptions on Earth each day is 20 Sandra records 162 volcanic eruptions during a period of one week. Sandra claims that there has been an increase in the mean number of volcanic eruptions per week. Test Sandra's claim at the \(5 \%\) level of significance.
AQA Further Paper 3 Statistics 2024 June Q5
5 marks Standard +0.3
5 The continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} \frac { 1 } { 6 } e ^ { \frac { x } { 3 } } & 0 \leq x \leq \ln 27 \\ 0 & \text { otherwise } \end{cases}$$ Show that the mean of \(X\) is \(\frac { 3 } { 2 } ( \ln 27 - 2 )\)
AQA Further Paper 3 Statistics 2024 June Q6
9 marks Standard +0.3
6 Over time it has been accepted that the mean retirement age for professional baseball players is 29.5 years old. Imran claims that the mean retirement age is no longer 29.5 years old.
He takes a random sample of 5 recently retired professional baseball players and records their retirement ages, \(x\). The results are $$\sum x = 152.1 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 7.81$$ 6
  1. State an assumption that you should make about the distribution of the retirement ages to investigate Imran's claim. 6
  2. Investigate Imran's claim, using the 10\% level of significance.
AQA Further Paper 3 Statistics 2024 June Q8
5 marks Moderate -0.3
8
16
256 2 The random variable \(T\) has an exponential distribution with mean 2 Find \(\mathrm { P } ( T \leq 1.4 )\) Circle your answer. \(\mathrm { e } ^ { - 2.8 }\) \(\mathrm { e } ^ { - 0.7 }\) \(1 - e ^ { - 0.7 }\) \(1 - \mathrm { e } ^ { - 2.8 }\) The continuous random variable \(Y\) has cumulative distribution function $$\mathrm { F } ( y ) = \left\{ \begin{array} { l r } 0 & y < 2 \\ - \frac { 1 } { 9 } y ^ { 2 } + \frac { 10 } { 9 } y - \frac { 16 } { 9 } & 2 \leq y < 5 \\ 1 & y \geq 5 \end{array} \right.$$ Find the median of \(Y\) Circle your answer. 2 \(\frac { 10 - 3 \sqrt { 2 } } { 2 }\) \(\frac { 7 } { 2 }\) \(\frac { 10 + 3 \sqrt { 2 } } { 2 }\) Turn over for the next question 4 Research has shown that the mean number of volcanic eruptions on Earth each day is 20 Sandra records 162 volcanic eruptions during a period of one week. Sandra claims that there has been an increase in the mean number of volcanic eruptions per week. Test Sandra's claim at the \(5 \%\) level of significance.
5 The continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} \frac { 1 } { 6 } e ^ { \frac { x } { 3 } } & 0 \leq x \leq \ln 27 \\ 0 & \text { otherwise } \end{cases}$$ Show that the mean of \(X\) is \(\frac { 3 } { 2 } ( \ln 27 - 2 )\) 6 Over time it has been accepted that the mean retirement age for professional baseball players is 29.5 years old. Imran claims that the mean retirement age is no longer 29.5 years old.
He takes a random sample of 5 recently retired professional baseball players and records their retirement ages, \(x\). The results are $$\sum x = 152.1 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 7.81$$ 6
  1. State an assumption that you should make about the distribution of the retirement ages to investigate Imran's claim. 6
  2. Investigate Imran's claim, using the 10\% level of significance.
AQA Further Paper 3 Statistics 2024 June Q9
11 marks Standard +0.3
9 A company owns three shops, A, B and C, which are based in different towns. Each shop gives a questionnaire to 250 of their customers, and every customer completes the questionnaire. One of the questions asks whether the customer rates the shop as good, satisfactory or poor. For shop A, 26\% of customers rate the shop as good and 38\% of customers rate the shop as poor. For shop B, 32\% of customers rate the shop as good and 40\% of customers rate the shop as satisfactory. Altogether, there are 210 good ratings and 261 satisfactory ratings. 9
  1. Complete the following table with the observed frequencies.
    \multirow{2}{*}{}Rating
    GoodSatisfactoryPoor
    \multirow{3}{*}{Shop}A
    B
    C
    9
  2. Carry out a test for association between shop and rating, using the 1\% level of significance.
AQA Further Paper 3 Statistics 2024 June Q16
Moderate -0.8
16
256 2 The random variable \(T\) has an exponential distribution with mean 2 Find \(\mathrm { P } ( T \leq 1.4 )\) Circle your answer. \(\mathrm { e } ^ { - 2.8 }\) \(\mathrm { e } ^ { - 0.7 }\) \(1 - e ^ { - 0.7 }\) \(1 - \mathrm { e } ^ { - 2.8 }\) The continuous random variable \(Y\) has cumulative distribution function $$\mathrm { F } ( y ) = \left\{ \begin{array} { l r } 0 & y < 2 \\ - \frac { 1 } { 9 } y ^ { 2 } + \frac { 10 } { 9 } y - \frac { 16 } { 9 } & 2 \leq y < 5 \\ 1 & y \geq 5 \end{array} \right.$$ Find the median of \(Y\) Circle your answer. 2 \(\frac { 10 - 3 \sqrt { 2 } } { 2 }\) \(\frac { 7 } { 2 }\) \(\frac { 10 + 3 \sqrt { 2 } } { 2 }\) Turn over for the next question 4 Research has shown that the mean number of volcanic eruptions on Earth each day is 20 Sandra records 162 volcanic eruptions during a period of one week. Sandra claims that there has been an increase in the mean number of volcanic eruptions per week. Test Sandra's claim at the \(5 \%\) level of significance.
5 The continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} \frac { 1 } { 6 } e ^ { \frac { x } { 3 } } & 0 \leq x \leq \ln 27 \\ 0 & \text { otherwise } \end{cases}$$ Show that the mean of \(X\) is \(\frac { 3 } { 2 } ( \ln 27 - 2 )\) 6 Over time it has been accepted that the mean retirement age for professional baseball players is 29.5 years old. Imran claims that the mean retirement age is no longer 29.5 years old.
He takes a random sample of 5 recently retired professional baseball players and records their retirement ages, \(x\). The results are $$\sum x = 152.1 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 7.81$$ 6
  1. State an assumption that you should make about the distribution of the retirement ages to investigate Imran's claim. 6
  2. Investigate Imran's claim, using the 10\% level of significance.
AQA Further Paper 3 Mechanics 2019 June Q1
1 marks Moderate -0.8
1 A spring has natural length 0.4 metres and modulus of elasticity 55 N
Calculate the elastic potential energy stored in the spring when the extension of the spring is 0.08 metres. Circle your answer. \(0.176 \mathrm {~J} \quad 0.44 \mathrm {~J} \quad 0.88 \mathrm {~J} \quad 1.76 \mathrm {~J}\)
AQA Further Paper 3 Mechanics 2019 June Q2
1 marks Easy -1.8
2 A particle has an angular speed of 72 revolutions per minute.
Find the angular speed in radians per second.
Circle your answer.
[0pt] [1 mark] \(\frac { 6 \pi } { 5 } \quad \frac { 12 \pi } { 5 } \quad 12 \pi \quad 24 \pi\)
AQA Further Paper 3 Mechanics 2019 June Q3
3 marks Standard +0.3
3 A disc, of mass \(m\) and radius \(r\), rotates about an axis through its centre, perpendicular to the plane face of the disc. The angular speed of the disc is \(\omega\).
A possible model for the kinetic energy \(E\) of the disc is $$E = k m ^ { a } r ^ { b } \omega ^ { c }$$ where \(a , b\) and \(c\) are constants and \(k\) is a dimensionless constant.
Find the values of \(a , b\) and \(c\).
AQA Further Paper 3 Mechanics 2019 June Q4
8 marks Moderate -0.3
4 In this question use \(g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) An inelastic string has length 1.2 metres.
One end of the string is attached to a fixed point \(O\).
A sphere, of mass 500 grams, is attached to the other end of the string.
The sphere is held, with the string taut and at an angle of \(20 ^ { \circ }\) to the vertical, touching the chin of a student, as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{f2470caa-0f73-4ec1-b08f-525c02ed2e67-04_739_511_676_762} The sphere is released from rest.
Assume that the student stays perfectly still once the sphere has been released.
4
  1. Calculate the maximum speed of the sphere.
    4
AQA Further Paper 3 Mechanics 2019 June Q5
11 marks Standard +0.8
5 The triangular region shown below is rotated through \(360 ^ { \circ }\) around the \(x\)-axis, to form a solid cone. \includegraphics[max width=\textwidth, alt={}, center]{f2470caa-0f73-4ec1-b08f-525c02ed2e67-06_328_755_415_644} The coordinates of the vertices of the triangle are \(( 0,0 ) , ( 8,0 )\) and \(( 0,4 )\).
All units are in centimetres. 5
  1. State an assumption that you should make about the cone in order to find the position of its centre of mass. 5
  2. Using integration, prove that the centre of mass of the cone is 2 cm from its plane face.
    5
  3. The cone is placed with its plane face on a rough board. One end of the board is lifted so that the angle between the board and the horizontal is gradually increased. Eventually the cone topples without sliding. 5
    1. Find the angle between the board and the horizontal when the cone topples, giving your answer to the nearest degree. 5
  5. (ii) Find the range of possible values for the coefficient of friction between the cone and the board.
AQA Further Paper 3 Mechanics 2019 June Q6
6 marks Standard +0.3
6 A ball moving on a smooth horizontal surface collides with a fixed vertical wall. Before the collision, the ball moves with speed \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and at an angle of \(40 ^ { \circ }\) to the wall. After the collision, the ball moves with speed \(5 \mathrm {~ms} ^ { - 1 }\) and at an angle of \(26 ^ { \circ }\) to the wall. Model the ball as a particle.
6
  1. Find the coefficient of restitution between the ball and the wall, giving your answer correct to two significant figures.
    6
  2. Determine whether or not the wall is smooth. Fully justify your answer.
AQA Further Paper 3 Mechanics 2019 June Q7
9 marks Challenging +1.8
7 A particle of mass 2.5 kilograms is attached to one end of a light, inextensible string of length 75 cm . The other end of this string is attached to a point \(A\). The particle is also attached to one end of an elastic string of natural length 30 cm and modulus of elasticity \(\lambda \mathrm { N }\). The other end of this string is attached to a point \(B\), which is 60 cm vertically below \(A\). The particle is set in motion so that it describes a horizontal circle with centre \(B\). The angular speed of the particle is \(8 \mathrm { rad } \mathrm { s } { } ^ { - 1 }\) Find \(\lambda\), giving your answer in terms of \(g\).
AQA Further Paper 3 Mechanics 2019 June Q8
11 marks Challenging +1.8
8 In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) A 'reverse' bungee jump consists of two identical elastic ropes. One end of each elastic rope is attached to either side of the top of a gorge. The other ends are both attached to Hannah, who has mass 84 kg
Hannah is modelled as a particle, as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{f2470caa-0f73-4ec1-b08f-525c02ed2e67-12_467_844_678_598} The depth of the gorge is 50 metres and the width of the gorge is 40 metres.
Each elastic rope has natural length 30 metres and modulus of elasticity 3150 N
Hannah is released from rest at the centre of the bottom of the gorge.
8
  1. Show that the speed of Hannah when the ropes become slack is \(30 \mathrm {~ms} ^ { - 1 }\) correct to two significant figures.
    8
  2. Determine whether Hannah is moving up or down when the ropes become taut again. [5 marks] \includegraphics[max width=\textwidth, alt={}, center]{f2470caa-0f73-4ec1-b08f-525c02ed2e67-14_2492_1721_217_150} Additional page, if required.
    Write the question numbers in the left-hand margin. Question number Additional page, if required.
    Write the question numbers in the left-hand margin. Question number Additional page, if required.
    Write the question numbers in the left-hand margin. Question number Additional page, if required.
    Write the question numbers in the left-hand margin.
AQA Further Paper 3 Mechanics 2020 June Q1
1 marks Moderate -0.8
1 A rigid rod, \(A B\), has mass 2 kg and length 4 metres.
Two particles of masses 5 kg and 3 kg are fixed to \(A\) and \(B\) respectively to create a composite body, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{b0d0c552-71cb-4e5a-b545-de8a9052def0-02_120_730_769_653} Find the distance of the centre of mass of the composite body from \(B\). Circle your answer.
1.5 metres
1.6 metres
2.4 metres
2.5 metres
AQA Further Paper 3 Mechanics 2020 June Q2
1 marks Standard +0.3
2 The tension, \(T\) newtons, in a spring is given by \(T = 20 e\), where \(e\) metres is the extension of the spring. Calculate the work done when the extension is increased from 0.2 metres to 0.4 metres. Circle your answer.
[0pt] [1 mark]
0.4 J 0.9 J 1.2 J 1.6 J
AQA Further Paper 3 Mechanics 2020 June Q3
2 marks Easy -1.8
3 The speed, \(v\), of a particle moving in a horizontal circle is given by the formula \(v = r \omega\) where: \(v =\) speed \(r =\) radius \(\omega =\) angular speed.
Show that the dimensions of angular speed are \(T ^ { - 1 }\) [0pt] [2 marks]
AQA Further Paper 3 Mechanics 2020 June Q4
8 marks Standard +0.3
4 A car has mass 1000 kg and travels on a straight horizontal road. The maximum speed of the car on this road is \(48 \mathrm {~ms} ^ { - 1 }\) In a simple model, it is assumed that the car experiences a resistance force that is proportional to its speed. When the car travels at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the magnitude of the resistance force is 600 newtons. 4
  1. Show that the maximum power of the car is 69120 W
    4
  2. Find the maximum acceleration of the car when it is travelling at \(25 \mathrm {~ms} ^ { - 1 }\) 4
  3. Find the maximum acceleration of the car when it is travelling at \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) 4
  4. Comment on the validity of the model in the context of your answers to parts (b) and (c).