Questions FP2 (1157 questions)

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CAIE FP2 2017 November Q3
3 Three uniform small smooth spheres \(A , B\) and \(C\) have equal radii and masses \(m , k m\) and \(m\) respectively, where \(k\) is a constant. The spheres are moving in the same direction along a straight line on a smooth horizontal surface, with \(B\) between \(A\) and \(C\). The speeds of \(A , B\) and \(C\) are \(2 u , u\) and \(\frac { 4 } { 3 } u\) respectively. The coefficient of restitution between any pair of the spheres is \(\frac { 1 } { 2 }\). After sphere \(A\) has collided with sphere \(B\), sphere \(B\) collides with sphere \(C\).
  1. Find an inequality satisfied by \(k\).
  2. Given that \(k = 2\), show that after \(B\) has collided with \(C\) there are no further collisions between any of the three spheres.
CAIE FP2 2017 November Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{9b520e69-a14e-47e5-97d7-998f5145844b-06_465_663_262_742} A small ring \(P\) of weight \(W\) is free to slide on a rough horizontal wire, one end of which is attached to a vertical wall at \(Q\). The end \(A\) of a thin uniform \(\operatorname { rod } A B\) of length \(2 a\) and weight \(\frac { 5 } { 2 } W\) is freely hinged to the wall at the point \(A\) which is a distance \(a\) vertically below \(Q\). A light elastic string of natural length \(2 a\) has one end attached to the ring \(P\) and the other end attached to the rod at \(B\). The string is at right angles to the rod and \(A , B , P\) and \(Q\) lie in a vertical plane. The system is in limiting equilibrium with \(A B\) making an angle \(\theta\) with the horizontal, where \(\sin \theta = \frac { 3 } { 5 }\) (see diagram).
  1. Find the tension in the string in terms of \(W\).
  2. Find the coefficient of friction between the ring and the wire.
  3. Find the magnitude of the resultant force on the rod at the hinge in terms of \(W\).
  4. Find the modulus of elasticity of the string in terms of \(W\).
    \includegraphics[max width=\textwidth, alt={}, center]{9b520e69-a14e-47e5-97d7-998f5145844b-08_862_698_260_721} A uniform picture frame of mass \(m\) is made by removing a rectangular lamina \(E F G H\) in which \(E F = 4 a\) and \(F G = 2 a\) from a larger rectangular lamina \(A B C D\) in which \(A B = 6 a\) and \(B C = 4 a\). The side \(E F\) is parallel to the side \(A B\). The point of intersection of the diagonals \(A C\) and \(B D\) coincides with the point of intersection of the diagonals \(E G\) and \(F H\). One end of a light inextensible string of length \(10 a\) is attached to \(A\) and the other end is attached to \(B\). The frame is suspended from the mid-point \(O\) of the string. A small object of mass \(\frac { 11 } { 12 } m\) is fixed to the mid-point of \(A B\) (see diagram).
CAIE FP2 2017 November Q6
6 A pair of fair dice is thrown repeatedly until a pair of sixes is obtained. The number of throws taken is denoted by the random variable \(X\).
  1. Find the mean value of \(X\).
  2. Find the probability that exactly 12 throws are required to obtain a pair of sixes.
  3. Find the probability that more than 12 throws are required to obtain a pair of sixes.
CAIE FP2 2017 November Q7
7 The random variable \(X\) has probability density function f given by $$\mathrm { f } ( x ) = \begin{cases} 0.2 \mathrm { e } ^ { - 0.2 x } & x \geqslant 0
0 & \text { otherwise } \end{cases}$$
  1. Find the distribution function of \(X\).
  2. Find \(\mathrm { P } ( X > 2 )\).
  3. Find the median of \(X\).
CAIE FP2 2017 November Q8
8 Members of a Statistics club are voting to elect a new president of the club. Members must choose to vote either by post or by text or by email. The method of voting chosen by a random sample of 60 male members and 40 female members is given in the following table.
\cline { 3 - 5 } \multicolumn{2}{c|}{}Method of voting
\cline { 3 - 5 } \multicolumn{2}{c|}{}PostTextEmail
\multirow{2}{*}{Gender}Male101238
\cline { 2 - 5 }Female52114
Test, at the \(1 \%\) significance level, whether there is an association between method of voting and gender.
CAIE FP2 2017 November Q9
9 The land areas \(x\) (in suitable units) and populations \(y\) (in millions) for a sample of 8 randomly chosen cities are given in the following table.
Land area \(( x )\)1.04.52.41.63.88.67.56.5
Population \(( y )\)0.88.44.21.62.210.24.25.2
$$\left[ \Sigma x = 35.9 , \Sigma x ^ { 2 } = 216.47 , \Sigma y = 36.8 , \Sigma y ^ { 2 } = 244.96 , \Sigma x y = 212.62 . \right]$$
  1. Find, showing all necessary working, the value of the product moment correlation coefficient for this sample.
  2. Using a \(1 \%\) significance level, test whether there is positive correlation between land area and population of cities.
    The land areas and populations for another randomly chosen sample of cities, this time of size \(n\), give a product moment correlation coefficient of 0.651 . Using a test at the \(1 \%\) significance level, there is evidence of non-zero correlation between the variables.
  3. Find the least possible value of \(n\), justifying your answer.
CAIE FP2 2017 November Q10
10 A factory produces bottles of an energy juice. Two different machines are used to fill empty bottles with the juice. The manager chooses a random sample of 50 bottles filled by machine \(X\) and a random sample of 60 bottles filled by machine \(Y\). The volumes of juice, \(x\) and \(y\) respectively, measured in appropriate units, are summarised by $$\Sigma x = 45.5 , \quad \Sigma ( x - \bar { x } ) ^ { 2 } = 19.56 , \quad \Sigma y = 72.3 , \quad \Sigma ( y - \bar { y } ) ^ { 2 } = 30.25$$ where \(\bar { x }\) and \(\bar { y }\) are the sample means of the volume of juice in the bottles filled by \(X\) and \(Y\) respectively.
  1. Find a 90\% confidence interval for the difference between the mean volume of juice in bottles filled by machine \(X\) and the mean volume of juice in bottles filled by machine \(Y\).
    A test at the \(\alpha \%\) significance level does not provide evidence that there is any difference in the means of the volume of juice in bottles filled by machine \(X\) and the volume of juice in bottles filled by machine \(Y\).
  2. Find the set of possible values of \(\alpha\).
CAIE FP2 2017 November Q11 EITHER
\includegraphics[max width=\textwidth, alt={}]{9b520e69-a14e-47e5-97d7-998f5145844b-18_552_588_438_776}
A particle \(P\) of mass \(m\) is free to move on the smooth inner surface of a fixed hollow sphere of radius \(a\). The centre of the sphere is \(O\). The points \(A\) and \(A ^ { \prime }\) are on the inner surface of the sphere, on opposite sides of the vertical through \(O\); the radius \(O A\) makes an angle \(\alpha\) with the downward vertical and the radius \(O A ^ { \prime }\) makes an angle \(\beta\) with the upward vertical. The point \(B\) is on the inner surface of the sphere, vertically below \(O\). The point \(B ^ { \prime }\) is on the inner surface of the sphere and such that \(O B ^ { \prime }\) makes an angle \(2 \beta\) with the upward vertical through \(O\) (see diagram). It is given that \(\cos \alpha = \frac { 1 } { 16 }\).
  1. \(P\) is projected from \(A\) with speed \(u\) along the surface of the sphere downwards towards \(B\). Subsequently it loses contact with the sphere at \(A ^ { \prime }\). Show that \(u ^ { 2 } = \frac { 1 } { 8 } a g ( 1 + 24 \cos \beta )\).
  2. \(P\) is now projected from \(B\) with speed \(u\) along the surface of the sphere towards \(B ^ { \prime }\). Subsequently it loses contact with the sphere at \(B ^ { \prime }\). Find \(\cos \beta\).
  3. In part (i), the reaction of the sphere on \(P\) when it is initially projected at \(A\) is \(R\). Find \(R\) in terms of \(m\) and \(g\).
CAIE FP2 2017 November Q11 OR
A large number of people attended a course to improve the speed of their logical thinking. The times taken to complete a particular type of logic puzzle at the beginning of the course and at the end of the course are recorded for each person. The time taken, in minutes, at the beginning of the course is denoted by \(x\) and the time taken, in minutes, at the end of the course is denoted by \(y\). For a random sample of 9 people, the results are summarised as follows. $$\Sigma x = 45.3 \quad \Sigma x ^ { 2 } = 245.59 \quad \Sigma y = 40.5 \quad \Sigma y ^ { 2 } = 195.11 \quad \Sigma x y = 218.72$$ Ken attended the course, but his time to complete the puzzle at the beginning of the course was not recorded. His time to complete the puzzle at the end of the course was 4.2 minutes.
  1. By finding, showing all necessary working, the equation of a suitable regression line, find an estimate for the time that Ken would have taken to complete the puzzle at the beginning of the course.
    The values of \(x - y\) for the sample of 9 people are as follows. $$\begin{array} { l l l l l l l l l } 0.2 & 0.8 & 0.5 & 1.0 & 0.2 & 0.6 & 0.2 & 0.5 & 0.8 \end{array}$$ The organiser of the course believes that, on average, the time taken to complete the puzzle decreases between the beginning and the end of the course by more than 0.3 minutes.
  2. Stating suitable hypotheses and assuming a normal distribution, test the organiser's belief at the \(2 \frac { 1 } { 2 } \%\) significance level.
CAIE FP2 2017 November Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{1651d08b-b20f-4f2e-9f47-0a1a5d0a839a-06_465_663_262_742} A small ring \(P\) of weight \(W\) is free to slide on a rough horizontal wire, one end of which is attached to a vertical wall at \(Q\). The end \(A\) of a thin uniform \(\operatorname { rod } A B\) of length \(2 a\) and weight \(\frac { 5 } { 2 } W\) is freely hinged to the wall at the point \(A\) which is a distance \(a\) vertically below \(Q\). A light elastic string of natural length \(2 a\) has one end attached to the ring \(P\) and the other end attached to the rod at \(B\). The string is at right angles to the rod and \(A , B , P\) and \(Q\) lie in a vertical plane. The system is in limiting equilibrium with \(A B\) making an angle \(\theta\) with the horizontal, where \(\sin \theta = \frac { 3 } { 5 }\) (see diagram).
  1. Find the tension in the string in terms of \(W\).
  2. Find the coefficient of friction between the ring and the wire.
  3. Find the magnitude of the resultant force on the rod at the hinge in terms of \(W\).
  4. Find the modulus of elasticity of the string in terms of \(W\).
    \includegraphics[max width=\textwidth, alt={}, center]{1651d08b-b20f-4f2e-9f47-0a1a5d0a839a-08_862_698_260_721} A uniform picture frame of mass \(m\) is made by removing a rectangular lamina \(E F G H\) in which \(E F = 4 a\) and \(F G = 2 a\) from a larger rectangular lamina \(A B C D\) in which \(A B = 6 a\) and \(B C = 4 a\). The side \(E F\) is parallel to the side \(A B\). The point of intersection of the diagonals \(A C\) and \(B D\) coincides with the point of intersection of the diagonals \(E G\) and \(F H\). One end of a light inextensible string of length \(10 a\) is attached to \(A\) and the other end is attached to \(B\). The frame is suspended from the mid-point \(O\) of the string. A small object of mass \(\frac { 11 } { 12 } m\) is fixed to the mid-point of \(A B\) (see diagram).
CAIE FP2 2017 November Q11 EITHER
\includegraphics[max width=\textwidth, alt={}]{1651d08b-b20f-4f2e-9f47-0a1a5d0a839a-18_552_588_438_776}
A particle \(P\) of mass \(m\) is free to move on the smooth inner surface of a fixed hollow sphere of radius \(a\). The centre of the sphere is \(O\). The points \(A\) and \(A ^ { \prime }\) are on the inner surface of the sphere, on opposite sides of the vertical through \(O\); the radius \(O A\) makes an angle \(\alpha\) with the downward vertical and the radius \(O A ^ { \prime }\) makes an angle \(\beta\) with the upward vertical. The point \(B\) is on the inner surface of the sphere, vertically below \(O\). The point \(B ^ { \prime }\) is on the inner surface of the sphere and such that \(O B ^ { \prime }\) makes an angle \(2 \beta\) with the upward vertical through \(O\) (see diagram). It is given that \(\cos \alpha = \frac { 1 } { 16 }\).
  1. \(P\) is projected from \(A\) with speed \(u\) along the surface of the sphere downwards towards \(B\). Subsequently it loses contact with the sphere at \(A ^ { \prime }\). Show that \(u ^ { 2 } = \frac { 1 } { 8 } a g ( 1 + 24 \cos \beta )\).
  2. \(P\) is now projected from \(B\) with speed \(u\) along the surface of the sphere towards \(B ^ { \prime }\). Subsequently it loses contact with the sphere at \(B ^ { \prime }\). Find \(\cos \beta\).
  3. In part (i), the reaction of the sphere on \(P\) when it is initially projected at \(A\) is \(R\). Find \(R\) in terms of \(m\) and \(g\).
CAIE FP2 2017 November Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{2ab1a594-6c37-4c78-b53c-33c13bf6eb21-06_465_663_262_742} A small ring \(P\) of weight \(W\) is free to slide on a rough horizontal wire, one end of which is attached to a vertical wall at \(Q\). The end \(A\) of a thin uniform \(\operatorname { rod } A B\) of length \(2 a\) and weight \(\frac { 5 } { 2 } W\) is freely hinged to the wall at the point \(A\) which is a distance \(a\) vertically below \(Q\). A light elastic string of natural length \(2 a\) has one end attached to the ring \(P\) and the other end attached to the rod at \(B\). The string is at right angles to the rod and \(A , B , P\) and \(Q\) lie in a vertical plane. The system is in limiting equilibrium with \(A B\) making an angle \(\theta\) with the horizontal, where \(\sin \theta = \frac { 3 } { 5 }\) (see diagram).
  1. Find the tension in the string in terms of \(W\).
  2. Find the coefficient of friction between the ring and the wire.
  3. Find the magnitude of the resultant force on the rod at the hinge in terms of \(W\).
  4. Find the modulus of elasticity of the string in terms of \(W\).
    \includegraphics[max width=\textwidth, alt={}, center]{2ab1a594-6c37-4c78-b53c-33c13bf6eb21-08_862_698_260_721} A uniform picture frame of mass \(m\) is made by removing a rectangular lamina \(E F G H\) in which \(E F = 4 a\) and \(F G = 2 a\) from a larger rectangular lamina \(A B C D\) in which \(A B = 6 a\) and \(B C = 4 a\). The side \(E F\) is parallel to the side \(A B\). The point of intersection of the diagonals \(A C\) and \(B D\) coincides with the point of intersection of the diagonals \(E G\) and \(F H\). One end of a light inextensible string of length \(10 a\) is attached to \(A\) and the other end is attached to \(B\). The frame is suspended from the mid-point \(O\) of the string. A small object of mass \(\frac { 11 } { 12 } m\) is fixed to the mid-point of \(A B\) (see diagram).
CAIE FP2 2017 November Q11 EITHER
\includegraphics[max width=\textwidth, alt={}]{2ab1a594-6c37-4c78-b53c-33c13bf6eb21-18_552_588_438_776}
A particle \(P\) of mass \(m\) is free to move on the smooth inner surface of a fixed hollow sphere of radius \(a\). The centre of the sphere is \(O\). The points \(A\) and \(A ^ { \prime }\) are on the inner surface of the sphere, on opposite sides of the vertical through \(O\); the radius \(O A\) makes an angle \(\alpha\) with the downward vertical and the radius \(O A ^ { \prime }\) makes an angle \(\beta\) with the upward vertical. The point \(B\) is on the inner surface of the sphere, vertically below \(O\). The point \(B ^ { \prime }\) is on the inner surface of the sphere and such that \(O B ^ { \prime }\) makes an angle \(2 \beta\) with the upward vertical through \(O\) (see diagram). It is given that \(\cos \alpha = \frac { 1 } { 16 }\).
  1. \(P\) is projected from \(A\) with speed \(u\) along the surface of the sphere downwards towards \(B\). Subsequently it loses contact with the sphere at \(A ^ { \prime }\). Show that \(u ^ { 2 } = \frac { 1 } { 8 } a g ( 1 + 24 \cos \beta )\).
  2. \(P\) is now projected from \(B\) with speed \(u\) along the surface of the sphere towards \(B ^ { \prime }\). Subsequently it loses contact with the sphere at \(B ^ { \prime }\). Find \(\cos \beta\).
  3. In part (i), the reaction of the sphere on \(P\) when it is initially projected at \(A\) is \(R\). Find \(R\) in terms of \(m\) and \(g\).
CAIE FP2 2018 November Q1
1 A particle \(P\) oscillates in simple harmonic motion between the points \(A\) and \(B\), where \(A B = 6 \mathrm {~m}\). The period of the motion is \(\frac { 1 } { 2 } \pi \mathrm {~s}\). Find the speed of \(P\) when it is 2 m from \(B\).
CAIE FP2 2018 November Q2
2 Two uniform small smooth spheres \(A\) and \(B\) have equal radii and masses \(5 m\) and \(2 m\) respectively. Sphere \(A\) is moving with speed \(u\) on a smooth horizontal surface when it collides directly with sphere \(B\) which is moving towards it with speed \(2 u\). The coefficient of restitution between the spheres is \(e\).
  1. Show that the speed of \(B\) after the collision is \(\frac { 1 } { 7 } u ( 1 + 15 e )\) and find an expression for the speed of \(A\).
    In the collision, the speed of \(A\) is halved and its direction of motion is reversed.
  2. Find the value of \(e\).
  3. For this collision, find the ratio of the loss of kinetic energy of \(A\) to the loss of kinetic energy of \(B\).
    \includegraphics[max width=\textwidth, alt={}, center]{89a83576-4568-46c8-8872-f59f2397627d-04_630_332_264_900} A uniform disc, of radius \(a\) and mass \(2 M\), is attached to a thin uniform rod \(A B\) of length \(6 a\) and mass \(M\). The rod lies along a diameter of the disc, so that the centre of the disc is a distance \(x\) from \(A\) (see diagram).
  4. Find the moment of inertia of the object, consisting of disc and rod, about a fixed horizontal axis \(l\) through \(A\) and perpendicular to the plane of the disc.
    The object is free to rotate about the axis \(l\). The object is held with \(A B\) horizontal and is released from rest. When \(A B\) makes an angle \(\theta\) with the vertical, where \(\cos \theta = \frac { 3 } { 5 }\), the angular speed of the object is \(\sqrt { } \left( \frac { 2 g } { 5 a } \right)\).
  5. Find the possible values of \(x\).
CAIE FP2 2018 November Q4
4 A uniform rod \(A B\) of length \(4 a\) and weight \(W\) is smoothly hinged to a vertical wall at the end \(A\). The rod is held at an angle \(\theta\) above the horizontal by a light elastic string. One end of the string is attached to the point \(C\) on the rod, where \(A C = 3 a\). The other end of the string is attached to a point \(D\) on the wall, with \(D\) vertically above \(A\) and such that angle \(A C D = 2 \theta\). A particle of weight \(\frac { 1 } { 2 } W\) is attached to the rod at \(B\). It is given that \(\tan \theta = \frac { 8 } { 15 }\).
  1. Show that the tension in the string is \(\frac { 17 } { 12 } W\).
  2. Find the magnitude and direction of the reaction at the hinge.
  3. Given that the natural length of the string is \(2 a\), find its modulus of elasticity.
CAIE FP2 2018 November Q5
5 The fixed points \(A\) and \(B\) are on a smooth horizontal surface with \(A B = 2.6 \mathrm {~m}\). One end of a light elastic spring, of natural length 1.25 m and modulus of elasticity \(\lambda \mathrm { N }\), is attached to \(A\). The other end is attached to a particle \(P\) of mass 0.4 kg . One end of a second light elastic spring, of natural length 1.0 m and modulus of elasticity \(0.6 \lambda \mathrm {~N}\), is attached to \(B\); its other end is attached to \(P\). The system is in equilibrium with \(P\) on the surface at the point \(E\).
  1. Show that \(A E = 1.4 \mathrm {~m}\).
    The particle \(P\) is now displaced slightly from \(E\), along the line \(A B\).
  2. Show that, in the subsequent motion, \(P\) performs simple harmonic motion.
  3. Given that the period of the motion is \(\frac { 1 } { 7 } \pi \mathrm {~s}\), find the value of \(\lambda\).
CAIE FP2 2018 November Q6
6 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} \frac { 1 } { 80 } \left( 3 \sqrt { } x - \frac { 8 } { \sqrt { } x } \right) & 4 \leqslant x \leqslant 16
0 & \text { otherwise } \end{cases}$$
  1. Find the distribution function of \(X\).
    The random variable \(Y\) is defined by \(Y = \sqrt { } X\).
  2. Find the probability density function of \(Y\).
CAIE FP2 2018 November Q7
7 The random variable \(T\) is the lifetime, in hours, of a particular type of battery. It is given that \(T\) has a negative exponential distribution with mean 500 hours.
  1. Write down the probability density function of \(T\).
  2. Find the probability that a randomly chosen battery of this type has a lifetime of more than 750 hours.
  3. Find the median value of \(T\).
CAIE FP2 2018 November Q8
8 The weekly salaries of employees at two large electronics companies, \(A\) and \(B\), are being compared. The weekly salaries of an employee from company \(A\) and an employee from company \(B\) are denoted by \(
) x\( and \)\\( y\) respectively. A random sample of 50 employees from company \(A\) and a random sample of 40 employees from company \(B\) give the following summarised data. $$\Sigma x = 5120 \quad \Sigma x ^ { 2 } = 531000 \quad \Sigma y = 3760 \quad \Sigma y ^ { 2 } = 375135$$
  1. The population mean salaries of employees from companies \(A\) and \(B\) are denoted by \(
    ) \mu _ { A }\( and \)\\( \mu _ { B }\) respectively. Using a \(5 \%\) significance level, test the null hypothesis \(\mu _ { A } = \mu _ { B }\) against the alternative hypothesis \(\mu _ { A } \neq \mu _ { B }\).
  2. State, with a reason, whether any assumptions about the distributions of employees' salaries are needed for the test in part (i).
CAIE FP2 2018 November Q9
9 There are a large number of students at a particular college. The heights, in metres, of a random sample of 8 students are as follows. $$\begin{array} { l l l l l l l l } 1.75 & 1.72 & 1.62 & 1.70 & 1.82 & 1.75 & 1.68 & 1.84 \end{array}$$ You may assume that heights of students are normally distributed.
  1. Test, at the \(5 \%\) significance level, whether the population mean height of students at this college is greater than 1.70 metres.
  2. Find a \(95 \%\) confidence interval for the population mean height of students at this college.
CAIE FP2 2018 November Q10
10 For a random sample of 10 observations of pairs of values \(( x , y )\), the equation of the regression line of \(y\) on \(x\) is \(y = 1.1664 + 0.4604 x\). It is given that $$\Sigma x ^ { 2 } = 1419.98 \quad \text { and } \quad \Sigma y ^ { 2 } = 439.68 .$$ The mean value of \(y\) is 6.24 .
  1. Find the equation of the regression line of \(x\) on \(y\).
  2. Find the product moment correlation coefficient.
  3. Test at the \(5 \%\) significance level whether there is evidence of positive correlation between the two variables.
CAIE FP2 2018 November Q11 EITHER
6 marks
A particle \(P\) of mass \(m\) is free to move on the smooth inner surface of a fixed hollow sphere of radius \(a\). The centre of the sphere is \(O\) and the point \(C\) is on the inner surface of the sphere, vertically below \(O\). The points \(A\) and \(B\) on the inner surface of the sphere are the ends of a diameter of the sphere. The diameter \(A O B\) makes an acute angle \(\alpha\) with the vertical, where \(\cos \alpha = \frac { 4 } { 5 }\), with \(A\) below the horizontal level of \(B\). The particle is projected from \(A\) with speed \(u\), and moves along the inner surface of the sphere towards \(C\). The normal reaction forces on the particle at \(A\) and \(C\) are in the ratio \(8 : 9\).
  1. Show that \(u ^ { 2 } = 4 a g\).
  2. Determine whether \(P\) reaches \(B\) without losing contact with the inner surface of the sphere. [6]
CAIE FP2 2018 November Q11 OR
A machine is used to produce metal rods. When the machine is working efficiently, the lengths, \(x \mathrm {~cm}\), of the rods have a normal distribution with mean 150 cm and standard deviation 1.2 cm . The machine is checked regularly by taking random samples of 200 rods. The latest results are shown in the following table.
Interval\(146 \leqslant x < 147\)\(147 \leqslant x < 148\)\(148 \leqslant x < 149\)\(149 \leqslant x < 150\)
Observed frequency122352
\(150 \leqslant x < 151\)\(151 \leqslant x < 152\)\(152 \leqslant x < 153\)\(153 \leqslant x < 154\)
6936152
As a first check, the sample is used to calculate an estimate for the mean.
  1. Show that an estimate for the mean from this sample is close to 150 cm .
    As a second check, the results are tested for goodness of fit of the normal distribution with mean 150 cm and standard deviation 1.2 cm . The relevant expected frequencies, found using the normal distribution function given in the List of Formulae (MF10), are shown in the following table.
    Interval\(x < 147\)\(147 \leqslant x < 148\)\(148 \leqslant x < 149\)\(149 \leqslant x < 150\)
    Observed frequency122352
    Expected frequency1.248.3230.9459.50
    \(150 \leqslant x < 151\)\(151 \leqslant x < 152\)\(152 \leqslant x < 153\)\(153 \leqslant x\)
    6936152
    59.5030.948.321.24
  2. Show how the expected frequency for \(151 \leqslant x < 152\) is obtained.
  3. Test, at the \(5 \%\) significance level, the goodness of fit of the normal distribution to the results.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE FP2 2018 November Q1
1 The point \(O\) is on the fixed horizontal line \(l\). Points \(A\) and \(B\) on \(l\) are such that \(O A = 0.1 \mathrm {~m}\) and \(O B = 0.5 \mathrm {~m}\), with \(A\) between \(O\) and \(B\). A particle \(P\) oscillates on \(l\) in simple harmonic motion with centre \(O\). The kinetic energy of \(P\) when it is at \(A\) is twice its kinetic energy when it is at \(B\). Find the amplitude of the motion.