SPS SPS FM (SPS FM) 2021 February

4. Ray is conducting a hypothesis test with the hypotheses
\(\mathrm { H } _ { 0 }\) : There is no association between time of day and number of snacks eaten
\(\mathrm { H } _ { 1 }\) : There is an association between time of day and number of snacks eaten
He calculates expected frequencies correct to two decimal places, which are given in the following table. Ray calculates his test statistic using \(\sum \frac { ( O - E ) ^ { 2 } } { E }\)

1. State, with a reason, the error Ray has made and describe any changes Ray will need to make to his test.
   [0pt] [3 marks]
2. Having made the necessary corrections as described in part (a), the correct value of the test statistic is 8.74 Complete Ray's hypothesis test using a \(1 \%\) level of significance.

5. The distance, \(X\) metres, between successive breaks in a water pipe is modelled by an exponential distribution. The mean of \(X\) is 25 The distance between two successive breaks is measured. A water pipe is given a 'Red' rating if the distance is less than \(d\) metres. The govemment has introduced a new law changing \(d\) to 2
Before the govermment introduced the new law, the probability that a water pipe is given a 'Red' rating was 0.05

1. Explain whether or not the probability that a water pipe is given a 'Red' rating has increased as a result of the new law.
2. Find the probability density function of the random variable \(X\).
3. After investigation, the distances between successive breaks in water pipes are found to have a standard deviation of 5 metres. Explain whether or not the use of an exponential model in parts (a) and (b) is appropriate.
   [0pt] [2 marks]

6. The continuous random variable \(X\) has the cumulative distribution function shown below. $$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < 0
\frac { 1 } { 62 } \left( 4 x ^ { 3 } + 6 x ^ { 2 } + 3 x \right) & 0 \leq x \leq 2
1 & x > 2 \end{array} \right.$$ The discrete random variable \(Y\) has the probability distribution shown below. The random variables \(X\) and \(Y\) are independent.
Find the exact value of \(\mathrm { E } \left( X ^ { 3 } + Y \right)\).
[0pt] [6 marks]

7. \section*{In this question you must show detailed reasoning.} On the manufacturer's website, it is claimed that the average daily electricity consumption of a particular model of fridge is 1.25 kWh (kilowatt hours). A researcher at a consumer organisation decides to check this figure. A random sample of 40 fridges is selected. Summary statistics for the electricity consumption \(x \mathrm { kWh }\) of these fridges, measured over a period of 24 hours, are as follows.
\(\Sigma x = 51.92 \quad \Sigma x ^ { 2 } = 70.57\) Carry out a test at the \(5 \%\) significance level to investigate the validity of the claim on the website.
[0pt] [10]

8. A student is investigating immunisation. He wonders if there is any relationship between the percentage of young children who have been given measles vaccine and the percentage who have been given BCG vaccine in various countries. He takes a random sample of 8 countries and finds the data for the two variables. The spreadsheet in Fig. 5.1 shows the values obtained, together with a scatter diagram which illustrates the data. \begin{figure}[h] \end{figure} The student carries out a test based on Spearman's rank correlation coefficient.

1. Calculate the value of Spearman's rank correlation coefficient.
2. Carry out a test based on this coefficient at the \(5 \%\) significance level to investigate whether there is any association between measles and BCG vaccination levels. The student then decides to investigate the relationship between number of doctors per 1000 people in a country and unemployment rate in that country (unemployment rate is the percentage of the working age population who are not in paid work). He selects a random sample of 6 countries. The spreadsheet in Fig. 5.2 shows the values obtained, together with a scatter diagram which illustrates the data. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{e88ba052-a95b-4741-95d5-6cc18672ce30-06_732_1552_534_255} \captionsetup{labelformat=empty} \caption{Fig. 5.2}
   \end{figure}
3. Use your calculator to write down the equation of the regression line of unemployment rate on doctors per 1000 .
4. Use the regression line to estimate the unemployment rate for a country with 2.00 doctors per 1000.
5. Comment briefly on the reliability of your answer to part(d). Name: □ 25 \({ ^ { \text {th } }\) February 2021} Instructions
   • Answer all the questions.
   • Write your answer to each question on file paper The question number(s) must be clearly shown.
   • Use black or blue ink. Pencil may be used for graphs and diagrams only.
   • You should clearly write your name at the top of each page.
   • You are permitted to use a scientific or graphical calculator in this paper.
   • Final answers should be given to a degree of accuracy appropriate to the context.
   • At the end you must upload your solutions to the mechanics questions to the Google classroom of your mechanics teacher before you leave the examination Google Meet.
   Information
   • The total mark for this paper is \(\mathbf { 5 6 }\) marks.
   • The marks for each question are shown in brackets.
   • You are reminded of the need for clear presentation in your answers.
   • You should allow approximately 65 minutes for this section of the test.
   1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{e88ba052-a95b-4741-95d5-6cc18672ce30-08_508_590_347_475} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{e88ba052-a95b-4741-95d5-6cc18672ce30-08_303_328_550_1347} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} A uniform plane figure \(R\), shown shaded in Figure 1, is bounded by the \(x\)-axis, the line with equation \(x = \ln 5\), the curve with equation \(y = 8 \mathrm { e } ^ { - x }\) and the line with equation \(x = \ln 2\). The unit of length on each axis is one metre. The area of \(R\) is \(2.4 \mathrm {~m} ^ { 2 }\)
   The centre of mass of \(R\) is at the point with coordinates \(( \bar { x } , \bar { y } )\).
6. Use algebraic integration to show that \(\bar { y } = 1.4\) Figure 2 shows a uniform lamina \(A B C D\), which is the same size and shape as \(R\). The lamina is freely suspended from \(C\) and hangs in equilibrium with \(C B\) at an angle \(\theta ^ { \circ }\) to the downward vertical.
7. Find the value of \(\theta\) \section*{2.} Two particles, \(A\) and \(B\), have masses \(3 m\) and \(4 m\) respectively. The particles are moving in the same direction along the same straight line on a smooth horizontal surface when they collide directly. Immediately before the collision the speed of \(A\) is \(2 u\) and the speed of \(B\) is \(u\). The coefficient of restitution between \(A\) and \(B\) is \(e\).
8. Show that the direction of motion of each of the particles is unchanged by the collision. After the collision with \(A\), particle \(B\) collides directly with a third particle, \(C\), of mass \(2 m\), which is at rest on the surface. The coefficient of restitution between \(B\) and \(C\) is also \(e\).
9. Show that there will be a second collision between \(A\) and \(B\). \section*{3.} A light elastic string with natural length \(l\) and modulus of elasticity \(k m g\) has one end attached to a fixed point \(A\) on a rough inclined plane. The other end of the string is attached to a package of mass \(m\). The plane is inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 5 } { 12 }\)
   The package is initially held at \(A\). The package is then projected with speed \(\sqrt { 6 g l }\) up a line of greatest slope of the plane and first comes to rest at the point \(B\), where \(A B = 3 l\).
   The coefficient of friction between the package and the plane is \(\frac { 1 } { 4 }\)
   By modelling the package as a particle,
10. show that \(k = \frac { 15 } { 26 }\)
11. find the acceleration of the package at the instant it starts to move back down the plane from the point \(B\).
    4. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{e88ba052-a95b-4741-95d5-6cc18672ce30-10_471_574_340_790} \captionsetup{labelformat=empty} \caption{Figure 4}
    \end{figure} A particle \(P\) of mass 0.75 kg is attached to one end of a light inextensible string of length 60 cm . The other end of the string is attached to a fixed point \(A\) that is vertically above the point \(O\) on a smooth horizontal table, such that \(O A = 40 \mathrm {~cm}\). The particle remains in contact with the table, with the string taut, and moves in a horizontal circle with centre \(O\), as shown in Figure 4. The particle is moving with a constant angular speed of 3 radians per second.
12. Find
    1. the tension in the string,
    2. the normal reaction between \(P\) and the table. The angular speed of \(P\) is now gradually increased.
13. Find the angular speed of \(P\) at the instant \(P\) loses contact with the table. \section*{5.} A particle \(P\) of mass 0.5 kg is moving along the positive \(x\)-axis in the direction of \(x\) increasing. At time \(t\) seconds \(( t \geqslant 0 ) , P\) is \(x\) metres from the origin \(O\) and the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resultant force acting on \(P\) is directed towards \(O\) and has magnitude \(k v ^ { 2 } \mathrm {~N}\), where \(k\) is a positive constant. When \(x = 1 , v = 4\) and when \(x = 2 , v = 2\)
14. Show that \(v = a b ^ { x }\), where \(a\) and \(b\) are constants to be found. The time taken for the speed of \(P\) to decrease from \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is \(T\) seconds.
15. Show that \(T = \frac { 1 } { 4 \ln 2 }\)