| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2012 |
| Session | June |
| Marks | 24 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of Pearson’s product-moment correlation coefficient |
| Type | State distributional assumption for test |
| Difficulty | Standard +0.3 This is a standard hypothesis test for correlation with straightforward calculation of PMCC from given data and comparison to critical values. The calculations are routine (sums, products, formula application) and the hypothesis test follows a standard template. While it requires careful arithmetic and understanding of the test procedure, it involves no novel problem-solving or conceptual difficulty beyond typical A-level statistics. |
| Spec | 5.09a Dependent/independent variables5.09c Calculate regression line6.02h Elastic PE: 1/2 k x^26.02i Conservation of energy: mechanical energy principle6.02j Conservation with elastics: springs and strings6.03k Newton's experimental law: direct impact |
| Day | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| \(x\) | 1.2 | 1.4 | 0.9 | 1.1 | 0.8 | 1.0 | 0.6 | 1.5 |
| \(y\) | 0.3 | 0.4 | 0.6 | 0.6 | 0.25 | 0.75 | 0.6 | 0.35 |
| Answer | Marks |
|---|---|
| 11 (a) | Resolve vertically at equilibrium with extn. |
| Answer | Marks | Guidance |
|---|---|---|
| V = ¼√(5gl) A.G. A1 | 4 | |
| 8 | [12] | |
| Page 8 | Mark Scheme: Teachers’ version | Syllabus |
| GCE A LEVEL – May/June 2012 | 9231 | 22 |
| Answer | Marks |
|---|---|
| (iii) | Find correlation coefficient r: Σx = 8⋅5, Σx 2 = 9⋅67, Σxy = 3⋅955, |
| Answer | Marks | Guidance |
|---|---|---|
| Valid method for reaching conclusion: Accept H0 if | r | < tabular value M1 |
| Answer | Marks |
|---|---|
| Deduce range of possible values of N: N ≥ 16 A1 | 4 |
| Answer | Marks |
|---|---|
| 3 | [12] |
Question 11:
--- 11 (a) ---
11 (a) | Resolve vertically at equilibrium with extn.
e: 4mge / l = mg [e = ¼l] B1
2 2
Use Newton’s Law at general point: m d x/dt = mg – 4mg(e+x)/l
[ or – mg + 4mg(e–x)/l ] M1
2 2
Simplify to give standard SHM eqn: d x/dt = – (4g/l) x A1
S.R.: Stating this without derivation
(max 3/4): (B1)
Find period T using SHM with ω = √(4g/l): T = 2π/√(4g/l) = π√(l/g) A.G. B1
2 2 2 2
Find speed v E at E using v = ω (A – x )
with x = 0: vE = ωl/8 = ¼√(gl) M1 A1
Find speed vP before striking plane (A.E.F.): vP = √(gl/16 + 14gl/16) = ¼√(15gl) M1 A1
Find comps. of speed V after striking plane: Parallel to plane: vP sin 30°
or ½ vP or √(15gl/64) B1
Normal to plane: ⅓ vP cos 30°
or ⅓(√3/2) vP or √(5gl/64) B1
2
Combine to find V: V = 15gl/64 + 5gl/64 = 5gl/16 M1
V = ¼√(5gl) A.G. A1 | 4
8 | [12]
Page 8 | Mark Scheme: Teachers’ version | Syllabus | Paper
GCE A LEVEL – May/June 2012 | 9231 | 22
(b) (i)
(ii)
(iii) | Find correlation coefficient r: Σx = 8⋅5, Σx 2 = 9⋅67, Σxy = 3⋅955,
Σy = 3⋅85, Σy 2 = 2⋅0775
r = (3⋅955 – 8⋅5 × 3⋅85/8) / √{(9⋅67 –
8⋅5 2 /8) (2⋅0775 – 3⋅85 2 /8)} M1 A1
(A0 if only 3 s.f. used) = –0⋅1356 / √(0⋅6387 × 0⋅2247) A1
= –0⋅1356 / (0⋅7992 × 0⋅4740)
[or –0⋅01695 / √(0⋅07984 × 0⋅02809)
= –0⋅01695 / (0⋅2826 × 0⋅1676) ]
= – 0⋅358 *A1
H0: ρ = 0, H1: ρ < 0 B1
State both hypotheses:
State or use correct tabular one-tail r value: r8, 2.5% = 0⋅707 *B1
Valid method for reaching conclusion: Accept H0 if |r| < tabular value M1
Correct conclusion (AEF, dep *A1, *B1): There is no negative correlation A1
Valid comment, consistent with values No effect of S on R (A.E.F.) B1√
Find critical tabular one-tail r value: r16, 5% = 0⋅426 or r15, 5% = 0⋅441 M1 A1
Deduce range of possible values of N: N ≥ 16 A1 | 4
5
3 | [12]Answer only one of the following two alternatives.
EITHER
A particle $P$ of mass $m$ is attached to one end of a light elastic string of modulus of elasticity $4mg$ and natural length $l$. The other end of the string is attached to a fixed point $O$. The particle rests in equilibrium at the point $E$, vertically below $O$. The particle is pulled down a vertical distance $\frac{3l}{4}$ from $E$ and released from rest. Show that the motion of $P$ is simple harmonic with period $\pi\sqrt{\left(\frac{l}{g}\right)}$.
[4]
At an instant when $P$ is moving vertically downwards through $E$, the string is cut. When $P$ has descended a further distance $\frac{5l}{4}$ under gravity, it strikes a fixed smooth plane which is inclined at 30° to the horizontal. The coefficient of restitution between $P$ and the plane is $\frac{1}{3}$. Show that the speed of $P$ immediately after the impact is $\frac{1}{3}\sqrt{(5gl)}$.
[8]
OR
A new restaurant $S$ has recently opened in a particular town. In order to investigate any effect of $S$ on an existing restaurant $R$, the daily takings, $x$ and $y$ in thousands of dollars, at $R$ and $S$ respectively are recorded for a random sample of 8 days during a six-month period. The results are shown in the following table.
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
Day & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
\hline
$x$ & 1.2 & 1.4 & 0.9 & 1.1 & 0.8 & 1.0 & 0.6 & 1.5 \\
\hline
$y$ & 0.3 & 0.4 & 0.6 & 0.6 & 0.25 & 0.75 & 0.6 & 0.35 \\
\hline
\end{tabular}
\begin{enumerate}[label=(\roman*)]
\item Calculate the product moment correlation coefficient for this sample. [4]
\item Stating your hypotheses, test, at the 2.5\% significance level, whether there is negative correlation between daily takings at the two restaurants and comment on your result in the context of the question. [5]
\end{enumerate}
Another sample is taken over $N$ randomly chosen days and the product moment correlation coefficient is found to be $-0.431$. A test, at the 5\% significance level, shows that there is evidence of negative correlation between daily takings in the two restaurants.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{2}
\item Find the range of possible values of $N$. [3]
\end{enumerate}
\hfill \mbox{\textit{CAIE FP2 2012 Q11 [24]}}