| 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 calculations. Students compute the PMCC using the formula (routine but tedious arithmetic), then perform a one-tailed test at 2.5% level using critical values from tables. The context is simple and the question structure is typical for Further Statistics, requiring only standard procedures without novel insight. |
| 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 | 21 |
| 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 | 21
(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 $\frac{3l}{4}$ 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{center}
\begin{tabular}{|l|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}
\end{center}
\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]}}