Moderate -0.5 This is a standard two-sample t-test with equal variances (pooled variance) followed by a one-sample t-test. Both parts require routine application of formulas from the specification with straightforward calculations. The question provides all necessary summary statistics and clearly states assumptions, making it easier than average A-level questions which often require more problem-solving or conceptual understanding.
Answer only one of the following two alternatives.
**EITHER**
One end of a light elastic spring, of natural length 0.8 m and modulus of elasticity 40 N, is attached to a fixed point \(O\). The spring hangs vertically, at rest, with particles of masses 2 kg and \(M\) kg attached to its free end. The \(M\) kg particle becomes detached from the spring, and as a result the 2 kg particle begins to move upwards.
\begin{enumerate}[label=(\roman*)]
\item Show that the 2 kg particle performs simple harmonic motion about its equilibrium position with period \(\frac{2\pi}{5}\) s. State the distance below \(O\) of the centre of the oscillations.
[7]
\item The speed of the 2 kg particle is 0.4 m s\(^{-1}\) when its displacement from the centre of oscillation is 0.06 m.
Find the amplitude of the motion.
[3]
\item Deduce the value of \(M\).
[4]
\end{enumerate]
**OR**
In a particular country, large numbers of ducks live on lakes \(A\) and \(B\). The mass, in kg, of a duck on lake \(A\) is denoted by \(x\) and the mass, in kg, of a duck on lake \(B\) is denoted by \(y\). A random sample of 8 ducks is taken from lake \(A\) and a random sample of 10 ducks is taken from lake \(B\). Their masses are summarised as follows.
\(\Sigma x = 10.56\) \(\quad\) \(\Sigma x^2 = 14.1775\) \(\quad\) \(\Sigma y = 12.39\) \(\quad\) \(\Sigma y^2 = 15.894\)
A scientist claims that ducks on lake \(A\) are heavier on average than ducks on lake \(B\).
\begin{enumerate}[label=(\roman*)]
\item Test, at the 10% significance level, whether the scientist's claim is justified. You should assume that both distributions are normal and that their variances are equal.
[9]
\item A second random sample of 8 ducks is taken from lake \(A\) and their masses are summarised as
\(\Sigma x = 10.24\) \(\quad\) and \(\quad\) \(\Sigma(x - \bar{x})^2 = 0.294\),
where \(\bar{x}\) is the sample mean. The scientist now claims that the population mean mass of ducks on lake \(A\) is greater than \(p\) kg. A test of this claim is carried out at the 10% significance level, using only this second sample from lake \(A\). This test supports the scientist's claim.
Find the greatest possible value of \(p\).
[5]
\end{enumerate]
Answer only one of the following two alternatives.
**EITHER**
One end of a light elastic spring, of natural length 0.8 m and modulus of elasticity 40 N, is attached to a fixed point $O$. The spring hangs vertically, at rest, with particles of masses 2 kg and $M$ kg attached to its free end. The $M$ kg particle becomes detached from the spring, and as a result the 2 kg particle begins to move upwards.
\begin{enumerate}[label=(\roman*)]
\item Show that the 2 kg particle performs simple harmonic motion about its equilibrium position with period $\frac{2\pi}{5}$ s. State the distance below $O$ of the centre of the oscillations.
[7]
\item The speed of the 2 kg particle is 0.4 m s$^{-1}$ when its displacement from the centre of oscillation is 0.06 m.
Find the amplitude of the motion.
[3]
\item Deduce the value of $M$.
[4]
\end{enumerate]
**OR**
In a particular country, large numbers of ducks live on lakes $A$ and $B$. The mass, in kg, of a duck on lake $A$ is denoted by $x$ and the mass, in kg, of a duck on lake $B$ is denoted by $y$. A random sample of 8 ducks is taken from lake $A$ and a random sample of 10 ducks is taken from lake $B$. Their masses are summarised as follows.
$\Sigma x = 10.56$ $\quad$ $\Sigma x^2 = 14.1775$ $\quad$ $\Sigma y = 12.39$ $\quad$ $\Sigma y^2 = 15.894$
A scientist claims that ducks on lake $A$ are heavier on average than ducks on lake $B$.
\begin{enumerate}[label=(\roman*)]
\item Test, at the 10% significance level, whether the scientist's claim is justified. You should assume that both distributions are normal and that their variances are equal.
[9]
\item A second random sample of 8 ducks is taken from lake $A$ and their masses are summarised as
$\Sigma x = 10.24$ $\quad$ and $\quad$ $\Sigma(x - \bar{x})^2 = 0.294$,
where $\bar{x}$ is the sample mean. The scientist now claims that the population mean mass of ducks on lake $A$ is greater than $p$ kg. A test of this claim is carried out at the 10% significance level, using only this second sample from lake $A$. This test supports the scientist's claim.
Find the greatest possible value of $p$.
[5]
\end{enumerate]
\hfill \mbox{\textit{CAIE FP2 2018 Q11 [28]}}