| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2010 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of Pearson’s product-moment correlation coefficient |
| Type | Calculate PMCC from summary statistics |
| Difficulty | Moderate -0.3 Part (a) tests basic interpretation of correlation coefficients (routine recall), while part (b) involves standard PMCC calculation from given summary statistics and a straightforward one-tailed hypothesis test. The formulas are provided in the specification and the calculation is mechanical with no conceptual challenges, making this slightly easier than average despite being Further Maths content. |
| Spec | 5.08a Pearson correlation: calculate pmcc5.08d Hypothesis test: Pearson correlation |
| Altitude, \(x\) | 10 | 30 | 50 | 70 | 90 | 100 | 350 |
| Number of deaths, \(y\) | 102 | 65 | 34 | 27 | 22 | 17 | 8 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| (i) State valid comment on each diagram (A.E.F.): Straight line with negative gradient | B1 | |
| (ii) (Almost) no linear relation | B1 | |
| (iii) Close to straight line with positive gradient | B1 | Subtotal: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Calculate correlation coefficient: \(r = (13040 - 700\times 275/7)/\sqrt{\{(149000 - 700^2/7)(17351 - 275^2/7)\}}\) | M1 A1 | |
| \(= -14460/\sqrt{(79000\times 6547.4285)} = -0.636\) | A1 | Subtotal: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State hypotheses: \(H_0: \rho = 0\), \(H_1: \rho < 0\) | B1 | |
| Compare with consistent tabular value: \(r_{7,\,5\%} = 0.669\) | B1 | |
| Valid method for reaching conclusion: Reject \(H_0\) if \(r < -\)tabular value | M1 | |
| Correct conclusion (needs correct values): No negative correlation (A.E.F.) | A1 | Total: 10 |
## Question 9:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| **(i)** State valid comment on each diagram (A.E.F.): Straight line with negative gradient | B1 | |
| **(ii)** (Almost) no linear relation | B1 | |
| **(iii)** Close to straight line with positive gradient | B1 | **Subtotal: 3** |
### Part (b)(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Calculate correlation coefficient: $r = (13040 - 700\times 275/7)/\sqrt{\{(149000 - 700^2/7)(17351 - 275^2/7)\}}$ | M1 A1 | |
| $= -14460/\sqrt{(79000\times 6547.4285)} = -0.636$ | A1 | **Subtotal: 3** |
### Part (b)(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| State hypotheses: $H_0: \rho = 0$, $H_1: \rho < 0$ | B1 | |
| Compare with consistent tabular value: $r_{7,\,5\%} = 0.669$ | B1 | |
| Valid method for reaching conclusion: Reject $H_0$ if $r < -$tabular value | M1 | |
| Correct conclusion (needs correct values): No negative correlation (A.E.F.) | A1 | **Total: 10** |
---9
\begin{enumerate}[label=(\alph*)]
\item The following are values of the product moment correlation coefficient between the $x$ and $y$ values of three different large samples of bivariate data. State what each indicates about the appearance of a scatter diagram illustrating the data.
\begin{enumerate}[label=(\roman*)]
\item - 1 ,
\item 0.02 ,
\item 0.92 .
\end{enumerate}\item In 1852 Dr William Farr published data on deaths due to cholera during an outbreak of the disease in London. The table shows the altitude (in feet, above the level of the river Thames) at which people lived and the corresponding number of deaths from cholera per 10000 people.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | c | }
\hline
Altitude, $x$ & 10 & 30 & 50 & 70 & 90 & 100 & 350 \\
\hline
Number of deaths, $y$ & 102 & 65 & 34 & 27 & 22 & 17 & 8 \\
\hline
\end{tabular}
\end{center}
$$\left[ \Sigma x = 700 , \Sigma x ^ { 2 } = 149000 , \Sigma y = 275 , \Sigma y ^ { 2 } = 17351 , \Sigma x y = 13040 . \right]$$
\begin{enumerate}[label=(\roman*)]
\item Calculate the product moment correlation coefficient.
\item Test, at the $5 \%$ significance level, whether there is evidence of negative correlation.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{CAIE FP2 2010 Q9 [10]}}