| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2017 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of Pearson’s product-moment correlation coefficient |
| Type | One-tailed test for positive correlation |
| Difficulty | Standard +0.8 This is a standard Further Maths hypothesis testing question requiring calculation of PMCC from summary statistics, conducting a one-tailed test with critical values, and finding minimum sample size for significance. While it involves multiple parts and careful arithmetic, the techniques are routine for FM students with no novel problem-solving required—slightly above average due to the FM context and part (iii) requiring reverse lookup of critical values. |
| Spec | 5.05c Hypothesis test: normal distribution for population mean5.08a Pearson correlation: calculate pmcc5.08d Hypothesis test: Pearson correlation |
| Land area \(( x )\) | 1.0 | 4.5 | 2.4 | 1.6 | 3.8 | 8.6 | 7.5 | 6.5 |
| Population \(( y )\) | 0.8 | 8.4 | 4.2 | 1.6 | 2.2 | 10.2 | 4.2 | 5.2 |
| Answer | Marks | Guidance |
|---|---|---|
| \(r = S_{xy}/\sqrt{(S_{xx}S_{yy})}\) with e.g. \(S_{xy} = 212.62 - 35.9 \times 36.8/8 = 47.48\) (or \(5.935\)); \(S_{xx} = 216.47 - 35.9^2/8 = 55.37\) (or \(6.921\)); \(S_{yy} = 244.96 - 36.8^2/8 = 75.68\) (or \(9.46\)) (all to 3 s.f.) | M1 A1 | Find correlation coefficient \(r\) (Insufficient working loses first A1) |
| \(r = 0.733\) | *A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0: \rho = 0\), \(H_1: \rho > 0\) | B1 | State both hypotheses (B0 for \(r\ldots\)) |
| EITHER: \(r_{8,\,1\%} = 0.789\) | (*B1) | State or use correct tabular one-tail \(r\)-value |
| Accept \(H_0\) if \( | r | <\) tab. \(r\)-value (AEF) |
| OR: \(t_r = r\sqrt{(n-2)/(1-r^2)} = 2.64\), \(t_{6,0.99} = 3.143\) | (*B1) | |
| Accept \(H_0\) if \( | t_r | <\) tab. \(t\)-value (AEF) |
| No positive correlation (AEF) | DA1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(r_{14,\,1\%} = 0.661\), \(r_{15,\,1\%} = 0.641\) so \(n_{min} = 15\) | M1 A1 | Find \(n_{min}\) from relevant two-tail tabular value[s]. SC: Award B1 for stating 15 without any justification |
## Question 9(i):
$r = S_{xy}/\sqrt{(S_{xx}S_{yy})}$ with e.g. $S_{xy} = 212.62 - 35.9 \times 36.8/8 = 47.48$ (or $5.935$); $S_{xx} = 216.47 - 35.9^2/8 = 55.37$ (or $6.921$); $S_{yy} = 244.96 - 36.8^2/8 = 75.68$ (or $9.46$) (all to 3 s.f.) | M1 A1 | Find correlation coefficient $r$ (Insufficient working loses first **A1**)
$r = 0.733$ | *A1 |
**Total: 3**
## Question 9(ii):
$H_0: \rho = 0$, $H_1: \rho > 0$ | B1 | State both hypotheses (**B0** for $r\ldots$)
EITHER: $r_{8,\,1\%} = 0.789$ | (*B1) | State or use correct tabular one-tail $r$-value
Accept $H_0$ if $|r| <$ tab. $r$-value (AEF) | M1 | State or imply valid method for conclusion
OR: $t_r = r\sqrt{(n-2)/(1-r^2)} = 2.64$, $t_{6,0.99} = 3.143$ | (*B1) |
Accept $H_0$ if $|t_r| <$ tab. $t$-value (AEF) | M1 |
No positive correlation (AEF) | DA1 |
**Total: 4**
## Question 9(iii):
$r_{14,\,1\%} = 0.661$, $r_{15,\,1\%} = 0.641$ so $n_{min} = 15$ | M1 A1 | Find $n_{min}$ from relevant two-tail tabular value[s]. SC: Award **B1** for stating 15 without any justification
**Total: 2**
---9 The land areas $x$ (in suitable units) and populations $y$ (in millions) for a sample of 8 randomly chosen cities are given in the following table.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | c | c | }
\hline
Land area $( x )$ & 1.0 & 4.5 & 2.4 & 1.6 & 3.8 & 8.6 & 7.5 & 6.5 \\
\hline
Population $( y )$ & 0.8 & 8.4 & 4.2 & 1.6 & 2.2 & 10.2 & 4.2 & 5.2 \\
\hline
\end{tabular}
\end{center}
$$\left[ \Sigma x = 35.9 , \Sigma x ^ { 2 } = 216.47 , \Sigma y = 36.8 , \Sigma y ^ { 2 } = 244.96 , \Sigma x y = 212.62 . \right]$$
(i) Find, showing all necessary working, the value of the product moment correlation coefficient for this sample.\\
(ii) Using a $1 \%$ significance level, test whether there is positive correlation between land area and population of cities.\\
The land areas and populations for another randomly chosen sample of cities, this time of size $n$, give a product moment correlation coefficient of 0.651 . Using a test at the $1 \%$ significance level, there is evidence of non-zero correlation between the variables.\\
(iii) Find the least possible value of $n$, justifying your answer.\\
\hfill \mbox{\textit{CAIE FP2 2017 Q9 [9]}}