| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2017 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear regression |
| Type | Calculate y on x from raw data table |
| Difficulty | Standard +0.3 This is a straightforward three-part statistics question requiring standard calculations: regression line equation, correlation coefficient, and hypothesis test. All procedures are routine textbook exercises with clear formulas. The arithmetic is manageable with only 5 data points. While it requires careful working and knowledge of multiple techniques, there's no conceptual difficulty or novel insight needed—just systematic application of learned methods. |
| Spec | 5.08a Pearson correlation: calculate pmcc5.08d Hypothesis test: Pearson correlation5.09c Calculate regression line |
| \(x\) | 1 | 2 | 4 | 5 | 8 |
| \(y\) | 7 | 5 | 8 | 6 | 4 |
| Answer | Marks |
|---|---|
| 10(i) | Σ x = 20, Σ y = 30, Σ xy = 111, Σ x2 = 110, Σ y2 = 190 |
| Answer | Marks | Guidance |
|---|---|---|
| xy xx | M1 A1 | Find reqd. values |
| (y – 6) = b (x – 4), y = – 0⋅3x + 7⋅2 | M1 A1 | Find gradient b in y –y = b (x –x) |
| Answer | Marks |
|---|---|
| Total: | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| 10(ii) | r = S / √(S S ) = – 9 / √(30 × 10) | |
| xy xx yy | M1 A1 | Find correlation coefficient r |
| = – 0⋅520 | *A1 | |
| Total: | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| 10(iii) | H : ρ = 0, H : ρ ≠ 0 | |
| 0 1 | B1 | State both hypotheses (B0 for r …) |
| Answer | Marks | Guidance |
|---|---|---|
| 5,10% | *B1 | State or use correct tabular two-tail r-value |
| Accept H if | r | < tab. value (AEF) |
| 0 | M1 | State or imply valid method for conclusion |
| No [non-zero] correlation (AEF) | DA1 | Correct conclusion (dep *A1, *B1) |
| Total: | 4 | |
| Question | Answer | Marks |
Question 10:
--- 10(i) ---
10(i) | Σ x = 20, Σ y = 30, Σ xy = 111, Σ x2 = 110, Σ y2 = 190
S = 111 – 20 × 30/5 = – 9 or – 1⋅8
xy
S = 110 – 202/5 = 30 or 6
xx
[S = 190 – 302/5 = 10 or 2 ]
yy
b = S / S = – 9/30 = – 3/10 or – 0⋅3
xy xx | M1 A1 | Find reqd. values
(y – 6) = b (x – 4), y = – 0⋅3x + 7⋅2 | M1 A1 | Find gradient b in y –y = b (x –x)
and hence eqn. of regression line (may be implied
by writing y = a + bx and finding a, b)
Total: | 4
--- 10(ii) ---
10(ii) | r = S / √(S S ) = – 9 / √(30 × 10)
xy xx yy | M1 A1 | Find correlation coefficient r
= – 0⋅520 | *A1
Total: | 3
--- 10(iii) ---
10(iii) | H : ρ = 0, H : ρ ≠ 0
0 1 | B1 | State both hypotheses (B0 for r …)
r = 0⋅805
5,10% | *B1 | State or use correct tabular two-tail r-value
Accept H if |r| < tab. value (AEF)
0 | M1 | State or imply valid method for conclusion
No [non-zero] correlation (AEF) | DA1 | Correct conclusion (dep *A1, *B1)
Total: | 4
Question | Answer | Marks | GuidanceA random sample of 5 pairs of values $(x, y)$ is given in the following table.
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$x$ & 1 & 2 & 4 & 5 & 8 \\
\hline
$y$ & 7 & 5 & 8 & 6 & 4 \\
\hline
\end{tabular}
\begin{enumerate}[label=(\roman*)]
\item Find, showing all necessary working, the equation of the regression line of $y$ on $x$. [4]
\item Find, showing all necessary working, the value of the product moment correlation coefficient for this sample. [3]
\item Test, at the 10% significance level, whether there is evidence of non-zero correlation between the variables. [4]
\end{enumerate}
\hfill \mbox{\textit{CAIE FP2 2017 Q10 [11]}}