| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2017 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of Pearson’s product-moment correlation coefficient |
| Type | Two-tailed test for any correlation |
| Difficulty | Standard +0.3 This is a straightforward application of standard formulas: the correlation coefficient equals the geometric mean of the regression coefficients, followed by a routine hypothesis test using the t-distribution with n-2 degrees of freedom. Both parts require only direct recall and substitution into well-known formulas, making it slightly easier than average for A-level Further Maths. |
| Spec | 5.08a Pearson correlation: calculate pmcc5.08d Hypothesis test: Pearson correlation |
| Answer | Marks | Guidance |
|---|---|---|
| 7(i) | r = √(0⋅46 × 0⋅93) = 0⋅654 | M1 A1 |
| Total: | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| 7(ii) | H : ρ = 0, H : ρ ≠ 0 | |
| 0 1 | B1 | State both hypotheses (B0 for r …) |
| Answer | Marks | Guidance |
|---|---|---|
| 12, 5% | B1 | State or use correct tabular two-tail r-value |
| Accept H if | r | > tab. value (AEF) |
| 1 | M1 | State or imply valid method for conclusion |
| Answer | Marks | Guidance |
|---|---|---|
| 0⋅654 [or 0⋅65] > 0⋅576 so there is non-zero correlation | A1 | Correct conclusion (AEF) |
| Total: | 4 |
Question 7:
--- 7(i) ---
7(i) | r = √(0⋅46 × 0⋅93) = 0⋅654 | M1 A1 | Find correlation coefficient r
Total: | 2
--- 7(ii) ---
7(ii) | H : ρ = 0, H : ρ ≠ 0
0 1 | B1 | State both hypotheses (B0 for r …)
r = 0⋅576
12, 5% | B1 | State or use correct tabular two-tail r-value
Accept H if |r| > tab. value (AEF)
1 | M1 | State or imply valid method for conclusion
(M0 if r or tab. value has magnitude > 1)
0⋅654 [or 0⋅65] > 0⋅576 so there is non-zero correlation | A1 | Correct conclusion (AEF)
Total: | 4A random sample of twelve pairs of values of $x$ and $y$ is taken from a bivariate distribution. The equations of the regression lines of $y$ on $x$ and of $x$ on $y$ are respectively
$$y = 0.46x + 1.62 \quad \text{and} \quad x = 0.93y + 8.24.$$
\begin{enumerate}[label=(\roman*)]
\item Find the value of the product moment correlation coefficient for this sample. [2]
\item Using a $5\%$ significance level, test whether there is non-zero correlation between the variables. [4]
\end{enumerate}
\hfill \mbox{\textit{CAIE FP2 2017 Q7 [6]}}