| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2008 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear regression |
| Type | Find means from regression lines |
| Difficulty | Moderate -0.3 This is a straightforward application of standard regression line properties: both lines pass through (x̄, ȳ) giving a simple simultaneous equation, and the correlation coefficient formula r = ±√(b₁b₂) is direct recall. The hypothesis test is routine with n=25. Slightly easier than average due to minimal problem-solving required, just applying memorized formulas. |
| Spec | 5.08a Pearson correlation: calculate pmcc5.08d Hypothesis test: Pearson correlation5.09a Dependent/independent variables5.09c Calculate regression line |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| (i) \(\bar{y} + 0.425\bar{x} = 1.28\) and \(\bar{x} + 0.516\bar{y} = 1.05\) | M1 | Formulate two equations for means |
| \(\bar{x} = 0.499\), \(\bar{y} = 1.068\) or \(1.07\) | M1 A1 | Solve for means; Part total: 3 |
| (ii) \(r^2 = 0.425 \times 0.516\); \(r = -0.468\) | M1; *A1 | Find correlation coefficient for sample; Part total: 2 |
| (iii) \(H_0: \rho = 0\), \(H_1: \rho \neq 0\) | B1 | State hypotheses |
| \(\rho \neq 0\) if \( | r | >\) tabular value |
| \(\rho_{25, 2.5\%} = 0.396\) | *B1 | Use of correct tabular value |
| Coefficient does differ from zero | A1 | Correct conclusion (A.E.F., dep *A1, *B1); Part total: 4 |
| Total: 9 |
## Question 8:
| Answer/Working | Mark | Guidance |
|---|---|---|
| **(i)** $\bar{y} + 0.425\bar{x} = 1.28$ and $\bar{x} + 0.516\bar{y} = 1.05$ | M1 | Formulate two equations for means |
| $\bar{x} = 0.499$, $\bar{y} = 1.068$ or $1.07$ | M1 A1 | Solve for means; **Part total: 3** |
| **(ii)** $r^2 = 0.425 \times 0.516$; $r = -0.468$ | M1; *A1 | Find correlation coefficient for sample; **Part total: 2** |
| **(iii)** $H_0: \rho = 0$, $H_1: \rho \neq 0$ | B1 | State hypotheses |
| $\rho \neq 0$ if $|r| >$ tabular value | M1 | Valid method for reaching conclusion |
| $\rho_{25, 2.5\%} = 0.396$ | *B1 | Use of correct tabular value |
| Coefficient does differ from zero | A1 | Correct conclusion (A.E.F., dep *A1, *B1); **Part total: 4** |
| | | **Total: 9** |
---8 The equations of the regression lines for a random sample of 25 pairs of data $( x , y )$ from a bivariate population are
$$\begin{array} { c c }
y \text { on } x : & y = 1.28 - 0.425 x , \\
x \text { on } y : & x = 1.05 - 0.516 y .
\end{array}$$
(i) Find the sample means, $\bar { x }$ and $\bar { y }$.\\
(ii) Find the product moment correlation coefficient for the sample.\\
(iii) Test at the $5 \%$ significance level whether the population correlation coefficient differs from zero.
\hfill \mbox{\textit{CAIE FP2 2008 Q8 [9]}}