| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2013 |
| Session | November |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear regression |
| Type | Relate two regression lines |
| Difficulty | Standard +0.3 This is a straightforward application of standard regression formulas: recognizing that r² equals the product of regression slopes, using the t-test for correlation with given critical values, finding means from the intersection of regression lines, and making a prediction. All steps are routine recall of A-level statistics techniques with no novel problem-solving required, though it does test understanding of the relationship between the two regression lines. |
| 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 | Marks | Guidance |
| \(r^2 = 4.21 \times 0.043 = 0.181\) or \(0.425^2\) | M1 A1 | Find sample coefficient using \(r^2 = b_1 b_2\) |
| \(r = 0.425\) | *A1 | |
| Subtotal: 3 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(H_0: \rho = 0\), \(H_1: \rho \neq 0\) | B1 | State both hypotheses |
| \(r_{10,\,5\%} = 0.549\) | *B1 | State or use correct tabular one-tail \(r\) value |
| Accept \(H_0\) if \( | r | <\) tabular value |
| There is no non-zero correlation | A1 | Correct conclusion (AEF, dep *A1, *B1) |
| Subtotal: 4 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\bar{x} = 7.72\) and \(\bar{y} = 31.6\) | M1 A1 | Solve regression eqns for mean values |
| Subtotal: 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(x = 6.46\) or \(0.751\) | B1 | Estimate \(x\) from either eqn. |
| Not reliable because e.g. value of \(r\) is small, or range of data is unknown, or two estimates of \(x\) very different | B1 | State valid comment on reliability |
| Subtotal: 2 marks | ||
| Total: 11 marks |
## Question 9(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $r^2 = 4.21 \times 0.043 = 0.181$ or $0.425^2$ | M1 A1 | Find sample coefficient using $r^2 = b_1 b_2$ |
| $r = 0.425$ | *A1 | |
| **Subtotal: 3 marks** | | |
## Question 9(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $H_0: \rho = 0$, $H_1: \rho \neq 0$ | B1 | State both hypotheses |
| $r_{10,\,5\%} = 0.549$ | *B1 | State or use correct tabular one-tail $r$ value |
| Accept $H_0$ if $|r| <$ tabular value | M1 | Valid method for reaching conclusion |
| There is no non-zero correlation | A1 | Correct conclusion (AEF, dep *A1, *B1) |
| **Subtotal: 4 marks** | | |
## Question 9(iii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\bar{x} = 7.72$ and $\bar{y} = 31.6$ | M1 A1 | Solve regression eqns for mean values |
| **Subtotal: 2 marks** | | |
## Question 9(iv):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $x = 6.46$ or $0.751$ | B1 | Estimate $x$ from either eqn. |
| Not reliable because e.g. value of $r$ is small, or range of data is unknown, or two estimates of $x$ very different | B1 | State valid comment on reliability |
| **Subtotal: 2 marks** | | |
| **Total: 11 marks** | | |
---9 For a random sample of 10 observations of pairs of values $( x , y )$, the equations of the regression lines of $y$ on $x$ and of $x$ on $y$ are
$$y = 4.21 x - 0.862 \quad \text { and } \quad x = 0.043 y + 6.36$$
respectively.\\
(i) Find the value of the product moment correlation coefficient for the sample.\\
(ii) Test, at the $10 \%$ significance level, whether there is evidence of non-zero correlation between the variables.\\
(iii) Find the mean values of $x$ and $y$ for this sample.\\
(iv) Estimate the value of $x$ when $y = 2.3$ and comment on the reliability of your answer.
\hfill \mbox{\textit{CAIE FP2 2013 Q9 [11]}}