# Question 3:

## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $r = \frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}} = \frac{15.1608}{\sqrt{6.90181 \times 45.304}}$ | M1 | For use of $\frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}$ |
| $= 0.8573\ldots$ awrt 0.857 | A1 | awrt 0.857 |

## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $H_0: \rho = 0$, $H_1: \rho > 0$ | B1 | Both hypotheses correct. Must be in terms of $\rho$. Must be attached to $H_0$ and $H_1$. Do not allow hypotheses in words on their own. |
| Critical value $5\% = 0.5494$ | B1 | Critical value of 0.5494 |
| Significant evidence to suggest positive correlation between MR and BMI | B1 | Correct conclusion rejecting $H_0$ which must mention positive correlation, MR and BMI. Must be consistent with their CV and their $r$, with $|\text{their CV}| < 1$ and $|\text{their } r| < 1$ |

## Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| MR and BMI measurements are normally (or bivariate normal) distributed | B1 | Correct assumption referring to MR and BMI needing to be normally distributed |

## Part (d):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Ranks for MR: 9 10 6 7 8 4 5 1 2 3 | B1 | Attempt to rank MR (at least four correct). Allow reverse ranks for MR: 2 1 5 4 3 7 6 10 9 8 |
| $\sum d^2 = 1+9+9+1+4+1+16+9+9+1 = 60$ | M1 | For finding the difference between each of the ranks and evaluating $\sum d^2$. Implied by $\sum d^2 = 60$ or for reverse ranks $\sum d^2 = 270$ |
| $r_s = 1 - \frac{6(60)}{10(99)}$ | M1 | Using $1 - \frac{6\sum d^2}{10(99)}$ with their $\sum d^2$ |
| $= 0.6363$ awrt $(\pm)0.636$ | A1 | awrt $(\pm)0.636$ |

## Part (e):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $[H_0: \rho = 0, \ H_1: \rho \neq 0]$ | — | |
| Critical value 0.6485 | B1 | Critical value of 0.6485 (or $-0.6485$ if $r_s < 0$) |
| There is insufficient evidence of a correlation between MR and DPA | B1 | Correct conclusion not rejecting $H_0$, must mention MR and DPA. Must be consistent with their CV and their $r_s$, with $|\text{their CV}| < 1$ and $|\text{their } r_s| < 1$ |

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