| Town | A | B | C | D | E | F | G | H | I | J |
|------|---|---|---|---|---|---|---|---|---|---|
| Pop | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| Empl | 2 | 1 | 3 | 5 | 4 | 6 | 10 | 8 | 9 | 7 |
| $\|d\|$ | 1 | 1 | 0 | 1 | 1 | 0 | 3 | 0 | 0 | 3 |
| $d^2$ | 1 | 1 | 0 | 1 | 1 | 0 | 9 | 0 | 0 | 9 |

| M1 |

$\sum d^2 = 22$ | M1A1 |

$r_s = 1 - \frac{6 \times 22}{10 \times 99} = \frac{143}{165} = 0.866$ | dM1 |

awrt **0.867** | A1 | (5)

**(b)** $H_0: \rho = 0$; $H_1: \rho > 0$ | B1 B1 |

CV = 0.6485 | B1 |

in critical region / significant/ reject $H_0$ | M1 |

evidence of positive correlation between population and no. of employees | A1 | (4)

**(c)** CV = 0.6319 | B1 |

[not in critical region / not significant/ do not reject $H_0$] | B1 |
No evidence of positive correlation | | (2)

**(d)** No evidence to suggest that as pop' increased the no. of employees increased linearly. Villages ranked highly for pop' were also ranked highly for the no. of employees. | B1 B1 | (2)

**ALT** Alternate for part (d) if different conclusions in part (b) and part (c): Data probably not (bivariate) normal therefore Spearman's coefficient is more suitable than the product moment correlation coefficient. | | **[Total 13]**

**Notes:**

**(a)** 1st M1 for an attempt to rank no of employees against the populations
2nd M1 for attempting $\sum d^2$ (must be using ranks) ft their ranks
1st A1 for 22
3rd dM1 dep on 1st M1 for use of the correct formula with their $\sum d^2$. If ans. is not correct an expr' is required.

**(b)** 1st B1 for both hypotheses in terms of $\rho$, $H_1$ must be one tail and compatible with their ranking
M1 for a correct statement relating their $r_s$ ($|r_s| < 1$) with their cv but cv must be such that $|cv| < 1$. Use of "association" is A0
A1 for a correct contextualised comment that is rejecting $H_0$. Must mention "population" and "no. of employees" and "positive correlation". Follow through their $r_s$ and their cv (provided it is $|cv| < 1$). Use of "association" is A0

**(c)** 1st B1 for 0.6319 2nd B1 does not require context just no positive correlation mentioned

**(d)** 1st B1 for a comment relating to pmcc
(i) no linear relationship or (ii) pmcc requires (joint) normal distribution
2nd B1 for a second comment relating to Spearman's
(i) there is a (non-linear) relationship between ranks.. or (ii) data not (joint) normal so Spearman's is better

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