## (i)

**Answer:** Graph with axes suitably labelled with some indication of linear scale provided. Points plotted correctly (8 points).

**Marks:** G1, G2, 1.0 | G1 | G0

**Guidance:** G1 For axes suitably labelled with some indication of linear scale provided. G2 for points plotted correctly. G1 if 8 points plotted correctly. G0 if two or more incorrectly plotted/omitted points. Special Case SC1 for points visibly correct on axes where no indication of scale has been provided. Allow axes reversed.

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## (ii)

**Answer:** 
$$\sum d^2 = 94$$

$$r_s = 1 - \frac{6\sum d^2}{n(n^2-1)} = 1 - \frac{6 \times 94}{9 \times 80} = 1 - 0.783 = 0.217$$
(to 3 s.f.) [allow 0.22 to 2 s.f.]

**Marks:** M1 | M1 | A1 | A1

**Guidance:** 
- M1 For ranking (allow all ranks reversed for either or both categories)
- NB No ranking or re-allocation of pairs scores 0/5
- M1 For $d^2$
- A1 For $\sum d^2$
- M1 for method for $r_s$, used
- A1 f.t. for $|r_s| < 1$ Allow 13/60 or $r_s = 1 - 1.217 = -0.217$ with reversed ranks

Push-up times ranked from highest (1st) to Lowest (9th) gives $\sum d^2 = 146$ which leads to -0.217. Allow both A marks.

| | A | B | C | D | E | F | G | H | I |
|---|---|---|---|---|---|---|---|---|---|
| 100m | 7 | 2 | 1 | 4 | 9 | 6 | 3 | 8 | 5 |
| Push up | 7 | 1 | 9 | 5 | 2 | 8 | 4 | 3 | 6 |
| d | 0 | 1 | -8 | -1 | 7 | -2 | -1 | 5 | -1 |
| $d^2$ | 0 | 1 | 64 | 1 | 49 | 4 | 1 | 25 | 1 |

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## (iii)

**Answer:**
$$H_0: \text{no association between 100m time and number of push-ups achieved in the population of long distance runners}$$

$$H_1: \text{some association between 100m time and number of push-ups achieved in the population of long distance runners}$$

Two tail test critical value at 5% level is 0.7000

Since $0.217 < 0.7000$, there is insufficient evidence to reject $H_0$

**Marks:** B1 | B1 | SC1 | B1 | M1 | A1

**Guidance:**
- B1 for $H_0$ in context (not x & y)
- B1 for $H_1$ in context (not x & y)
- SC1 for both correct but no context provided
- B1 for population SOI. NB $H_0$ $H_1$ not ito $\rho$. Do not condone the use of the word 'correlation' in place of 'association'. Population' should be mentioned to award B1, unless clear, unambiguous alternative wording is used.
- B1 for ±0.7000 (or ±0.6000 only if $H_1$ indicates a 1-tailed test is intended)
- M1 for sensible comparison with c.v leading to a conclusion seen, provided $|r_s| < 1$
- NOTE The comparison can be in the form of a diagram as long as it is clear and unambiguous.

Sensible comparison: e.g. − 0.217 > − 0.7000 is 'sensible' whereas − 0.217 < 0.7000 is 'not sensible'. Allow -0.7000 < 0.217 < 0.7000
Reversed inequality sign e.g. 0.217 > 0.7000 etc gets max M1 A0.

Also, if the c.v. comes from the p.m.e.c. table (0.6664 for 2-tailed test and 0.5822 for a 1-tailed test) award max M1 A0.

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# Question 1(iv)

**Answer:** It is appropriate to carry out a hypothesis test based on the product moment correlation coefficient when the underlying population has a **bivariate Normal distribution**.

The scatter diagram does not appear to be roughly elliptical

so the Spearman coefficient is more appropriate

**Marks:** E1 | E1 | E1dep

**Guidance:**
- E1 Do not accept 'both Normally distributed'
- E1 Allow reasonable alternatives e.g. in this case, one variable is discrete so pmcc invalid.
- E1dep E1 dependent on previous E1

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