# Question 1:

## Part (ii)

| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0: \rho = 0$ | B1 | For $H_0$, $H_1$ in symbols. Hypotheses in words must refer to population. Do not allow alternative symbols unless clearly defined as the population correlation coefficient. |
| $H_1: \rho > 0$ (one-tailed test) | | |
| where $\rho$ is the population correlation coefficient | B1 | For defining $\rho$. Condone omission of "population" if correct notation $\rho$ is used, but if $\rho$ is defined as the **sample** correlation coefficient then award **B0**. Allow "$\rho$ is the pmcc". |
| For $n = 60$, 5% critical value $= 0.2144$ | B1 | For critical value |
| Since $0.665 > 0.2144$, the result is significant. | M1 | For sensible comparison leading to a conclusion provided that $|r| < 1$. Sensible comparison: e.g. $0.665 > 0.2144$ is 'sensible' whereas $0.665 > -0.2144$ is 'not sensible'. Reversed inequality e.g. $0.665 < 0.2144$ gets max M1 A0. |
| Thus we have sufficient evidence to reject $H_0$ | A1 | For reject $H_0$ o.e. FT their $r$ and critical value from 5% 1-tail column. |
| There is sufficient evidence at the 5% level to **suggest** that there is **positive** correlation between FEV1 before and after the two-week course. | E1 | For correct, **non-assertive** conclusion in context (allow '$x$ and $y$' for context). E0 if $H_0$ and $H_1$ not stated, reversed or mention a value other than zero for $\rho$ in $H_0$. |
| | **[6]** | |

## Part (iii)

| Answer | Marks | Guidance |
|--------|-------|----------|
| The underlying population must have a bivariate Normal distribution. | B1 | Condone "bivariate Normal distribution", "underlying bivariate Normal distribution", but **do not allow** "the **data** have a bivariate Normal distribution" |
| Yes, since the scatter diagram appears to have a roughly elliptical shape. | E1 | Condone 'oval' or suitable diagram |
| | **[2]** | |

## Part (iv)

| Answer | Marks | Guidance |
|--------|-------|----------|
| The significance level is the probability of rejecting the null hypothesis when in fact it is true. | E1* | For "probability of rejecting $H_0$" or "probability of a significant result". |
| | E1dep* | For "when $H_0$ is true" |
| | **[2]** | |

## Part (v)

| Answer | Marks | Guidance |
|--------|-------|----------|
| $\sum x = 43.62 + 0.45 = 44.07$ | B1 | For $\sum x$ or $\sum y$ or $\sum xy$ |
| $\sum y = 55.15 - 0.45 = 54.70$ | | |
| $\sum xy = 40.66$ | | |
| $\sum x^2 = 32.68 + 1 - 0.55^2 = 33.3775$ | B1 | For $\sum x^2$ or $\sum y^2$ (to 2 dp) |
| $\sum y^2 = 51.44 - 1 + 0.55^2 = 50.7425$ | | |
| | B1 | For all correct (ignore $n$) |
| | **[3]** | |

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