## (a)(i)
$H_0: \mu_0 = 0$ (or $\mu = \mu_I$)

$H_1: \mu_0 \neq 0$ (or $\mu_0 \neq \mu_I$)

where $\mu_0$ is "mean for II – mean for I"

Normality of differences is required. | B1, B1, B1 | Both. Hypotheses in words only must include "population". For adequate verbal definition. Allow absence of "population" if correct notation $\mu$ is used, but do NOT allow "$\bar{X} = \bar{X}_B$" or similar unless $\bar{X}$ is clearly and explicitly stated to be a population mean.

## (a)(ii)
MUST be PAIRED COMPARISON $t$ test.

Differences are: 10.0, 26.8, 42.7, 2.4, –14.9, –2.0, 16.3, 11.5

$\bar{d} = 11.6$, $s_{n-1} = 17.707$

Test statistic is $\frac{11.6 - 0}{17.707/\sqrt{8}} = 1.852(92)$.

Refer to $t_7$.

Double-tailed 5% point is 2.365.

Not significant.

Seems there is no difference between the mean yields of the two types of plant. | B1, M1, A1, M1, A1, A1, A1, A1 | $s_n = 16.563$ but do NOT allow this here or in construction of test statistic, but FT from there. Allow $c's$ $\bar{d}$ and/or $s_{n-1}$. Allow alternative: $0 + (c's$ 2.365) $\times \frac{17.707}{\sqrt{8}}$ (= 14.806) for subsequent comparison with $\bar{d}$. (Or $\bar{d} - (c's$ 2.365) $\times \frac{17.707}{\sqrt{8}}$ (=-3.206) for comparison with 0.) c.a.o. but ft from here in any case if wrong. Use of $0 - \bar{d}$ scores M1A0, but ft. No ft from here if wrong. No ft from here if wrong. ft only c's test statistic. ft only c's test statistic. Special case: ($t_6$ and 2.306) can score 1 of these last 2 marks if either form of conclusion is given.

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# Question 3 (continued):

## (b)
| Diff | Rank of \|diff\| | –5, 4, –14, –3, 6, 1, –11, –8, –7, –9 | 4, 3, 10, 2, 5, 1, 9, 7, 6, 8 |

$W_+ = 1 + 3 + 5 = 9$ (or $W_- = 2 + 4 + 6 + 7 + 8 + 9 + 10 = 46$)

Refer to tables of Wilcoxon paired (single sample) statistic for $n = 10$.

Lower (or upper if 46 used) double-tailed 5% point is 8 (or 47 if 46 used).

Result is not significant.

No evidence to suggest the tasters differ on the whole. | M1, M1, A1, B1, M1, A1, A1, A1 | For differences. ZERO in this section if differences not used. For ranks. FT from here if ranks wrong. (Condone if not combined as fully as shown above, but require top two cells combined as a minimum.) No ft from here if wrong. i.e. a 2-tail test. No ft from here if wrong. ft only c's test statistic. ft only c's test statistic.

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