The question asks for two different quantities to be calculated ??S_{tt}?? and ??S_{gt}??, recall that:
$$S_{xx} = \sum \left( x_{i} - \bar{x} \right)^{2} =\sum x_{i}^{2} - \frac{1}{N} \left ( \sum x_{i} \right )^{2}$$
$$S_{xy} = \sum \left( x_{i} - \bar{x} \right) \left( y_{i} - \bar{y} \right) = \sum x_{i} y_{i} - \frac{1}{N} \left(\sum x_{i} \right) \left( \sum y_{i} \right ) $$

In this case we are asked to find ??S_{tt}??:
$$S_{tt} = \sum \left( t_{i} - \bar{t} \right)^{2} =\sum t_{i}^{2} - \frac{1}{N} \left ( \sum t_{i} \right )^{2}$$

We can substitute in for ??N??, ??\sum t_{i}^{2}?? and ??(\sum t_{i})^{2}??:
$$S_{tt} = \sum t_{i}^{2} - \frac{1}{N} \left ( \sum t_{i} \right )^{2}$$
$$S_{tt} = 8702 - \frac{1}{10} \times (258)^{2}$$
$$S_{tt} = 8702 - \frac{1}{10} \times 66564$$
$$S_{tt} = 8702 - 6656.4$$
$$S_{tt} = 2045.6$$

We can substitute in for ??N??, ??\sum g_{i}??, ??\sum t_{i}?? and ??\sum g_{i} t_{i}??:
$$S_{gt} = \sum g_{i} t_{i} - \frac{1}{N} \left(\sum g_{i} \right) \left( \sum t_{i} \right )$$
$$S_{gt} = 1550.2 - \frac{1}{10} \times 63.6 \times 258 $$
$$S_{gt} = 1550.2 - 1640.88$$
$$S_{gt} = -90.68$$

Question 1. (b)

Next, we're asked to calculate the product moment correlation coefficient, ??r??, recall the formula:
$$ r = \frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}$$

And for this problem it can be written in terms of ??g?? and ??t?? as:
$$ r = \frac{S_{gt}}{\sqrt{S_{gg}S_{tt}}}$$

Note that we have already calculated ??S_{gt}?? and ??S_{tt}?? in part 1. (a) and ??S_{gg}?? is given in the question, so this part of the question is just substituting values in:
$$ r = \frac{-90.68}{\sqrt{7.864 \times 2045.6}}$$
$$ r = -0.714956...$$
$$ r = -0.715 \: \mathrm{to \: 3 \: s.f.}$$

Do remember to show the result to 3 significant figures, as above and as requested by the question.

Question 1. (c)

We would expect the correlation between ??v?? and ??g?? to be positive and you could give either of the below reasons for this:

Would expect more revision to result in a higher grade; or

Because a high ??v?? corresponds to a a low ??t?? and because a low ??t?? corresponds to a high ??g??, would expect a high ??v?? to correspond to a high ??g??.

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