A random sample of 50 salmon was caught by a scientist - Edexcel - A-Level Maths Statistics - Question 1 - 2011 - Paper 1

The product moment correlation coefficient (r) is calculated using the formula:

$r = \frac{n(\sum lw) - (\sum l)(\sum w)}{\sqrt{(n \sum l^2 - (\sum l)^2)(n \sum w^2 - (\sum w)^2)}}$

Substituting the values:

• (n = 50)
• (\sum lw = 29330.5)
• (\sum l = 4027)
• (\sum w = 357.1)

Calculating:

1. Numerator:

   $50 \cdot 29330.5 - (4027)(357.1) = 1466525 - 1431497.7 = 35027.3$

2. Denominator:

   For (l:)
   $50 \cdot 327754.5 - 4027^2 = 16387725 - 16217629 = 17096$

   For (w:)
   $50 \cdot 289.6 - 357.1^2 = 14480 - 12712.41 = 768.5$

3. Therefore,

   $r = \frac{35027.3}{\sqrt{17096 \cdot 768.5}}\approx 0.572$

Thus, rounding to three significant figures, (r \approx 0.572).

The coefficient (r \approx 0.572) indicates a moderate positive correlation between the length and weight of the salmon. This suggests that as the length of the salmon increases, the weight tends to increase as well. Such a correlation implies that length can be a useful predictor of weight in this species.