A random sample of 50 salmon was caught by a scientist. He recorded the length l cm and weight w kg of each salmon. The following summary statistics were calculated from these data. $\sum l = 4027$ $\sum l^2=327754.5$ $\sum w = 357.1$ $\sum lw=29330.5$ $S_{ww} = 289.6$ (a) Find $S_l$ and $S_w$. (b) Calculate, to 3 significant figures, the product moment correlation coefficient between $l$ and $w$. (c) Give an interpretation of your coefficient.

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

Question 1

A random sample of 50 salmon was caught by a scientist. He recorded the length l cm and weight w kg of each salmon. The following summary statistics were calculated... show full transcript

Worked Solution & Example Answer:A random sample of 50 salmon was caught by a scientist - Edexcel - A-Level Maths Statistics - Question 1 - 2011 - Paper 1

Step 1

Find $S_l$ and $S_w$

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Answer

To find the sum of squares for length and weight, we use the formulas:

For (S_l):

$S_l = \frac{\sum l^2 - \frac{(\sum l)^2}{n}}{n}$
- Calculating, we have:
$S_l = \frac{327754.5 - \frac{4027^2}{50}}{50} = \frac{327754.5 - 323464.14}{50} = 569.66$
For (S_w):

$S_w = \frac{\sum w^2 - \frac{(\sum w)^2}{n}}{n}$

Using the sum for (S_{ww}):
- Therefore, we find: $S_w = \frac{289.6}{50}$
- The calculated values are: (S_l \approx 569.66) and (S_w \approx 289.6).

Step 2

Calculate, to 3 significant figures, the product moment correlation coefficient between $l$ and $w$

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Answer

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:

Numerator:

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

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

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

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

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

Step 3

Give an interpretation of your coefficient

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Answer

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.

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

Worked Solution & Example Answer:A random sample of 50 salmon was caught by a scientist - Edexcel - A-Level Maths Statistics - Question 1 - 2011 - Paper 1

Find \(S_l\) and \(S_w\)

Calculate, to 3 significant figures, the product moment correlation coefficient between \(l\) and \(w\)

Give an interpretation of your coefficient

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