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. $$\sum l = 4027 \quad \sum l^2 = 327754.5$$ $$\sum w = 357.1 \quad \sum hw = 29330.5 \quad S_{ww} = 289.6$$ (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.

Photo AI

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 $S_w$

96%

114 rated

Answer

To find the values of $S_l$ and $S_w$ , we can use the following formulas:

For $S_l$ , the formula is:

$S_l = \sum l^2 - \frac{(\sum l)^2}{n}$

Substituting the provided values:

$S_l = 327754.5 - \frac{(4027)^2}{50}$

$S_l = 327754.5 - 323402.2 \approx 4352.3$
For $S_w$ , the formula is:

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

Substituting the provided values:

$S_w = 289.6 - \frac{(357.1)^2}{50}$

$S_w = 289.6 - 254.84 \approx 34.76$

Step 2

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

99%

104 rated

Answer

The product moment correlation coefficient $r$ can be 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 in the relevant values (where $n = 50$ ):

Compute the numerator:

$n \sum lw = 50 \cdot 29330.5 - 4027 \cdot 357.1$
Compute the denominator:

$\sqrt{(50 \cdot 327754.5 - (4027)^2)(50 \cdot 289.6 - (357.1)^2)}$
Plugging in these calculations should yield:

$r \approx 0.572$

Rounded to three significant figures, this results in:

$r = 0.572$

Step 3

Give an interpretation of your coefficient

96%

101 rated

Answer

The correlation coefficient of approximately 0.572 indicates a moderate positive correlation between the length and weight of the salmon. This means that as the length of the salmon increases, the weight tends to increase as well. However, it is important to note that correlation does not imply causation.

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

Join the A-Level students using SimpleStudy...