Crickets make a noise - Edexcel - A-Level Maths Statistics - Question 4 - 2008 - Paper 2

The formula for the product moment correlation coefficient, $r$, is given by:

$r = \frac{n \sum (tv) - \sum t \cdot \sum v}{\sqrt{(n \sum t^2 - (\sum t)^2)(n \sum v^2 - (\sum v)^2)}}$

Substituting the values:

$r = \frac{15 \cdot 18 267.169 - 401.3 \cdot 25.08}{\sqrt{(15 \cdot 10 922.81 - (401.3)^2)(15 \cdot 42.3356 - (25.08)^2)}}$

Calculating gives: $r \approx 0.808$

$t$ is the explanatory variable because we can control temperature, but not the frequency of noise or any equivalent comment.

The relationship is expected to be linear; as temperature increases, we expect the pitch, $v$, to change systematically.

To find the coefficients $a$ and $b$:

1. Calculate $b$ using the formula:

   $b = \frac{n\sum(tv) - \sum t \cdot \sum v}{n\sum t^2 - (\sum t)^2}$

   Which gives: $b \approx 0.669$

2. Then find $a$:

   $a = \bar{v} - b \bar{t}$

   Using our averages we find: $a \approx 1.4$

Plugging $t = 19$ into the regression equation:

$v = a + bt$

$v \approx 1.4 + 0.669 \cdot 19$

This results in:

$v \approx 1.649$