A biologist is studying the behaviour of bees in a hive - Edexcel - A-Level Maths Statistics - Question 1 - 2016 - Paper 1

The response variable is ( w ) (the average number of wiggles) because it is the outcome that depends on the independent variable ( d ) (the distance from the hive).

The product moment correlation coefficient ( r ) can be calculated as follows:

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

Substituting the known values:

• Compute ( \sum (d \cdot w) ), you would need to calculate this from the provided distance and wiggles data.
• The resulting value will be ( r \approx 0.994 ).

The regression equation can be determined using:

$w = a + bd$

Where:

• ( a = \bar{w} - b \bar{d} )
• ( b = \frac{S_{dw}}{S_d} )

Using the calculated values, plug them into the formulas. The resulting equation will be:

$w \approx 0.722 + 0.0142d$

For a distance of 350 m, substitute ( d = 350 ) into the regression equation:

$w \approx 0.722 + 0.0142 \times 350 \approx 5.7 \text{ wiggles}$

The reliability of the estimate is uncertain because 350 m is outside the range of the data used to create the regression model (the data ranges from 50 m to 650 m). Extrapolating beyond the existing data can lead to less accurate predictions.