A scientist is researching whether or not birds of prey exposed to pollutants lay eggs with thinner shells - Edexcel - A-Level Maths Statistics - Question 3 - 2012 - Paper 2

A linear regression model may be appropriate because the scatter diagram reveals a roughly linear relationship between the pollutant level ($p$) and the thinning of the shell ($t$). The points appear to cluster around a straight line, which suggests that increases in $p$ are associated with increases in $t$, indicating a strong correlation.

To calculate $S_p$ and $S_{pp}$:

1. For $S_p$: Sum the pollutant levels:

   $S_p = 3 + 8 + 30 + 25 + 15 + 2 = 83$

2. For $S_{pp}$: Compute the sum of squares of pollutant levels:

   $S_{pp} = 3^2 + 8^2 + 30^2 + 25^2 + 15^2 + 2^2 = 9 + 64 + 900 + 625 + 225 + 4 = 1827$

To find the regression line of $t$ on $p$, use the formulas for the slope ($b$) and intercept ($a$). The slope is calculated as:

$b = \frac{S_{pt}}{S_{pp}}$

Substituting in the values:

$S_{pt} = \sum{(p_i - \bar{p})(t_i - \bar{t})}$

The equation of the line will be:

$t = a + bp$

Calculate the mean values $(\bar{p}, \bar{t})$. Then, plot this point on the scatter diagram. Draw the regression line that represents the relationship based on the calculated slope and intercept.

To estimate the minimum thinning that likely results in the death of a chick, find the corresponding value of $t$ when $p=16$. Use the derived regression equation and solve for $t$ when $p = 16$, leading to

$t = a + b imes 16$