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 is appropriate because the scatter plot shows a linear pattern, indicating a potential linear relationship between the pollutant level $p$ and the thinning of the shells $t$. The points appear to cluster around a straight line, suggesting that as the pollutant levels increase, the thinning of the shell tends to increase as well.

To calculate $S_p$, we use the formula:

$S_p = \sum p = 3 + 8 + 30 + 25 + 15 + 12 = 93$

To calculate $S_{pp}$ using the square of sum of $p$,

$S_{pp} = \sum p^2 = 3^2 + 8^2 + 30^2 + 25^2 + 15^2 + 12^2 = 1967$

To find the regression equation, we first calculate the slope $b$:

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

where $S_{pt}$ is the sum of the product of deviations of $p$ and $t$. Then, calculate $a$ using:

$a = \bar{t} - b\bar{p}$.

Thus, the regression line would be expressed as $t = a + bp$.

Calculate the mean values: $\bar{p}$ and $\bar{t}$.

Then, plot the point $(\bar{p}, \bar{t})$ on the scatter diagram. Finally, using the slope $b$ determined in part (d), draw the regression line from the point $(\bar{p}, \bar{t})$ such that it reflects the relationship found in the earlier calculations.

From earlier studies, it is stated that pollutant levels above 16 are associated with the risk of chick mortality. Use the regression equation from part (d) to estimate the corresponding value of $t$, the thinning of the shell, when $p = 16$. Input this into the equation to estimate the minimum thinning.