Answer 1:
Here I added a linear regression to the data
you plotted, giving an equation in the form
y = mx + b
We invert this equation (x = [y  b] / m)
to find an equation for the year in terms
of the egg density. We must be careful to
restrict our analysis only to eggs which have
~0.3 > density > ~ 0.175 to fall within our
interpolating limits.
To find an error associated with this
interpolated year, we calculate the standard
deviation in the residuals of the plot. To do
this, we find estimated values for the age each
known egg, and subtract the true age. Here we
find that our curve is off by about 40 years
(mean) for each egg.
We're looking for a confidence interval, such
that we're 95% sure that our egg has an age
within a certain range. Invoking the assumption
that our errors are normally distributed
(wikipedia 'normal distribution'), which seems
to be appropriate in this case, a 95% confidence
interval encompasses values up to 2 standard
deviations away from the calculated value. In
this case, +/ 41.2 years.
Therefore, given an egg of density 0.250, you
could be 95% sure it came from sometime between
1720 and 1800.
Best,
