= Solution
The expected activity is $E_t=\alpha+\beta(e^{-t}-c)$. Under normal errors, least squares is maximum likelihood, and invariance of maximum-likelihood estimation gives
$$
\widehat E_t=\widehat\alpha+\widehat\beta(e^{-t}-c).
$$
The covariance term vanishes by part (ii), so
$$
\boxed{\operatorname{Var}(\widehat E_t)=\sigma^2\left[
\frac1n+\frac{(e^{-t}-c)^2}{\sum_i(e^{-t_i}-c)^2}\right].}
$$
This is uncertainty in the estimated mean activity; the <variance> of a future individual observation would include its additional measurement-error <variance>.
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