We consider the problem of constructing confidence intervals for linear functionals β t x of a non-negative vector x given discrete indirect noisy observations y = Kx +. The intervals are defined by the range of the functional over constrained projected contours of a quadratic data misfit function. Rust and O'Leary (1994 J. Comput. Graph. Stat. 3 67) derived similar intervals but we show that some conditions are required for the intervals to have the correct coverage and provide valid intervals for the cases when the conditions are not met. The problem is reduced to two simple cases: for β > 0 a closed formula for the intervals can be used on reduced one-dimensional data without the need to solve an optimization problem. For β ≯ 0 the optimization is reduced to a two-dimensional problem with K = I but a non-diagonal covariance matrix.