The situation considered is that in which measurement of the characteristic of interest is not exact but subject to appreciable error. The error is assumed to be unbiased and independent of the actual value of the characteristic measured. The population and error variances,σ 2 andσ e 2 , are assumed to be such thatσ/σ e has a known lower limit which is greater than zero. The probability distributions involved are assumed to be normal while the actual values and measurement errors each form a random sample. For suitable specified acceptable and unacceptable fractions defective, and forσ e assumed known and unknown, this paper presents one-sided acceptance inspection criteria which are optimum in a specified sense, and which have the property that the producer’s and consumer’s risks have specified upper bounds.