What instrument is best used to accurately check the diameter of a hole with a tolerance of plus or minus 3000ths?

The best instrument for accurately checking the diameter of a hole with a tolerance of plus or minus 3000ths is a Vernier Caliper. This tool is designed for precise measurement, and it can measure both internal and external dimensions effectively. The Vernier scale allows the user to measure with a high degree of accuracy, typically to 0.02 mm or 0.001 inches, which is well-suited for the tolerance specified in the question.

Micrometers offer higher precision suitable for smaller dimensions, but they may not fit well when measuring diameters of larger holes due to their limited opening range. Dial indicators are generally used for measuring displacement or differences in height rather than direct diametric measurement, making them less appropriate for this task. Caliper gauges, while useful, may not provide the needed accuracy for such specific tolerances as effectively as a Vernier Caliper can.

Overall, for the specified task of measuring a diameter with a tolerance of plus or minus 3000ths, a Vernier Caliper emerges as the ideal choice due to its combination of accuracy, versatility for hole measurements, and ease of use.