A process that is in control has the following characteristics.

To determine the portion of the product expected to meet specification, we use the process capability. Given:

Mean (X̄) = 101

Range (R) = 4

Sample size (n) = 5

Specification = 100 ± 3 (i.e., 97 to 103)

First, calculate the standard deviation (σ) using the average range (R̄) and a factor (d2) from standard tables for n = 5, where d2 = 2.326:

σ = R̄ / d2 = 4 / 2.326 ≈ 1.72

Next, convert the specification limits to Z-scores:

Lower Z = (97 - 101) / 1.72 ≈ -2.33 Upper Z = (103 - 101) / 1.72 ≈ 1.16

Using standard normal distribution tables, the area between these Z-scores corresponds to the portion within specification:

Portion within specification ≈ P(-2.33 < Z < 1.16) ≈ 0.954 or 95.4%

References:

Montgomery, D.C. (2009). Introduction to Statistical Quality Control. Wiley.

ASQ. (n.d.). Process Capability Indices. Retrieved from https://asq.org/quality-resources/process-capability