Sample size determination for C_p comparisons

Comparison of quality for products (supplies and goods) is extremely important for manufacturers and consumers. Based on correct comparisons, manufacturers and consumers can find better suppliers to cooperate and better merchandise to purchase, respectively. Quality is often measured and compared by process capability indices, among which C_p is very efiective, simple to apply, and particularly useful for the first round of comparison. In practice, C_p is unknown and should be estimated from observations. Let dCpi denote the maximum likelihood estimator obtained from normal process, Xi, with index value C_pi; i = 1; 2. If dCp1 > (<)d C_p2 is observed, we will conclude that C_p1 > (<) C_p2 and decide that X1 is better (worse) than X2. Given a small and positive number, , there is no need to make comparison when (1-€) C_p2 < C_p1 < (1+€) C_p2 since C_p1 is close to C_p2. It is desirable to observe dCp1 <d C_p2 with high probability when (1 + €) C_p2 > C_p1 and with low probability when (1-€) C_p2 < C_p1. Given 0 < 1, 2 < 1, based on the table constructed from P(dC_p1 >d C_p2), we demonstrate how to find the smallest sample size needed to ensure observing dCp1 d C_p2 with probability greater than 1 €1 when (1 + €) C_p2 < C_p1 and smaller than 2 when (1 €) C_p2 < C_p1.