Page 232 - Six Sigma Demystified

212 Six SigMa DemystifieD

representative of the sample as a whole). The normal distribution provides a
good approximation to the binomial distribution when the sample size is large
and when np and n(1 – p) are both greater than 5.

Measure Stage
• To estimate process average error rate (for baseline estimates) when insuf-
ficient data exist to establish process control

Analyze Stage

• To compare error rates of samples from different process conditions

Methodology

Calculate an average error rate ˆ p of n sample units.
Calculate the confidence interval as

1
1
ˆ ( p − ˆ ) p ˆ ( p − ˆ ) p
ˆ p − Z α /2 n ≤ p ≤ ˆ p + Z α /2 n

Based on the assumption that the samples are from a population with a
normal distribution, we use the normal distribution to determine the z values
based on a confidence level. For a 95% confidence level, α = 0.05, so α/2 =
0.025. From Appendix 1, z α/2 = 1.96.
For example, there were 14,248 orders processed during the third week of
June. A sample of 100 orders processed during that week was randomly selected.
Twenty-four orders in the sample were found to have one or more critical
defects. The confidence interval is calculated as

p(1 − p) p(1 − p)
p + Z ≤ p ≤ p + Z
n 1 −α /2 n
α /2
.
× .
. 0 24 × 0 76 0 24 × 0 76.
. 0 24 + − ( 1 .96 ) ≤ p ≤ 0 24 1 96. + .
100 100
0 16 ≤ p ≤ 0 34.
.