Page 231 - Six Sigma Demystified

Part 3 s i x s i g m a to o l s 211

One-Sided Example at 95% Confidence

Lower (results in calculated value of 25.084) = 25.7 – TINV(0.1, 24) × 1.8/SQRT(25)

Interpretation

A 95% confidence limit on the mean, for example, indicates that in 95 per-
cent of the samples, the confidence interval will include the true population
mean µ (pronounced “mu”). A true value of the population mean so large (or
small) so as to fall outside the confidence interval is likely to happen only 5
percent of the time. We see from the calculation that as the number of sam-
ples n increases, the confidence interval gets smaller. That is, we have more
confidence in the value of the true mean when we take a larger sample.
Notice that the confidence interval when σ is unknown (using the t tables)
is wider than when σ is known (using the z tables) because we lose some
statistical confidence when we estimate the standard deviation. An addi-
tional parameter of the Student’s t distribution is the degrees of freedom 
(pronounced “nu”), which equals n – 1. Statistically, we say that we have lost
a degree of freedom in estimating the standard deviation using the sample
data.
A given sample lying within the confidence interval does not provide evi-
dence of process stability, which must be verified with an SPC chart. Confi-
dence intervals are applicable only to static populations and not to
processes.

Confidence interval on Proportion

When we sample from a population and have historical evidence of the popu-
lation standard deviation, we can estimate the confidence interval of the mean
at a given confidence level. A confidence interval is a tool of statistical inference,
where we use sample statistics (such as a sample average X or a sample stan-
dard deviation s) to infer properties of a population (such as its mean µ or
standard deviation σ).

When to Use

A key assumption is that the population has a normal distribution and is both
constant (it does not change over time) and homogeneous (a given sample is