Page 146 - Six Sigma Demystified

Chapter 6 a n a ly z e S tag e 127

There are some predictable problems that can occur with hypothesis testing
that should be considered, as outlined in the “Hypothesis Test” section of Part
3. Notably, samples must be random and representative of the population
under investigation. In surveys, low response rates typically would provide
extreme value estimates (i.e., the subpopulation of people who have strong
opinions one way or the other) that are not representative of the total popula-
tion. Samples must be from a stable population. If the population is changing
over time, then estimates will be biased, with associated increases in alpha and
beta risk. Statistical process control (SPC) charts provide an indication of sta-
tistical stability. Many of the hypothesis tests, as well as their associated alpha
and beta risk, depend on the normality of the population. If the population is
significantly nonnormal, then the tests are not meaningful. Goodness- of- fit tests
are used to verify this assumption. Nonparametric tests can be used if the popu-
lations are significantly nonnormal.
Some tests additionally require equal variance, which can be tested using
equality- of- variance tests. If the populations do not have equal variances, then
the data can be transformed (see “Transformation” in Part 3).
It’s important to note that a failure to reject a null hypothesis is not an
acceptance of the null hypothesis. Rather, it means that there is not yet ample
proof that the hypothesis should be rejected. Each of the tests uses a stated
alpha value, where alpha is the probability of observing samples this extreme if
the null hypothesis is true. In most situations, an alpha value of 0.05 is used,
providing a small chance (5 in 100) that samples this extreme (or worse) would
occur if the null hypothesis is true. Since we reject the null hypothesis, then we
also could state that there are 5 chances in 100 of incorrectly rejecting a true
null hypothesis. Furthermore, if n investigators are independently researching

the issue, the probability that at least one researcher (incorrectly) rejects the
n
null hypothesis is 1 – (1 – α) . For example, the chance that 1 of 10 researchers
(i.e., n = 10), each with an alpha risk of 0.05, will (incorrectly) reject the true
null hypothesis is 40 percent! Consider this the next time the headlines in your
newspaper report the “surprising results of a new study.” Would the unsurpris-
ing results of the other nine researchers warrant a headline? The alpha risk
demonstrates the need for independent replication of analysis results.
The beta risk is the probability of not rejecting a false null hypothesis. Usu-
ally, the power of the test (the probability of correctly rejecting the false null
hypothesis) is more interesting. It provides a quantitative reminder that even
though the test is not rejected, the null hypothesis still may be false.
What influences the ability to correctly reject the false null hypothesis? Larger