STATISTICAL TECHNIQUES IN SOFTWARE SIX SIGMA AND DESIGN FOR SIX SIGMA (DFSS) - Software Design for Six-Sigma: A Roadmap for Excellence - page 138

The used test statistics in hypothesis testing depends on the hypothesized parameter
and the data collected. In practical comparison studies, most tests involve comparisons
of a mean performance with a certain value or with another software mean. When
the variance (

2 ) is known, which rarely is the case in real-world applications, Z 0 is
used as a test statistic for the null hypothesis H 0 :

σ

µ = µ 0 , assuming that the observed
population is normal or the sample size is large enough so that the CLT applies. Z 0 is
computed as follows:

y − µ 0

σ/ √ n

Z 0 = (6.6)

The null hypothesis H 0 : µ = µ 0 would be rejected if | Z 0 | > Z α /2 when H a : µ =

µ 0 , Z 0 < − µ>µ 0 .

Depending on the test situation, several test statistics, distributions, and com-
parison methods also can be used at several hypothesis tests. Let us look at some
examples.
For the null hypothesis, H 0 :

Z α when H a : µ<µ 0 , and Z 0 > Z α when H a :

µ 1 = µ 2 , Z 0 is computed as follows:

y 1 − y 2

Z 0 =

(6.7)

1 2 2 2

n 2

σ

n 1 + σ

The null hypothesis H 0 : µ 1 = µ 2 would be rejected if | Z 0 | > Z α /2 when H a : µ 1 =

µ 2 , Z 0 < − Z α when H a : µ 1 <µ 2 , and Z 0 > Z α when H a : µ 1 > µ 2 .

2 ) is unknown, which is typically the case in real-world
applications, t 0 is used as a test statistic for the null hypothesis H 0 :

When the variance ( σ

µ = µ 0 and t 0 is

computed as follows:

y − µ 0

t 0 =

/ √ n

(6.8)

s

The null hypothesis H 0 : µ = µ 0 would be rejected if | t 0 | > t α /2, n − 1 when H a : µ

= µ 0 ,t 0 < − t α , n − 1 when H a : µ<µ 0 , and t 0 > t α , n − 1 when H a : µ>µ 0 .

For the null hypothesis H 0 : µ 1 = µ 2 , t 0 is computed as:

y 1 − y 2

t 0 = s 1 2

n 1 +

(6.9)

s 2 2
n 2

|

t 0 | >

Similarly, the null hypothesis H 0 : µ 1 = µ 2 would be rejected if t α /2 ,v when

H a : µ 1 = µ 2 , t 0 < − t α , v when H a : µ 1 <µ 2 , and t 0 > t α , v when H a : µ 1 >µ 2 , where

v 2.

The discussed examples of null hypotheses involved the testing of hypothe-
ses about one or more population means. Null hypotheses also can involve other

= n 1 + n 2 −