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In Depth Information

Exhibit 6.3
Selection of data for z-test

is usually the case, but you are permitted to designate other differences if you are

hypothesizing a speciﬁc difference in the sample means. For example, consider the

situation in which management is willing to purchase the training, but only if it

results in some minimum increase in scores. The desired difference in scores could

be tested as the
Hypothesized Mean Difference
.

The variances for the variables can be estimated to be the variances of the sam-

ples, as long as the samples are greater than approximately 30 observations. Recall

earlier that I suggested that a sample size of at least 30 was advantageous—this

is why! We can also use the variance calculated for the entire population at SC

(Table 6.2—Total SC VAR

=

77.31) since it is available, but the difference in the

calculated z-statistics is very minor: z-statistic using the sample variance is 2.7375

and 2.7475 for the known variance of SC. Next, we choose an

value of 0.05, but

you may want to make this smaller if you want to be very cautious about rejecting

true null hypotheses. Finally, this test of hypothesis is known as a
two-tail
test since

we are not speculating on whether one speciﬁc sample mean will be greater than the

other mean. We are simply positing a difference in the alternative. This is important

in the application of a critical z-value for possible rejection of the null hypothe-

sis. In cases where you have evidence that one mean is greater than another, then a

one-tail
test is appropriate. The critical z-values,
z-Critical one-tail
and
z-Critical

two-tail
, and p-values,
P(Z<

α

z) two-tail
, are provided when

the analysis is complete. These values represent our thresholds for the test.

=

z) one-tail
and
P(Z<

=

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