Statistics for the Behavioral Sciences by Frederick J. Gravetter, Larry B. Wallnau ISBN 10: 1305504917 ISBN 13: 9781305504912

SECTION 6.5 | Looking Ahead to Inferential Statistics 185
FIGURE 6.21
Using probability to evaluate a treatment
effect. Values that are extremely
unlikely to be obtained from the original
population are viewed as evidence
of a treatment effect.
Middle 95%
High probability values
(scores near m 5 400)
indicating that the treatment
has no effect
m 5 400
z 5 21.96 z 5 11.96
Extreme 5%
Scores that are very unlikely
to be obtained from the original population
and therefore provide evidence of a treatment effect
Figure 6.21 shows the original population from our hypothetical research study. Note
that most of the scores are located close to μ = 400. Also note that we have added boundaries
separating the middle 95% of the distribution from the extreme 5% or 0.0500 in the two
tails. Dividing the 0.0500 in half produces a proportion of 0.0250 in the right-hand tail and
0.0250 in the left-hand tail. Using column C of the unit normal table, the z-score boundaries
for the right and left tails are z = +1.96 and z = –1.96, respectively. If we are selecting
an individual from the original untreated population, then it is very unlikely that we would
obtain a score beyond the z = ±1.96 boundaries.
The boundaries set at z = ±1.96 provide objective criteria for deciding whether our
sample provides evidence that the treatment has an effect. Specifically, if our sample is
located in the tail beyond one of the ±1.96 boundaries, then we can conclude:
1. The sample is an extreme value, nearly 2 standard deviations away from average,
and therefore is noticeably different from most individuals in the original
population.
2. The sample is a very unlikely value with a very low probability if the treatment has
no effect.
Therefore, the sample provides clear evidence that the treatment has had an effect.
LEARNING CHECK
1. For a normal distribution with μ = 100 and σ = 10, which of the following accurately
describes the location of X = 80?
a. It is an extreme, very unlikely score.
b. It is a low score, but not extreme or unlikely.
c. It is a fairly representative score.
d. None of the other options is an accurate description.