396 CHAPTER 6 Inferential Statistics
Again, in the U.S. judicial justice system, it is assumed (or at least
hoped ) thatαis very small, which means thatβ can be large (too
large for many people’s comfort).
Let’s move now to a simple, but relatively concrete example. Assume
that a sample of 60 bolts was gathered from the manufacturing plant
whose claim was that the bolts they produce have a mean diameter of
8.1 mm. Suppose that you knew that the standard deviation of the
bolts wasσ = 0.04 mm. (As usual, it’s unreasonable to assume that
you would know this in advance!) The result of the sample of 60 bolts
is thatx = 8. 117 .This doesn’t look so bad; what should you do?
We proceed by checking how significantly this number is away from
the mean, as following. First, notice that thetest statistic(a random
variable!)
Z =
X−μ
σ/√
60
,
whereμrepresents the hypothesized mean, will be approximately nor-
mally distributed with mean 0 and variance 1. The observed value of
this test statistic is then
z =
x−μ
σ/√
60
≈ 3. 29.
Whoa! Look at this number; it’s over three standard deviations
away from the mean ofZand hence is way out in the right-hand tail of
the normal distribution.^28 The probability for us to have gotten such
a large number under the correct assumption thatH 0 : μ= 8.1 were
true is very small (roughly .1%). This suggests strongly that we reject
this null hypothesis!
(^28) The probabilityP(|Z| ≥ 3 .29) of measuring a value this far from the mean is often called the
P-value of the outcome of our measurement, and the smaller theP-value, the more significance we
attribute to the result. In this particular case,P(|Z| ≥ 3 .29)≈ 0 .001, which means that before
taking our sample and measuring the sample mean, the probability that we would have gotten
something this far from the true mean is roughly one-tenth of one percent!