SECTION 6.5 Hypothesis Testing 397
Continuing the above example, assume more realistically that we
didn’t know in advance the variance of the population of bolts, but
that in the sample of 60 bolts we measure a sample standard deviation
ofsx=.043. In this case sample statistic
T =
X−μ
Sx/√
60
has thetdistribution with 59 degrees of freedom (hence is very approx-
imately normal). The observed value of this sample statistic is
t =x−μ
sx/√
60
≈ 3. 06.
As above, obtaining this result would be extremely unlikely if the hy-
pothesisH 0 : μ= 8.1 were true.
Having treated the above two examples informally, we shall, in the
subsequent sections give a slightly more formal treatment. As we did
with confidence intervals, we divide the treatment into the cases of
known and unknown variances, taking them up individually in the next
two sections.
Exercises
- In leaving for school on an overcast April morning you make a
judgement on the null hypothesis: The weather will remain dry.
The following choices itemize the results of making type I and type
II errors. Exactly one is true; which one?
(A)
Type I error: get drenched
Type II error: needlessly carry around an umbrella(B)
Type I error: needlessly carry around an umbrella
Type II error: get drenched(C)
Type I error: carry an umbrella, and it rains
Type II error: carry no umbrella, but weather remains dry(D)
Type I error: get drenched
Type II error: carry no umbrella, but weather remains dry