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= (.0005, .0041).
" Hypothesis Testing. To test H0 : " = "0 vs. H1 : " = "0, we consider
the test statistic TW under the null hypothesis that " = "0. Let tW
denote the observed value of TW . Then a level-± test is to reject H0 if
and only if
P = P (|TW | e" |tW |) d" ±,
which is equivalent to rejecting H0 if and only if
|tW | e" t1-±/2(½).
Æ
This test is called Welch s approximate t-test.
" Example (continued). To test H0 : " = 0 vs. H1 : " = 0, we compute
(.0167 - .0144) - 0
.
tW = = 2.59.
.00422/57 + .00242/12
Since |2.59| > 2.05, we reject H0 at significance level ± = .05. (The
.
significance probability is P = .015.)
" In the preceding example, the sample pooled variance is s2 = .00402.
Hence, from the corresponding example in the preceding subsection,
we know that using Student s 2-sample t-test would have produced
.
a (misleading) significance probability of p = .067. Here, Student s
test produces a significance probability that is too large; however, the
reverse is also possible.
" Example. Suppose that n1 = 5, x = 12.00566, and s2 = 590.80 × 10-8;
¯
1
suppose that n2 = 4, y = 11.99620, and s2 = 7460.00 × 10-8. Then
¯
2
. .
tW = 2.124, ½ = 3.38, and to test H0 : " = 0 vs. H1 : " = 0 we obtain
Æ
.
a significance probability of P = .1135. In contrast, if we perform
Student s 2-sample t-test instead of Welch s approximate t-test, then
.
we obtain a (misleading) significance probability of P = .0495. Here
Student s test produces a significance probability that is too small,
which is precisely what we want to avoid.
212 CHAPTER 10. 2-SAMPLE LOCATION PROBLEMS
" In general:
If n1 = n2, then t = tW .
If the population variances are (approximately) equal, then t and
tW will tend to be (approximately) equal.
If the larger sample is drawn from the population with the larger
variance, then t will tend to be less than tW . All other things
equal, this means that Student s test will tend to produce signif-
icance probabilities that are too large.
If the larger sample is drawn from the population with the smaller
variance, then t will tend to be greater than tW . All other things
equal, this means that Student s test will tend to produce signif-
icance probabilities that are too small.
If the population variances are (approximately) equal, then ½ will
Æ
be (approximately) n1 + n2 - 2.
It will always be the case that ½ d" n1 + n2 - 2. All other things
Æ
equal, this means that Student s test will tend to produce signif-
icance probabilities that are too large.
" Conclusions:
If the population variances are equal, then Welch s approximate
t- test is approximately equivalent to Student s 2-sample t-test.
If the population variances are unequal, then Student s 2-sample
t-test may produce misleading significance probabilities.
If you get just one thing out of this course, I d like it to be
that you should never use Student s 2-sample t-test. (Erich L.
Lehmann)
10.2 The 2-Sample Location Problem for a Gen-
eral Shift Family
10.3 The Symmetric Behrens-Fisher Problem
10.4 Exercises
Chapter 11
k-Sample Location Problems
" We now generalize our study of location problems from 2 to k popu-
lations. Because the problem of comparing k location parameters is
considerably more complicated than the problem of comparing only
two, we will be less thorough in this chapter than in previous chapters.
11.1 The Normal k-Sample Location Problem
" Assume that Xij
This is sometimes called the fixed effects model for the oneway analysis
of variance (anova). The assumption of equal variances is sometimes
called the assumption of homoscedasticity.
11.1.1 The Analysis of Variance
" The fundamental problem of the analysis of variance is to test the null
hypothesis that all of the population means are the same, i.e.
H0 : µ1 = · · · = µk,
against the alternative hypothesis that they are not all the same. No-
tice that the statement that the population means are not identical
does not imply that each population mean is distinct. We stress that
the analysis of variance is concerned with inferences about means, not
variances.
213
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