(i) deciding whether or not two methods for determining the atomic weight of carbon differ, and
(ii) deciding whether or not babies of mothers who smoke have lower birth weights than babies of mothers who do not smoke.
and that are mutually independent. We
also assume that
,
,
, and
are
unknown. We do not assume that
. Let
denote the difference in population means, sometimes called the
shift parameter.
The distribution of this Welch's test statistic is approximately a t-distribution with the following number of degrees of freedom:
Remarks: When m=n (i.e. when the sample sizes are equal), tW equals Student's 2-sample t-statistic. The Welch test typically uses fewer degrees of freedom than the Student test.
A -level confidence interval for
is
where is the critical value.
Method A | Method B |
12.0072 | 11.9853 |
12.0064 | 11.9949 |
12.0054 | 11.9985 |
12.0016 | 12.0061 |
12.0077 |
Hypotheses: Let denote the population mean for
method A, and
the population mean for method B. We want
to test
vs.
, and we note that this is equivalent to testing:
vs.
where
.
DATA: The data are from a study of the birth weights in pounds of the children of 40 mothers who smoked at least one pack a day during pregnancy and 39 mothers who did not smoke at all.
10cChildren of Nonsmoking Mothers | |||||||||
8.3 | 7.9 | 9.6 | 7.1 | 6.8 | 10.2 | 7.3 | 8.8 | 8.0 | 9.5 |
5.9 | 10.1 | 8.2 | 8.7 | 9.6 | 12.3 | 8.1 | 7.3 | 7.8 | 6.6 |
9.1 | 7.4 | 6.8 | 7.5 | 8.2 | 6.6 | 7.9 | 8.4 | 8.9 | 10.4 |
9.0 | 7.5 | 8.2 | 8.7 | 7.0 | 10.8 | 9.9 | 8.8 | 12.3 |
10cChildren of Smoking Mothers | |||||||||
8.1 | 6.5 | 7.3 | 6.8 | 7.9 | 8.4 | 6.2 | 7.8 | 9.1 | 6.7 |
8.8 | 7.5 | 7.0 | 7.3 | 9.6 | 5.6 | 8.0 | 6.9 | 7.1 | 7.9 |
10.3 | 7.4 | 4.9 | 7.3 | 8.1 | 6.2 | 9.9 | 5.7 | 8.6 | 7.4 |
8.2 | 10.8 | 6.8 | 7.4 | 8.9 | 5.9 | 7.2 | 7.9 | 8.0 | 6.6 |
Hypotheses: Let denote the population mean for
the nonsmokers, and let
denote the population mean for
the smokers. We want to test
vs.
. Note that this is equivalent to
testing
vs.
, where
.
IF CASE<40 THEN LET CIG$='NO'
IF CASE>39 THEN LET CIG$='YES'