104 TESTS OF HYPOTHESES ON POPULATION MEANS
Table 8.2 Hemoglobin Levels for Acyanotic and Cyanotic ChildrenAc yanotic C yanotic
Number X1 (g/cm3) Number XZ (g/cm3)
1 2 3 4 5 6 7 8 910
11
12
13
14
15
16
17
18
1913.1
14.0
13.0
14.2
11.0
12.2
13.1
11.6
14.2
12.5
13.4
13.5
11.6
12.1
13.5
13.0
14.1
14.7
12.81 2 3 4 5 6 7 8 910
11
1215.6
16.8
17.6
14.8
15.9
14.6
13.0
16.5
14.8
15.1
16.1
18.1
- Xi = 13.03 X2 = 15.74
S: = 1.0167 S; = 1.9898
- Xi = 13.03 X2 = 15.74
From Table A.2, the area to the left of z = 3.99 is 1 .OOOO, so that the area to the right
of 3.99, correct to four decimal places, is .OOOO, or in other words, < .00005. The
area to the right of 7.35 is certainly < .00005. The probability of - xz being
> 2.71 is < .00005, and the probability of - F2 being smaller than -2.71 is
< .00005. Summing the area in the two tails, P < .0001.
For any reasonable values of a, the conclusion is that there is a real difference in
mean hemoglobin level between acyanotic and cyanotic children. In making this test,
we are assuming that we have two independent simple random samples and that the
computed z has a normal distribution.8.2.2 Testing for Equality of Means When 0 Is Unknown
Usually, the variances of the two populations are unknown and must therefore be
estimated from the samples. The Student t test for determining if two population
means are significantly different when the variances are unknown is one of the most
commonly used statistical tests. In making this test, we assume that we have simple
random samples chosen independently from two populations and that observations in
each population are normally distributed with equal variances. The Student t test is
used either when we take samples from two distinct populations or when we sample