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Example 2.11

Use the nitrate data to test the null hypothesis that H o : η = 8.0 and H a : η ≠ 8.0. Here the alternate
hypothesis considers deviations on both the positive and negative sides of the population mean,
which makes this a two-sided test. Both the lower and upper tail areas of the t reference
distribution must be used. Because of symmetry, these tail areas are equal. For a test at the α =
0.05 significance level, the sum of the upper and lower tail areas equals 0.05. The area of each
tail is α 2 = 0.05 2 = 0.025. For α 2 = 0.025 and ν = 26, t ν=26,α /2=0.025 = ± 2.056 . The computed
t value is the same as in Example 2.9; that is, t = –1.852. The computed t value is not outside the
range of the critical t values. There is insufficient evidence to reject the null hypothesis at the stated
level of significance.

Notice that the hypothesis tests in Examples 2.10 and 2.11 reached different conclusions although they
used the same data, the same significance level, and the same null hypothesis. The only difference was
the alternate hypothesis. The two-sided alternative hypothesis stated an interest in detecting both negative
and positive deviations from the assumed mean by dividing the rejection probability α between the two
tails. Thus, a decision to reject the null hypothesis takes into account differences between the sample
mean and the assumed population mean that are both significantly smaller and significantly larger than
zero. The consequence of this is that in order to be declared statistically significant, the deviation must
be larger in a two-sided test than in a one-sided test.
Is the correct test one-sided or the two-sided? The question cannot be answered in general, but often
the decision-making context will indicate which test is appropriate. In a case where a positive deviation
is undesirable but a negative deviation is not, a one-sided test would be indicated. Typical situations
would be (1) judging compliance with an environmental protection limit where high values indicate a
violation, and (2) an experiment intended to investigate whether adding chemical A to the process
increases the efficiency. If the experimental question is whether adding chemical A changes the efficiency
(either for better or worse), a two-sided test would be indicated.

Confidence Intervals
Hypothesis testing can be overdone. It is often more informative to state an interval within which the
value of a parameter would be expected to lie. A 1 – α confidence interval for the population mean can
be constructed using the appropriate value of t as:

ys y t α/2 < η <– y + s y t α/2
have ν = n – 1 degrees of freedom. This confidence interval is bounded by a lower
where t α/2 and s y
and an upper limit. The meaning of the 1 – α confidence level is “If a series of random sets of n obser-
vations is sampled from a normal distribution with mean η and fixed σ, and a 1 – α confidence interval
y ± s y t α 2 is constructed from each set, a proportion, 1 – α, of these intervals will include the value η
and a proportion, α, will not” (Box et al., 1978). (Another interpretation, a Bayesian interpretation, is
that there is a 1 – α probability that the true value falls within this confidence interval.)

Example 2.12
The confidence limits for the true mean of the test specimens are constructed for α/2 = 0.05/2 =
0.025, which gives a 95% confidence interval. For t ν =26,α/2=0.025 = 2.056, = 7.51 and y s y = 0.266,
the upper and lower 95% confidence limits are:
(
(
7.51 2.056 0.266) < η < 7.51 + 2.056 0.266)
–
6.96 < η < 8.05
© 2002 By CRC Press LLC