102 TESTS OF HYPOTHESES ON POPULATION MEANS
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0 2.576A2.79 tFigure 8.3 One-sided test of HO : p 5 12.0.
If a two-sided test is performed, the hypothesis is HO : p = 12.0 and is rejected
if the sample mean is either too small or too large. The area to the right of 2.79 is
doubled to include the lower tail and thus .01 < P < .02; again the test would be
rejected. Alternatively, the computed value of t = 2.79 can be compared with the
tabled value t[.975] = 2.11 with 17 d.f., and since 2.79 > 2.11, the null hypothesis
of equal means would be rejected. Using a two-sided test with cr = .05 and t[.975]
implies that one-half of the rejection region is in the upper or right tail and one-half is
in the lower tail. In making this t test, we are assuming that we have a simple random
sample and that the observations are normally distributed. The normality assumption
can be checked using the methods given in Section 6.4. If the data are not normally
distributed, transformations can be considered (see Section 6.5). Note that the t test
is sensitive to extreme outliers so the observations should be examined for outliers.
Computer programs generally used do not include the test using a known 0, but
automatically use the calculated standard deviation and perform the t test. The t test
is often called the Student t test. The programs usually provide the results only for
two-sided tests. Minitab will provide confidence intervals and tests of hypothesis for
a single sample using z.
Some programs will allow the user to test for any numerical value of po such as
po = 12.0. Some programs will only allow the user to test that ,uo = 0. When
using one of these programs, the user should have the program create a new variable
by subtracting the hypothesized po from each of the original observations. In the
example, the new variable would have observations that are X - 12.0. The t test is
then performed using the new variable and HO : p = 0. One of the advantages of
using computer programs is that they can perform the calculations and give accurate
P values for any d.f. In other words, the user does not have to say that the P value is
between two limits such as .01 < P < .02 or that P < .02 but, instead, can say that
P equals a particular value.