Introduction to a variety of useful statistical ideas and techniques - Improving the User Experience through Practical Data Analytics - page 39

SIDEBAR: HOW MANY TAILS?—cont'd

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A one-tail test has a p- value that is half of the p -value of the corresponding two-
tail test . We offer this to you without proof; it is the case! SPSS does not know
whether we are testing “4.10” as a one-tail test or a two-tail test, and that is why
SPSS is careful to tell us that the “Sig.” ( p -value) it is providing is for a two-tail test.
So, the relevant p -value for you to consider is half of 0.107, which equals 0.0535,
which is a small amount above 0.05, telling us that, if H0 is true, the data we have
(with its mean = 4.50, with n = 10, and the variability in the data) is not (by a tad!!)
as rare an event as required to reject H0: μ ≤ 4.1, when using 0.05 as a cutoff point.
Thus we can say that we do not have suficient proof that the true mean satisfaction
rating of the new design exceeds 4.10.
Consider a slightly different scenario that yields different results. If we go back to the
original data and change one of the 4's to a 5 (say, the second data value) getting data of

(5 , 5 , 5 , 5 , 3,4,5,5,4,5),
with sample mean now equal to 4.6, we would get the output in Figure 1.25 . It shows
that the mean is now 4.60 (see, again, the horizontal arrow), and the two-tail p -value
is 0.050 (again, see vertical arrow):
So, the p -value for you to consider is 0.025 (half of the 0.050), and now, with your
X-bar = 4.60, we would conclude that we do have suficient evidence to reject H0,
and conclude that the new design does , indeed, have a true mean satisfaction rating
above 4.10, the known mean satisfaction rating of the current design.