Page 237 - Statistics for Dummies

Chapter 14: Claims, Tests, and Conclusions
“how far is far” by looking at where your test statistic ends up on the distribu-
tion that it came from. When testing one population mean, under certain condi-
tions the distribution of comparison is the standard normal (Z-) distribution,
which has a mean of 0 and a standard deviation of 1; I use it throughout this
section as an example. (See Chapter 9 to find out more about the Z-distribution.)
If your test statistic is close to 0, or at least within that range where most of the
results should fall, then you don’t have much evidence against the claim (H )
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based on your data. If your test statistic is out in the tails of the standard normal
distribution (see Chapter 9 for more on tails), then your evidence against the claim
(H ) is great; this result has a very small chance of happening if the claim is true. In
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other words, you have sufficient evidence against the claim (H ), and you reject H .
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But how far is “too far” from 0? As long as you have a normal distribution or
a large enough sample size, you know that your test statistic falls somewhere
on a standard normal distribution (see Chapter 11). If the null hypothesis (H )
is true, most (about 95%) of the samples will result in test statistics that lie
roughly within 2 standard errors of the claim. If H is the not-equal-to alterna- o o 221
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tive, any test statistic outside this range will result in H being rejected. See
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Figure 14-1 for a picture showing the locations of your test statistic and their
corresponding conclusions. In the next section, you see how to quantify the
amount of evidence you have against H .
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Figure 14-1:
Decisions Reject H O Reject H O
Fail to reject H O Fail to reject H O
for H : not-
a
equal-to.
-2 0 +2
Note that if the alternative hypothesis is the less-than alternative, you reject
H only if the test statistic falls in the left tail of the distribution (below –1.64).
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Similarly, if H is the greater-than alternative, you reject H only if the test sta-
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tistic falls in the right tail (above 1.64).
Defining a p-value
A p-value is a probability associated with your test statistic. It measures the
chance of getting results at least as strong as yours if the claim (H ) were true.
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In the case of testing the population mean, the farther out your test statistic is
on the tails of the standard normal (Z-) distribution, the smaller your p-value
will be, the less likely your results were to have occurred, and the more
evidence you have against the claim (H ).
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