Paul T. von Hippel
Department of Sociology & Initiative in Population Research
Ohio State University
300 Bricker Hall
190 N. Oval Mall
Columbus, OH 43210
von-hippel.1@osu.edu
614 688-3768
357 words
Critical value. In a HYPOTHESIS TEST, we decide whether the data are extreme
enough to cast doubt on the NULL HYPOTHESIS. If the data are extreme, we reject the null
hypothesis; if the data are moderate, we accept it. A threshold between extreme and
moderate data, between accepting and rejecting the null hypothesis, is called a critical
value.
Suppose we flip a coin 10 times. An obvious null hypothesis is that the coin is
fair, or equally likely to come up heads or tails. If the null hypothesis is true, we will
probably flip close to 5 heads. A sensible test would accept the null hypothesis if the
number of heads is close to 5 (moderate), and reject the null hypothesis if the number of
heads is far from 5 (extreme). For example, a test might reject the null hypothesis if we
flip 9 heads or more, or if we flip 1 head or fewer. For this test, 1 and 9 are the critical
values.
Critical values are related to SIGNIFICANCE LEVELS. The test above has a
significance level of .0215: under the null hypothesis, there is a .0215 probability that the
number of heads will be at least as extreme as the critical values. (This probability is
calculated using the BINOMIAL DISTRIBUTION.) We can use the critical values to calculate
the significance level, as we did here. It is more common, however, to choose the
significance level first, and then calculate the corresponding critical values.
Paul von Hippel
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Tests using critical values are equivalent to tests using P VALUES. That is,
rejecting a hypothesis because of extreme data is equivalent to rejecting the hypothesis
because of a small p value. For example, suppose 10 coin flips produced 10 heads. 10 is
more extreme than the critical value of 9, so we would reject the null hypothesis using the
critical value. But we would also reject the null hypothesis if we used the TWO-TAILED p
value: the p value for 10 heads is.002, which is less than the significance level of .0215.
As this example illustrates, decisions based on p values will always agree with decisions
based on critical values.
PAUL T. VON HIPPEL
Paul von Hippel
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