It is noted that the national rate of 0.25 isnot includedin that confidence
interval.
5.4 BRIEF NOTES ON THE FUNDAMENTALS
A statistical hypothesis is a statement about a probability distribution or its
underlying parameter(s), or a statement about the relationship between proba-
bility distributions and their parameters. If the hypothesis specifies a probabil-
ity density function (pdf) completely, it is calledsimple; otherwise, it iscom-
posite. The hypothesis to be tested, thenull hypothesis, is denoted byH 0 ;itis
always stated in the null form, indicating no di¤erence or no relationship
between distributions or parameters. A statistical test is a decision-making
process that examines a set or sets of sample data and on the basis of expecta-
tion underH 0 leads to a decision as to whether or not to rejectH 0. An alter-
native hypothesis, which we denote byHA, is a hypothesis that in some sense
contradicts the null hypothesisH 0. A null hypothesis is rejected if and only if
there is su‰ciently strong evidence from the data to support its alternative.
5.4.1 Type I and Type II Errors
Since a null hypothesis may be true or false and our possible decisions are
whether to reject or not reject it, there are four possible combinations. Two of
the four combinations are correct decisions, but there are two possible ways to
commit an error:
1.Type I:A trueH 0 is wrongly rejected.
2.Type II:A falseH 0 is not rejected.The probability of a type I error is usually denoted byaand is commonly
referred to as thesignificance levelof a test. The probability of a type II error
(for a specific alternativeHA) is denoted bybandð 1 bÞis called thepowerof
the test. The general aim in hypothesis testing is to use statistical tests that
makeaandbas small as possible. This goal requires a compromise because
these actions are contradictory; in a statistical data analysis, we fixaat some
specific conventional level—say, 0.01 or 0.05—and use the test that minimizes
bor, equivalently, maximizes the power.
5.4.2 More about Errors andpValues
Given that type I error is more important or clear-cut, statistical tests are
formed according to theNeyman–Pearson framework.Begin by specifying a
small numbera>0 such that probabilities of type I error greater thanaare
undesirable. Then restrict attention to tests which have the probability of
rejection underH 0 less than or equal toa; such tests are said to to havelevel of
BRIEF NOTES ON THE FUNDAMENTALS 203