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Here's what you'll find in this section:
In the previous chapter, we found that by computing a confidence interval, we could obtain a range of likely values for the population parameter we're estimating. Not only that, but we could do a heuristic ``test'' to see if claims were correct by seeing if the confidence interval captured the claimed value. For example, a manufacturer claims that the average lifetime of an electronic component is 32 hours. We could take a sample of electronic components of size n and measure their lifetime. By measuring the sample mean and variance, we can compute a 95% confidence interval. If 32 fell within our interval, we said we would believe the claim of the manufacturer. If it didn't fall within the interval, we wouldn't believe the claim. Hypothesis testing/ is a formal way of testing claims such as these and is closely related to confidence intervals.
Hypothesis testing in science is a lot like the criminal court system in the United States. How do we decide guilt?
Science, in general, operates by disproving/ unsatisfactory
hypotheses and proposing new-and-improved hypotheses which are testable.
The approach we take in statistics is exactly this scientific method. We
start with a hypothesis which we assume/ is correct. We call
this the null hypothesis/ or 
, and our goal is to reject
in favor of the alternative hypothesis, 
.
The kind of errors we can make are
when
is really true.
when
is really false.
/ or fail to reject/
(in the same sense, a jury can only find someone ``guilty'' or ``not
guilty,'' not ``innocent''). Some books will call the latter
accepting
, but we will try to be careful in using terminology.
In the one and two sample situation, we will always have three forms
of
:
Note that hypotheses are always about population parameters. The
first hypothesis above, 
, is called a two-sided/ or two-tailed/ test, while
the second and third tests are one-sided/ or one-tailed/
hypotheses.
: The accepted explanation, status quo. This is what we're
trying to disprove.
: What the researcher or scientist thinks might really be going
on, a (possibly) better explanation than the null.
in favor of
. We do this by calculating a test statistic/ and
comparing its value with a value from a table in the book, the
critical value.
. We can set up the rejection region before computing our test
statistic.
.
.
when
is really true.
when
is really false.
We will generally use the following steps in hypothesis testing:
and
.
.
Often, statisticians will report their test result as a p-value.
The p-value indicates the chance that one would obtained a test
statistic which is more extreme than the observed one when the 
is true. The rule is always that we reject
if 
The formula for p-value is given in the next section. See the Tests
of Significance concept lab for more about p-values.
The formulas for the 11 cases considered in the `Calculating Tests of Hypotheses' concept lab are given in the table at the end of this chapter. For some examples, see the chapter for that lab.
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