Topic 8 and
Topic 9 presented
the statistical procedures that permit researchers to
use a single sample mean to test hypotheses about a
population. These statistical procedures were based
on a few basic notions, which are summarized as follows:
- A sample mean is expected to more or less approximate
its population mean. This permits us to use the
sample mean to test a hypothesis about the population
mean.
- The standard error provides a measure of how well
a sample mean approximates the population mean.
If we know the population standard deviation or
variance, the standard error formula is:
If we don't know the population standard deviation
or variance we use the sample's standard deviation
or variance to obtain an estimate of the
standard error. The formula for the estimate of
the standard error is:
- To quantify our inferences about the population,
we compare the obtained sample mean with the hypothesized
population mean. If we know the population standard
deviation or variance we compute a z-test statistic.
The formula is:
If we don't know the population standard deviation
or variance we compute a t-test statistics. The
formula is:
Review of Hypothesis Testing
The overall process of hypothesis testing is the same
whether or not we know the population standard deviation/variance:
We have the same four steps.
The details of some of the steps differ: The method
of determining the critical region depends which one-sample
test we are using, and, of course, we way we calculate
the (estimate of the) standard error differs for T
and Z.
- State the Hypotheses:
This step is the same for both one-sample tests.
We actually state two hypotheses:
- Ho: The null hypothesis This states there
is no effect (two-tail), or that the effect
is not in the direction we anticipate (one-tail).
- H1: The alternative hypothesis. This hypothesis
states that there is an effect (two-tail), or
that the effect is in an anticipated direction
(one-tail).
- (Classical Approach): Set the decision criteria.
- Specify the significance level.
This is the same for both one-sample tests.
Often times we state
- Determine the critical value. The details
depend on the test:
Z-Test: We use the alpha-level to
find the critical Z value in the Z table.
T-Test: We use the alpha-level and
the degrees of freedom to find the
critical T value in the T table.
Notice that there is a new complication
in using T: There isn't just one T-distribution
that we use to determine the critical value
of T. There is a whole family of distributions.
The distribution depends on the "degrees
of freedom".
For the one-sample T-test, the degrees of
freedom is simply equal to one less than the
sample size. That is:
- Gather Data.
This step is the same for both one-sample tests.
- Evaluate the Null Hypothesis .
- Classical approach (used when doing
problems by hand):
- Determine the (estimate of the) standard
error of the mean.
- Calculate the test-statistic value (this
is the "observed" test statistic value).
- Determine if the observed test-statistic
value is in the critical region.
- Decision: If the observed test-statistic
value is in the critical region, reject
the null hypothesis Ho. If it isn't, do
not reject the null hypothesis.
- Contemporary Approach (used when doing
problems by computer):
- Determine the observed test-statistic
value and its exact significance level.
- Report the values and interpret their
implications for the null hypothesis.
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