B. Weaver (31Oct2005)
Probability & Hypothesis Testing
1
Probability and Hypothesis Testing
1.1 PROBABILITY AND INFERENCE
The area of
descriptive statistics
is concerned with meaningful and efficient ways of presenting
data. When it comes to
inferential statistics
, though, our goal is to make some statement about a
characteristic of a population based on what we know about a sample drawn from that
population. Generally speaking, there are two kinds of statements one can make. One type
concerns
parameter estimation
, and the other
hypothesis testing
.
Parameter Estimation
In parameter estimation, one is interested in determining the magnitude of some population
characteristic. Consider, for example an economist who wishes to estimate the average monthly
amount of money spent on food by unmarried college students. Rather than testing all college
students, he/she can test a sample of college students, and then apply the techniques of inferential
statistics to estimate the population parameter. The conclusion of such a study would be
something like:
The
probability
is 0.95 that the population mean falls within the interval of £130£150.
Hypothesis Testing
In the hypothesis testing situation, an experimenter wishes to test the hypothesis that some
treatment has the effect of changing a population parameter. For example, an educational
psychologist believes that a new method of teaching mathematics is superior to the usual way of
teaching. The hypothesis to be tested is that all students will perform better (i.e., receive higher
grades) if the new method is employed. Again, the experimenter does not test everyone in
the population. Rather, he/she draws a sample from the population. Half of the subjects are
taught with the Old method, and half with the New method.
Finally, the experimenter compares
the mean test results of the two groups.
It is not enough, however, to simply state that the mean
is higher for New than Old (assuming that to be the case). After carrying out the appropriate
inference test, the experimenter would hope to conclude with a statement like:
The
probability
that the NewOld mean difference is due to chance (rather than to the
different teaching methods) is less than 0.01.
Note that in both parameter estimation and hypothesis testing, the conclusions that are drawn
have to do with
probabilities
. Therefore, in order to really understand parameter estimation and
hypothesis testing, one has to know a little bit about basic probability.
1.2 RANDOM SAMPLING
Random sampling is important because it allows us to apply the laws of probability to sample
data, and to draw inferences about the corresponding populations.
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Probability & Hypothesis Testing
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Sampling With Replacement
A sample is random if
each member of the population is equally likely to be selected each
time a selection is made
. When
N
is small, the distinction between
with
and
without
replacement is very important. If one samples with replacement, the probability of a particular
element being selected is constant from trial to trial (e.g., 1/10 if
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 Spring '09
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 Probability, Probability theory, B. Weaver

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