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10.2 Tests of significance
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10.2 TESTS OF SIGNIFICANCE
Confidence intervals are one of the two most common types of statistical
inference. Use a confidence interval when your goal is to estimate a popu
lation parameter. The second common type of inference, called tests
of
sig
nificance, has a different goal: to assess the evidence provided by data about
some claim concerning a population. Here is the reasoning of statistical
tests in a nutshell.
EXAMPLE 10.8
I'M A GREAT FREETHROW SHOOTER
I claim that I make 80% of my basketball free throws. To test my claim, you ask me to shoot
20 free throws.
I
make only 8 of the 20. "Aha!" you say. "Someone who makes 80% of his
free throws would almost never make only 8 out of 20. So I don't believe your claim."
Your reasoning is based on asking what would happen if my claim were true and we
repeated the sample of 20 free throws many timesI would almost never make as few as
8. This outcome is so unlikely that it gives strong evidence that my claim is not true.
You can say how strohg the evidence against my claim is by giving the probability
that I would make as few as 8 out of 20 free throws if I really make 80% in the long
run. This probability is 0.0001. I would make as few as 8 of 20 only once in 10,000 tries
in the long run if my claim to make 80% is true. The small probability convinces you
that my claim is false.
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Chapter 10
Introduction to Inference
significance test
null hypothesis
Significance tests use an elaborate vocabulary, but the basic idea is simple:.
an outcome that would rarely happen if a claim were true is good evidence
that the claim is not true.
The reasoning of tests of significance
The reasoning of statistical tests, like that of confidence intervals, is based on ask
ing what would happen if we repeated the sample or experiment many times.
Here is the first example we will explore.
EXAMPLE 10.9
SWEETENING COLAS
Diet colas use artificial sweeteners to avoid sugar. These sweeteners gradually lose their
sweetness over time. Manufacturers therefore test new colas for loss of sweetness before
marketing them. Trained tasters sip the cola along with drinks of standard sweetness
and score the cola on a "sweetness score" of
1
to 10. The cola is then stored for a month
at high temperature to imitate the effect of four months' storage at room temperature.
Each taster scores the cola again after storage. This is a matched pairs experiment. Our
data are the differences (score before storage minus score after storage) in the tasters'
scores. The bigger these differences, the bigger the loss of sweetness.
Here are the sweetness losses for a new cola, as measured by 10 trained tasters:
Most are positive. That is, most tasters found a loss of sweetness; But the losses are
small, and two tasters (the negative scores) thought the cola gained sweetness. Are these
data good evidence that the cola lost sweetness in storage?
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 Fall '08
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