10/18/2010
1
One Sample t-test
1
Hypothesis Testing with z-tests
•
Z-tests provided an easy way to illustrate
the logic of one-sample hypothesis tests,
but it was necessary to know both the
mean (
μ
) and standard deviation (
σ
) of the
population to use the test… in most cases
this is not realistic
•
When the value of
σ
is not known, we will
estimate
σ
from the population and use a
different statistical test called the
the t-test
2
Overview of the t-test
•
The t-test is almost identical to the z-test, but the
sample standard deviation (s) is used in the
formula rather than a known population standard
deviation (
σ
)
•
Like the z-test, it converts a sample mean into a
test statistic (here a t-score instead of a z-score)
and compares it to a critical value on a
distribution associated with the null hypothesis to
determine the probability value associated with
the statistic
3

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10/18/2010
2
History of the t-statistic
•
A statistician, William Sealy Gossett,
worked as a chemist at the Guinness
Brewing Company doing quality control
studies for brewing beer
using small
samples
of barley
•
After looking at his results over numerous
experiments, he noticed that the means
did not follow always normal curve, as
predicted by the central limit theorem
4
History of the t-statistic
•
Instead of following a normal curve, the
sampling distributions for smaller samples
of data (such as the barley varieties
Gossett studied at the brewery) were
somewhat flatter than the normal curve
and the shape of the distributions
depended on the size of the sample
–
These sampling distributions were t-
distributions
5
History of the t-statistic
•
Gossett wanted to publish his findings, but
due to contractual agreements with
Guinness he was not allowed
–
A researcher at Guinness had previously
published a paper exposing trade secrets, so
Guinness prohibited its employees from
publishing any papers regardless of content
–
So Gossett published his results under a fake
name, “Student”, and called his statistic
Student’s t statistic
6