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One sample problems
Statistics 2126
introduction
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•
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Yeah z tests are great
You can do statistical tests!
You simply need  and 
Ok the mean is easy, the standard
deviation, not so much
• What to do, what to do…
A thought…
• Well let’s look at the z formula
z
x  
( / n )
• What if we could substitute something
for ?

Introducing….
• The t statistic!
x  

t
(s / n )
• You do just that, put s in for 
Powerful technique
• Think about this
• You now don’t really need to know
anything
• A theoretical value for the population
mean
• And a calculated sample standard
deviation
The critical value of t
• Unlike z, t changes depending on
sample size
• It has a certain number of ‘degrees of
freedom’
• In our case, n-1 so if we have 22
subjects we have 21 df
• We look up the critical value with 21 df
An example
• Four black capped chickadees in a
memory experiment with two
alternatives.
• Their average percentage correct was
82.33
• Standard deviation was 12.5
• Are they better than chance?
x  

t
(s / n )
82.33  50
t
12.5 /2
32.33
t
6.25
 5.17
Now look up the critical value
• We have 3 degrees
of freedom
• Let’s use an alpha of
.05 (as usual)
• Just use the table
• Critical value is 2.35
• Our obtained value
is 5.17
• Reject H0
Confidence intervals? Why
not…
  x  t(s / n )
  82.33  3.18(12.5 /2)
  82.33  3.18(6.25)
  82.33  3.18(6.25)
  82.33  19.875
62.455    102.205
Again, just think about this
• That is incredible, we can estimate, we
know that accuracy of the estimate and
we do not need to know ANYTHING
about the population beyond some
theoretical population mean
• (which is usually pretty easy to figure
out)