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Introduction to the t statistic (Chapter 9)
 In chapter 8 we learned to use Z-scores and the unit normal
table to find critical regions for hypothesis testing
 The problem with Z-scores is that they require knowledge of the
population standard deviation so that we can compute the
standard error

x

n
but
 is often unknown
 Without the SEM, we can’t estimate the amount of error
between the sample mean and the population mean
 The solution to this problem is to use the t statistic
 The t statistic uses the sample standard deviation s rather than

s
ss
n 1
 Using s we can estimate SEM ( sx or SM)
s
sx 

n
 Therefore, sx is used instead of
unknown
s2
n
 when the population  is
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 Instead of using
 a t statistic
Z
t
2-
x

x
we can compute
x
sx
 sample mean – population mean / estimate of error
 Note: when you do have
 , always use Z-score rather than t
 Recall that df represents the number of scores in a sample that
are free to vary and that this is always n-1 for a sample because
knowing the sample mean places a restriction on the value of 1
score in the sample
Also
 Just as the sample size influences the SEM, (larger n, the less
error), the greater the df for a sample, the better the sample
represents  and the better the t statistic approximates the Zscore (population)
 T distribution: Table 9.1
note: df, 1-tail, 2-tail
 Full t table in Appendix B-2
 p 225 para 2 - when between df in table use larger t value
(smaller df)
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Hypothesis testing:
 Same steps for hypothesis testing as outlined in Ch 8.(remember
step 5)
 If the t statistic falls in the critical region (exceeds the critical
value of t) then reject Ho – if not, retain Ho
 Example 9.1 p 227
 In the literature (step 5)
 There is a tendency for the birds to avoid the eye spots and
spend more time in the plain side of the box t(8)=6.0, p<0.05
 Don’t worry about Cohen’s d on the next test or about r2
Non-directional and directional hypothesis testing with the t
statistic:
 Same issues as for the Z stat
Assumptions of t test:
1) Sample values are independent (orthogonal) usually met by
random sampling
2) Sample must be normal but violation is usually not a problem
especially if sample is large  30
Versatility of t test:
1)
 not needed from population
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2)
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 often not required, depending upon question – for example
with animal eye example,  can be worked out logically
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