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KNR 445
Statistics
Hyp-tests
Slide 1
Stage 5: The test statistic!
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 So, we insert that threshold value, and now we are
asked for some more values…
The sample mean
The sample size
The population SD
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KNR 445
Statistics
Hyp-tests
Slide 2
Stage 5: The test statistic!
 Why do we need these three? Because now we
have to convert our difference score to a score on
the distribution of sample means
 Remember this?
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XX
Ζ
SD
The purpose of this statistic
was to convert a raw score
difference (from the mean)
by scaling it according to the
spread of raw scores in the
distribution of raw scores
KNR 445
Statistics
Hyp-tests
Slide 3
Stage 5: The test statistic!
The purpose of this statistic is the same, but it converts a sample
mean difference (from µ) by scaling it according to the spread of
all sample means in the distribution of sample means
Sample
mean
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X 
Z
SE X
Population
mean
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Variability of
sample
means
KNR 445
Statistics
Hyp-tests
Slide 4
Stage 5: The test statistic!
 Understanding influences on the distribution of
sample means…we’ll use the applet again
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Note sample
size…
& note spread of
sample means
KNR 445
Statistics
Hyp-tests
Slide 5
Stage 5: The test statistic!
 Understanding influences on the distribution of
sample means…we’ll use the applet again
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As sample size
goes up…
Spread of
sample means
goes down
KNR 445
Statistics
Hyp-tests
Slide 6
Stage 5: The test statistic!
 Understanding influences on the distribution of
sample means…
 That means that the test statistic has to take sample
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size into account
 Other influences are mean difference (sample –
population) and variability in the population
 How do you think each of these things influence the
test statistic?
 This will help you understand why the test statistic looks like it does
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KNR 445
Statistics
Hyp-tests
Slide 7
Stage 5: The test statistic!
A closer look: to understand how the mean difference, population
variance, and sample size affect the test statistic, we need to look
at the SEM in more detail
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Sample
mean
X 
Z
SE X
Population
mean
Variability of
sample
means
KNR 445
Statistics
Hyp-tests
Slide 8
Stage 5: The test statistic!
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Population
standard
deviation
SE X 
So…can you see the influences?

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Sample
size
n
2
3
X 
Z
SE X
KNR 445
Statistics
Hyp-tests
Slide 9
Stage 5: The test statistic!
 To calculate, then…
 First the standard error of the mean:

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13.62 13.62
SE X 


 1.8534
n
54 7.3484
 Now the test statistic itself:
X   51.88  49.52
Z

 1.273
SE X
1.8534
KNR 445
Statistics
Hyp-tests
Slide 10
Stage 5: The test statistic!
 For you to practice, I’ve provided a simple excel
file that does the calculation bit for you…
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KNR 445
Statistics
Hyp-tests
Slide 11
Stage 6: The comparison & decision
 Do we fail to reject the null? Or reject the null?
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KNR 445
Statistics
Hyp-tests
Slide 12
3 ways of phrasing the decision…
 What is the probability of obtaining a Zobs = 1.273
if the difference is attributable only to random
sampling error?
 Is the observed probability (p) less than or equal
to the -level set?
 Is p   ?
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KNR 445
Statistics
Hyp-tests
Slide 13
Reporting the Results
 The observed mean of our treatment group was
51.88 ( 13.62) pages per employee per week. The
z-test for one sample indicates that the difference
between the observed mean of 51.88 and the
population average of 49.52 was not statistically
significant (Zobs = 1.27, p > 0.1). Our sample of
employees did not use significantly more paper
than the norm.
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KNR 445
Statistics
Hyp-tests
Slide 14
Do not reject H0 vs. Accept H0
 Accept infers that we are sure Ho is valid
 Do not reject implies that this time we are unable
to say with a high enough degree of confidence
that the difference observed is attributable to
anything other than sampling error.
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