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KNR 445
Statistics
Hyp-tests
Slide 1
Introduction to
Hypothesis Testing
1
The z-test
KNR 445
Statistics
Hyp-tests
Slide 2
Stage 1: The null hypothesis
 If you do research via the deductive method, then
you develop hypotheses
 From 497 (intro to research methods):
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Deduction
KNR 445
Statistics
Hyp-tests
Slide 3
Stage 1: The null hypothesis
 The null hypothesis
 The hypothesis of no difference
 Need for the null: in inferential stats, we test the
empirical evidence for grounds to reject the null
 Understanding this is the key to the whole thing…
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 The distribution of sample means, and its variation
 Time for a digression…
using this applet:

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http://onlinestatbook.com/stat_sim/sampling_dist/index.html
KNR 445
Statistics
Hyp-tests
Slide 4
The distribution of sampling means
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 Let’s look at this applet…
This is the
population from
which you draw
the sample
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4
Here’s one sample
(n=5)
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Here’s the sample
mean for the
sample
KNR 445
Statistics
Hyp-tests
Slide 5
The distribution of sampling means
 Let’s look at this applet…
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If we take a 1,000
more samples, we
get a distribution of
sample means. Note
that it looks
normally distributed,
but its variation
alters with sample
size (for later)
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3
KNR 445
Statistics
Hyp-tests
Slide 6
The distribution of sampling means
 Let’s look at this applet…
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For now, the
important thing
to note is that
some sample
means are more
likely than
others, just as
some scores are
more likely than
others in a
normal
distribution
KNR 445
Statistics
Hyp-tests
Slide 7
Stage 1: The null hypothesis
 Knowing that the distribution of sample means
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has certain characteristics (later, with the zstatistic) allows us to state with some certainty
how likely it is that a particular sample mean is
“different from” the population mean
 Thus we test for this “statistical oddity”
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 If it’s sufficiently odd (different), we reject the null
 If we reject the null, we conclude that our sample is not from the
original population, and is in some way different to it (i.e. from
another population)
KNR 445
Statistics
Hyp-tests
Slide 8
Stage 1: The null hypothesis
 We’re going to use this applet as an example:
http://www.ltcconline.net/greenl/java/Statistics/HypTestMean/HypTestMean.htm
 (You can open it and follow along, but it will be a
different example to the one I follow)
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KNR 445
Statistics
Hyp-tests
Slide 9
Stage 1: The null hypothesis
 Example of the null:
 You’re looking for an overall population to compare to
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KNR 445
Statistics
Hyp-tests
Slide 10
Stage 1: The null hypothesis
 Example of the null:
 So the null is the assumption that our sample mean is
equal to the overall population mean
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KNR 445
Statistics
Hyp-tests
Slide 11
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Stage 2: The alternative hypothesis
 Also known as the experimental hypothesis (HA,
H1)
 Two types:
 1-tailed, or directional
 Your sample is expected to be either more than, or less than, the
population mean
 Based on deduction from good research (must be justified)
 2-tailed, or non-directional
 You’re just looking for a difference
 More exploratory in nature
 Default in SPSS
KNR 445
Statistics
Hyp-tests
Slide 12
Stage 2: The alternative hypothesis
 Example of the alternative hypothesis
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HA can be that
you expect the
sample mean to
be less than the
null, greater than
the null, or just
different…which
is it here?
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KNR 445
Statistics
Hyp-tests
Slide 13
Stage 2: The alternative hypothesis
 So, here our HA: µ > 49.52. Now, next…
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2
What the heck is
that?
KNR 445
Statistics
Hyp-tests
Slide 14
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Stage 3: Significance threshold (α)
 How do we decide if our sample is “different”?
 It’s based on probability
 Recall normal distribution & z-scores
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KNR 445
Statistics
Hyp-tests
Slide 15
Stage 3: Significance threshold (α)
 Notice the fact that distances from the mean are
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marked by certain probabilities in a normal
distribution
KNR 445
Statistics
Hyp-tests
Slide 16
Stage 3: Significance threshold (α)
 Our distribution of sample means is similarly
defined by probabilities
 So, we can use this to make estimates of how
likely certain sample means are to be derived from
the null population
 What we are saying here is that:
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 Sample means vary
 The question is whether the variation is due to chance,
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or due to being from another population
 When the variation exceeds a certain probability (α), we
reject the null (see applet again)
KNR 445
Statistics
Hyp-tests
Slide 17
Stage 3: Significance threshold (α)
 When the variation exceeds a certain probability
(α), we reject the null…
Sample means of these sizes are
unusual. How unusual is dictated
by the normal distribution’s pdf
(probability density function)
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KNR 445
Statistics
Hyp-tests
Slide 18
Stage 3: Significance threshold (α)
 When the variation exceeds a certain probability
(α), we reject the null…
Convention in the social sciences
has become to reject the null
when the probability of the
variation is less than 0.05.
This gives us our significance
level (α = .05)
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KNR 445
Statistics
Hyp-tests
Slide 19
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Stage 4: The critical value of Z
 How do we obtain this probability?
 Every test uses a distribution
 The z-test uses the z-distribution
 So we use probabilities from the z distribution…
 …and then we convert the difference between the sample and
population means to a z-statistic for comparison
 First, we need that probability – we can use tables for
this…or an applet…let’s do the tables thing for now
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KNR 445
Statistics
Hyp-tests
Slide 20
Stage 4: The critical value of Z
 For our example:
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This is α (= .10)
KNR 445
Statistics
Hyp-tests
Slide 21
Stage 4: The critical value of Z
 For our example:
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 α = 0.1, and the hypothesis is 1-tailed, so our
distribution would look like this
Fail to reject the null
1 - α (= .90)
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Rejection region
α (= .10)
Z score for the α
(= .10) threshold
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KNR 445
Statistics
Hyp-tests
Slide 22
Stage 4: The critical value of Z
 For our example:
 However, the tables only show half the distribution
(from the mean onwards), so we would have this:
Area referred to
in the table
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Rejection region
α (= .10)
Z score for the α
(= .10) threshold
KNR 445
Statistics
Hyp-tests
Slide 23
Stage 4: The critical value of Z
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• So, we need to find a
probability of 0.40
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1. Locate the number
nearest to .4 in the
table
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6. Break!
2. Then look across to
the “Z” column for
the value of Z to the
nearest tenth (= 1.2)
3. Then look up the
column for the
hundredths (.08)
4. So, z ≈ 1.28 (& a bit)
5. …and it means what?
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