Transcript T-Test 1.

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t-tests
Slide 1
Introduction to
Hypothesis Testing
The z-test
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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):
Deduction
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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…
 The distribution of sample means, and its variation
 Time for a digression…
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Slide 4
The distribution of sampling means
 Let’s look at this applet…
This is the
population from
which you draw
the sample
Here’s one sample
(n=5)
Here’s the sample
mean for the
sample
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Slide 5
The distribution of sampling means
 Let’s look at this applet…
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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Slide 6
The distribution of sampling means
 Let’s look at this applet…
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
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Slide 7
Stage 1: The null hypothesis
 Knowing that the distribution of sample
means has certain characteristics (later, with
the z-statistic) 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”
 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)
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Slide 8
Stage 1: The null hypothesis
 Example of the null:
 You’re looking for an overall population to
compare to
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Slide 9
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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Slide 10
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
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Slide 11
Stage 2: The alternative hypothesis
 Example of the alternative hypothesis
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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Slide 12
Stage 2: The alternative hypothesis
 So, here our HA: µ > 49.52. Now, next…
What the heck is
that?
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Slide 13
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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Slide 14
Stage 3: Significance threshold (α)
 Notice the fact that distances from the mean
are marked by certain probabilities in a
normal distribution
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Slide 15
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:
 Sample means vary
 The question is whether the variation is due to
chance, or due to being from another population
 When the variation exceeds a certain probability
(α), we reject the null (see applet again)
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Slide 16
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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Slide 17
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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Slide 18
Stage 4: The critical value of Z
 How do we use 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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Slide 19
Stage 4: The critical value of Z
 For our example:
This is α (= .10)
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Slide 20
Stage 4: The critical value of Z
 For our example:
 α = 0.1, and the hypothesis is 1-tailed, so our
distribution would look like this
Fail to reject the null
1 - α (= .90)
Rejection region
α (= .10)
Z score for the α
(= .10) threshold
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Slide 21
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
Rejection region
α (= .10)
Z score for the α
(= .10) threshold
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Slide 22
Stage 4: The critical value of Z
• So, we need to
find a probability
of 0.40
1. Locate the
number nearest to
.4 in the table
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 ≈ 1.285
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Slide 23
Stage 5: The test statistic!
 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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Slide 24
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?
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
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Slide 25
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
X 
Z
SE X
Population
mean
Variability of
sample
means
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Slide 26
Stage 5: The test statistic!
 Understanding influences on the distribution
of sample means…we’ll use the applet again
Note sample
size…
& note spread of
sample means
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Slide 27
Stage 5: The test statistic!
 Understanding influences on the distribution
of sample means…we’ll use the applet again
As sample size
goes up…
Spread of
sample means
goes down
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Slide 28
Stage 5: The test statistic!
 Understanding influences on the distribution
of sample means…
 That means that the test statistic has to take
sample 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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Slide 29
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
Sample
mean
X 
Z
SE X
Population
mean
Variability of
sample
means
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Slide 30
Stage 5: The test statistic!
Population
standard
deviation
SE X 
So…can you see the influences?

Sample
size
n
X 
Z
SE X
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Slide 31
Stage 5: The test statistic!
 To calculate, then…
 First the standard error of the mean:

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
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Slide 32
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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Slide 33
Stage 6: The comparison & decision
 Do we fail to reject the null? Or reject the null?
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Slide 34
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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Slide 35
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.
Notice this would change
if  changed
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Slide 36
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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Slide 37
Note: Z & t-tests
 Same concept, different assumptions
 Can only use z-tests if you know population SD
 You usually don’t – SPSS does not even provide the
test
 So SPSS uses t-test instead
 t-test more robust against departures from normality (doesn’t
affect the accuracy of the p-estimate as much)
 T-test estimates population SD from sample SD
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Slide 38
Note: Z & t-tests
 To estimate pop SD from sample SD, the
sample SD is inflated a little…
( x  x)
s   es t 
n 1
2
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Slide 39
Note: Z & t-tests
 To estimate standard error from sample
SD, use the estimated SD again, thus…
s
sX 
n
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Slide 40
Note: Z & t-tests
 This is important
 Size of estimated SE obviously depends
on both SD of sample, and sample size
s
sX 
n
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Slide 41
Testing in SPSS
STEP 1: Choose the
procedure. SPSS uses the
one sample t-test instead of
the z-test. It’s similar (see
previous notes). I used the
midterm data for this
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Slide 42
Testing in SPSS
STEP 3: Choose a population
mean value to test it against
(SPSS doesn’t have a clue
what population your testing
against, right?)
STEP 2: Choose a variable
to test
STEP 4: Choose “OK” (you could also go
into options and change the confidence
interval size – the default is 95%)
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Slide 43
And you get this…
T-Test
2. Here’s the standard error
3. If you think of the equation,
Statistics
it’s obvious a meanOne-Sample
difference
this big would result in a
significant difference,
right?
N
Mean
Std. Deviation
Average
pupil/teacher ratio
50
16.8580
1. Here’s the important bit
– the statistical outcome
(big difference)
t
Average
pupil/teacher ratio
-34.763
Std. Error
Mean
2.2664
.3205
One-Sample Test
Tes t Value = 28
df
49
Sig. (2-tailed)
Mean
Difference
.000
-11.1420
95% Confidence
Interval of the
Difference
Lower
Upper
-11.7861
-10.4979
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Slide 44
Quittin’ time