B. Weaver (27May2011)
z and ttests .
..
1
Hypothesis Testing Using
z and ttests
In hypothesis testing, one attempts to answer the following question:
If the null hypothesis is
assumed to be true, what is the probability of obtaining the observed result, or any more extreme
result that is favourable to the alternative hypothesis?
1
In order to tackle this question, at least in
the context of z and ttests, one must first understand two important concepts:
1)
sampling
distributions of statistics, and 2) the central limit theorem.
Sampling Distributions
Imagine drawing (with replacement) all possible samples of size
n
from a population, and for
each sample, calculating a statistice.g., the sample mean.
The frequency distribution of those
sample means would be the sampling distribution of the mean (for samples of size
n
drawn from
that particular population).
Normally, one thinks of sampling from relatively large populations.
But the concept of a
sampling distribution can be illustrated with a small population.
Suppose, for example, that our
population consisted of the following 5 scores:
2, 3, 4, 5, and 6.
The
population mean = 4
, and
the
population standard deviation
(dividing by N) =
1.414
.
If we drew (with replacement) all possible samples of 2 from this population, we would end up
with the 25 samples shown in Table 1.
Table 1:
All possible samples of n=2 from a population of 5 scores.
First
Second
Sample
First
Second
Sample
Sample #
Score
Score
Mean
Sample #
Score
Score
Mean
1
2
2
2
14
4
5
4.5
2
2
3
2.5
15
4
6
5
3
2
4
3
16
5
2
3.5
4
2
5
3.5
17
5
3
4
5
2
6
4
18
5
4
4.5
6
3
2
2.5
19
5
5
5
7
3
3
3
20
5
6
5.5
8
3
4
3.5
21
6
2
4
9
3
5
4
22
6
3
4.5
10
3
6
4.5
23
6
4
5
11
4
2
3
24
6
5
5.5
12
4
3
3.5
25
6
6
6
13
4
4
4
Mean of the sample means =
4.000
SD of the sample means =
1.000
(SD calculated with division by N)
1
That probability is called a
p
value.
It is a really a
conditional probability
it is conditional on the null hypothesis
being true.
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View Full DocumentB. Weaver (27May2011)
z and ttests .
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2
The 25 sample means from Table 1 are plotted below in Figure 1 (a histogram).
This distribution
of sample means is called the
sampling distribution of the mean
for samples of n=2 from the
population of interest (i.e., our population of 5 scores).
Figure 1:
Sampling distribution of the mean for samples of n=2 from a population of N=5.
I suspect the first thing you noticed about this figure is peaked in the middle, and symmetrical
about the mean.
This is an important characteristic of sampling distributions, and we will return
to it in a moment.
You may have also noticed that the standard deviation reported in the figure legend is 1.02,
whereas I reported SD = 1.000 in Table 1.
Why the discrepancy?
Because I used the population
SD formula (with division by N) to compute SD = 1.000 in Table 1, but SPSS used the sample
SD formula (with division by n1) when computing the SD it plotted alongside the histogram.
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 Fall '08
 Robert
 Normal Distribution, Null hypothesis, Student's ttest, B. Weaver

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