Occupational Biomechanics
KNR 445StatisticsHyp-testsSlide 1Introduction to Hypothesis
TestingThe z-test1
KNR 445StatisticsHyp-testsSlide 2Stage 1: The null hypothesisIf
you do research via the deductive method, then you develop
hypothesesFrom 497 (intro to research methods):
Deduction1
KNR 445StatisticsHyp-testsSlide 3Stage 1: The null hypothesisThe
null hypothesisThe hypothesis of no differenceNeed for the null: in
inferential stats, we test the empirical evidence for grounds to
reject the nullUnderstanding this is the key to the whole thingThe
distribution of sample means, and its variationTime for a
digressionusing this
applet:http://onlinestatbook.com/stat_sim/sampling_dist/index.html
123
KNR 445StatisticsHyp-testsSlide 4The distribution of sampling
meansLets look at this applet
This is the population from which you draw the sampleHeres one
sample (n=5)Heres the sample mean for the sample1234
KNR 445StatisticsHyp-testsSlide 5
The distribution of sampling meansLets look at this appletIf 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)123
KNR 445StatisticsHyp-testsSlide 6
The distribution of sampling meansLets look at this appletFor
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 distribution1
KNR 445StatisticsHyp-testsSlide 7Stage 1: The null
hypothesisKnowing 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 meanThus we test for this
statistical oddityIf its sufficiently odd (different), we reject
the nullIf 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)123
Stage 1: The null hypothesisWere 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)KNR 445StatisticsHyp-testsSlide 81
KNR 445StatisticsHyp-testsSlide 9Stage 1: The null
hypothesisExample of the null:Youre looking for an overall
population to compare to
1
KNR 445StatisticsHyp-testsSlide 10Stage 1: The null
hypothesisExample of the null:So the null is the assumption that
our sample mean is equal to the overall population mean
1
KNR 445StatisticsHyp-testsSlide 11Stage 2: The alternative
hypothesisAlso known as the experimental hypothesis (HA, H1)Two
types:1-tailed, or directionalYour sample is expected to be either
more than, or less than, the population meanBased on deduction from
good research (must be justified)2-tailed, or non-directionalYoure
just looking for a differenceMore exploratory in natureDefault in
SPSS12
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 differentwhich is it here?KNR
445StatisticsHyp-testsSlide 12Stage 2: The alternative
hypothesis12
KNR 445StatisticsHyp-testsSlide 13Stage 2: The alternative
hypothesisSo, here our HA: > 49.52. Now, next
What the heck is that?12
KNR 445StatisticsHyp-testsSlide 14Stage 3: Significance
threshold ()How do we decide if our sample is different?Its based
on probabilityRecall normal distribution & z-scores
12
KNR 445StatisticsHyp-testsSlide 15Stage 3: Significance
threshold ()Notice the fact that distances from the mean are marked
by certain probabilities in a normal distribution
1
KNR 445StatisticsHyp-testsSlide 16Stage 3: Significance
threshold ()Our distribution of sample means is similarly defined
by probabilitiesSo, we can use this to make estimates of how likely
certain sample means are to be derived from the null populationWhat
we are saying here is that:Sample means varyThe question is whether
the variation is due to chance, or due to being from another
populationWhen the variation exceeds a certain probability (), we
reject the null (see applet again)123
KNR 445StatisticsHyp-testsSlide 17Stage 3: Significance
threshold ()When the variation exceeds a certain probability (), we
reject the null
Sample means of these sizes are unusual. How unusual is dictated
by the normal distributions pdf (probability density function)
1
KNR 445StatisticsHyp-testsSlide 18Stage 3: Significance
threshold ()When the variation exceeds a certain probability (), 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 ( = .05)1
KNR 445StatisticsHyp-testsSlide 19Stage 4: The critical value of
ZHow do we obtain this probability?Every test uses a
distributionThe z-test uses the z-distributionSo we use
probabilities from the z distributionand then we convert the
difference between the sample and population means to a z-statistic
for comparisonFirst, we need that probability we can use tables for
thisor an appletlets do the tables thing for now
12
KNR 445StatisticsHyp-testsSlide 20Stage 4: The critical value of
ZFor our example:
This is (= .10)1
KNR 445StatisticsHyp-testsSlide 21Stage 4: The critical value of
ZFor our example: = 0.1, and the hypothesis is 1-tailed, so our
distribution would look like this
Rejection region (= .10)Fail to reject the null 1 - (= .90)Z
score for the (= .10) threshold 123
KNR 445StatisticsHyp-testsSlide 22Stage 4: The critical value of
ZFor our example:However, the tables only show half the
distribution (from the mean onwards), so we would have this:
Area referred to in the tableRejection region (= .10)Z score for
the (= .10) threshold 1
KNR 445StatisticsHyp-testsSlide 23Stage 4: The critical value of
Z
So, we need to find a probability of 0.40Locate the number
nearest to .4 in the tableThen look across to the Z column for the
value of Z to the nearest tenth (= 1.2)Then look up the column for
the hundredths (.08)So, z 1.28 (& a bit)1234
5. and it means what?6. Break!
