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CHAPTER 21
Inferential Statistical Analysis
Understanding probability
The idea of probability is central to
inferential statistics.
It means the chance or likelihood of
a particular event occurring, or
the proportion of times it will
occur in a long series of trials.
Statistical significance levels are
probabilities.
Student Activity 1
Testing hypotheses
Formulate null and alternative hypotheses
Select statistical technique and test statistic
Decide on required level of significance
Collect data
Use test to determine significance level of
results
Decide whether to reject the null hypothesis
Relate this decision to your research
question, and draw conclusions
Student Activity 2
What statistics do for you
Statistical tests don’t prove anything –
they merely indicate probabilities of
statements about parent populations
being true.
It is possible to obtain false positives
and false negatives.
The power of a statistical test is 1–β:
ie 1–the probability of a false positive.
Many statistics assume a
normal curve
Two examples of normal curves
Defining a normal curve
A normal curve is a theoretical distribution.
It is defined by two parameters: its mean
and its standard deviation.
The standard deviation (σ) indicates the
variability of observations about the
mean.
5% of the area under a normal curve lies
more than approx. 2σ from the mean.
1% of the area under a normal curve lies
more than approx. 2.5σ from the mean.
Normal curves and probability
Probability areas on a normal curve
What statistics do for you
You can use statistics to:
•
estimate population values from a
sample
• test for differences between samples
• test whether variables are related
• find a line which best fits your data
Estimating population values
Sampling error is the difference
between the (unknown) population
value and your sample statistic.
This is why you can only estimate
population values from sample
measures.
You can work out the probability that
they will fall within a certain range.
Testing for differences
Key questions:
Are samples paired or unpaired?
A1
↓
A2
B1 C1 D1 E1
↓
↓
↓
↓
B2 C2 D2 E2
F1 G1
↓
↓
F2 G2
H1 I1
↓ ↓
H2 I2
F
O
H
Q
or
A
J
B
K
C
L
D
M
E
N
G
P
I
R
Does the direction of difference
matter?
Student Activity 3
Testing for association
Correlation does not mean causation!
A and B might be related because
• A caused B
• B caused A
• both are related to C
• the effect is due to chance
Choosing statistical tests
The appropriate statistical test will
depend upon:
• the question you are asking
• the nature of your variables
• the size of your sample.
Tests are parametric or nonparametric.
Parametric tests
Parametric tests are more powerful
– ie more likely to identify a
difference as significant.
They assume interval or ratio data,
a normal distribution and random
samples with similar variance.
If these conditions are not met or
samples are too small, they can
mislead.
Guidelines for statistical testing
Choose an adequate sample.
Choose an appropriate statistical test.
Use a one-tailed test only if justified.
Check that any computer output
makes sense.
Ensure that your conclusions are
justified by the tests, and do not
mislead.
Student Activity 4