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URBDP 591 A Lecture 12:
Statistical Inference
Objectives
• Sampling Distribution
• Principles of Hypothesis Testing
• Statistical Significance
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Inferential Statistics
• Based on probability theory.
• The question that inferential statistics
answers is whether the difference
between the sample results and the
population results is too great to be due
to chance alone.
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Sampling Distributions
• A sampling distribution is the probability
distribution of sample statistics.
• A sampling distribution involves sample
statistics that are distributed around a
population parameter.
• The central limit theorem.
– (1) the sampling distribution will approximate a
normal distribution,
– (2) the mean of the sampling distribution will be
equal to the population parameter.
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The Central Limit Theorem
• For any population, no matter what its shape
or form, the distribution of means taken from
that population with a constant sample size
will be normally distributed if the sample size
is sufficiently large (i.e., at least 30).
• The standard deviation of the sampling
distribution, or standard error, is equal to the
standard deviation of the population divided
by the square root of the sample size
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Hypothesis testing
• Step 1: Set up hypothesis
– you should determine whether it is 1-tailed or 2-tailed test
• Step 2: Compute test statistics
• Step 3: Determine p-value of the test statistic
– for a pre-determined a, you can find the corresponding
critical limit
• Step 4: Draw conclusion
– reject H0 if p-value < alpha (ie greater than the critical
limit)
– accept H0 if p-value > alpha (ie less than the critical limit)
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B-IBI = 60 (no change despite development)
B-IBI
60
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Test Statistics
Very general statement:
obtained difference
Test statistic = difference expected by chance
Basic form of all tests of significance:
sample statistic – hypothesized population parameter
Test statistic =
standard error of the distribution of the test statistic
e.g., z-score for use with sample means:
Z
X
=
X -m
s
X
X
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Hypothesis tests
discrepanc y in some measure from what is expected under H 0
Test statistic =
sampling variation in the measure
difference between tw o sample means
e.g. t =
standard error of the difference
F=
variance of group means
sampling variance within groups
difference between observed and expected counts
 =
variabilit y in counts
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Interpreting p-values
• p-value quantifies the role of chance
• Large p-value
Result may be due to chance
• Small p-value
Result unlikely to be due to chance.
Conclude that a true and statistically
significant difference exists
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The logic of statistical testing
1. Assume the observed effect is due to chance (This is null hypothesis - written H0.)
2. Find the probability of obtaining the observed effect or bigger
when H0 is true. - (The p value)
3. If p is small then it is implausible that the effect is due to chance
and we reject H0. (We call this result statistically significant.)
4. If p is large then the effect could be due to chance and we retain
H0 as plausible. We call this result statistically not significant.
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Drawing conclusion for one-tailed tests
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Drawing conclusion for two-tailed tests
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A single population mean
– Suppose we want to study the effect of
development on bird species richness on a
randomly selected number of sites (n=100).
– We measure species richness (X) after
development, and mean species richness is 9.
– Assume X follows a Normal distribution with a
S.D. of 4.
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Step 1: Set up hypothesis
H0 : m0 = 8
H1 : m0  8
This is a 2-tailed test.
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Step 2: Compute test statistics
x= (x1+x2+...+x100)/100 = 9
If x ~ N(m0 ,s2), then x ~ N(m0 ,s2/n),
z
x - m0
=
s/ n
It follows a Normal
distribution with mean
0 and variance 1
If s is known to be 4, then
test statistics z = (9.0 - 8.0) / (4.0 / 100 )
= 2.5
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Step 3: Determine p-value
• For a=0.05, Z0.05/2 = 1.96 (from Normal table)
Since z-value = 2.5 > 1.96, so p-value < 0.05
• For a=0.01, Z0.01/2 = 2.58 (from Normal table)
Since z-value = 2.5 < 2.58, so p-value > 0.01
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Step 4: Draw conclusion
• We reject H0 at 5% level as p-value<0.05
and conclude that bird species richness is
significantly different from 8 at 5%
significance level.
• Notice that we have to accept H0 at 1%
level as p-value>0.01 and conclude that
bird species richness is not statistically
different from 8 at 1% significant level.
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Difference between two population means
– Suppose we want to study the effect of two
development patterns A and B on bird species
richness. We randomly select 52 sites which
will be developed with high density
development (development type A), and low
density development (development type B).
– We measure species richness (X) after
development. The means for treatments A and
B are 9 and 8 respectively.
– Assume Normal distribution, and the S.D. for
treatments A and B are 4 and 4.5.
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Step 1: Set up hypothesis
H 0: m A = m B
H 1: m A  m B
or
H0: mA - mB = 0
H 1: m A - m B  0
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Step 2: Compute test statistics
SE( xA - xB ) = sA2/nA + sB2/nB
= 8.51
Test statistic is:
z = [ ( xA - xB ) - (mA - mB) ] / SE( xA - xB )
= [ (90 - 80) - 0 ] / 8.51
=1.18
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Step 3: Determine p-value
• For a=0.05, Z0.05/2 = 1.96
Since z-value = 1.18 < 1.96, so p-value > 0.05
Step 4: Draw conclusion
– We accept H0 at 5% level as pvalue>0.05, and conclude that bird
species richness after the two treatments
are not statistically different at 5%
significance level. In other words, the
effects of the two treatments are not
statistically different.
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One-tailed vs. two-tailed tests
– If there is an effect, the effect may either be
positive (get better) or negative (get worse).
– Two-tailed test is to study the existence of the
effect in either direction (ie. positive or negative
effect).
– One-tailed test is to study the existence of the
effect in one direction (eg. positive effect).
– Provided that we have a priori knowledge about it
– Directional hypothesis
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Statistical significance vs. practical importance
• If the sample size is unnecessarily large...
– Differences may be established as statistically
significant, and yet be too small to have any
practical consequences.
• The optimum sample size…
– is just large enough to detect differences of a size
which the researcher believes to be of practical
importance. This firstly involves a professional
assessment of how large a difference is important,
followed by a power analysis to determine the
required sample size.
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Statistical significance vs. practical importance
• If the sample size is unnecessarily large...
– Differences may be established as statistically
significant, and yet be too small to have any
practical consequences.
• The optimum sample size…
– is just large enough to detect differences of a size
which the researcher believes to be of practical
importance. This firstly involves a professional
assessment of how large a difference is important,
followed by a power analysis to determine the
required sample size.
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