Chapter 8 McGrew
Elements of Inferential Statistics
Dave Muenkel
Geog 3000
Outline
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Classical Hypothesis Testing
P-Value Hypothesis Testing
One Sample Difference of Means Test
One Sample Difference of Proportions Test
Issues in Inferential Testing / Test Selection
Hypothesis
• A statistical hypothesis is simply a claim about a
population that can be put to the test by drawing a
random sample
Hypothesis Testing in Geography
• Make statements regarding unknown population
parameter values based on sample data in order to:
- Refine Spatial Models
- Develop Laws and Theories
• A properly created sample is essential to Inferential
Statistics
Classical Hypothesis Test
• Steps:
– State Null Hypothesis - Statement regarding the value of an unknown
parameter. Usually implies no association between explanatory and
response variable.
– State Alternative Hypothesis - Statement contradictory to the null
hypothesis.
– Select Test Statistic - Quantity based on sample data and null
hypothesis used to test between null and alternative hypotheses
– Select Rejection Criteria – The value of the test statistic in which we
reject the null or the alternative hypothesis
– Calculate the Test Statistic
– Make a Decision regarding the Hypothesis
State the Hypothesis
The null hypothesis, Ho: Specifies hypothesized values
for one or more of the population parameters
The alternative hypothesis, HA: A statement which says
that the population parameter is something other
than the value specified by the null hypothesis
Null and Alternative Hypothesis
The typical claim is that  is equal to some value H
(hypothesized mean). This claim of equality is called the
Null Hypothesis.
Ho: 1 - 2 = 0, or Ho: 1 = 2
The Alternative Hypothesis is the alternate Hypothesis
and expresses the condition for rejecting the Null
Hypothesis.
HA: 1 - 2  0, or HA: 1  2
The two Hypotheses are mutually exclusive
Example Hypotheses
• H0: μ1 = μ2
• HA: μ1 ≠ μ2
– Two-sided test
• HA: μ1 > μ2
– One-sided test
Type I and Type II Error
State of the World
If Ho is true
Ho Accepted
Correct decision
Pr = 1- 
Ho Rejected
Type I error
Pr = 
If Ho is false
Type II error
Probability = 
Correct decision
Probability = 1 - 
Select the Statistical Test
(www.wikipedia.org)
Statistical Symbols
(www.wikipedia.org)
Select Level of Significance
• If we want to have only a 5% probability of rejecting
H0 if it is really true, then we say our significance
level is 5%
Select Rejection Criteria
Calculate Test Statistic
Test Statistic:
Z
X  0
S/ n
Make a Decision
The rejection of the null hypothesis implies the
acceptance of the alternative hypothesis
Involves
Estimation
Hypothesis Testing
Purpose
To make decisions about population
characteristics
Compare Test Statistic to
Rejection Region
Upper-Tailed
Lower-Tailed
Two-Tailed
Make Decision on Hypothesis
fail to reject
reject
/2
(1  )
reject
/2
P-value
• The smallest α the observed sample would reject H0
• If H0 is true, probability of obtaining a result as
extreme or more extreme than the actual sample
• Is based on a model
Normal, t, binomial, etc.
Determining Statistical Significance: PValue Method
• Compute the exact p-value (X.XX)
• Compare to the predetermined α-level (0.05)
• If p-value < predetermined α-level
– Reject H0
– Results are statistically significant
• If p-value > predetermined α-level
– Do not reject H0
– Results are not statistically significant
Difference of Means / Proportions Test
• Used to compare a mean / proportion from a
random sample to the mean of a population.
• Used to compare a mean / proportion from a
random sample to the mean of a population.
• Assume Normal Distribution
• For Large Samples use Z-Score
• For small samples less than 30, use Students t
distribution
One sample difference of means z test
Degrees of Freedom
• the number of values in the final calculation of a
statistic that are free to vary
• the minimal number of values which should be
specified to determine all the data points
• whenever a parameter must be estimated to
calculate a test statistic, a degree of freedom is lost
Inferential Test Selection
Consider
- population of interest
- investigative variables
- sample data
- inference about population based on sample data
- reliability measure for the inference
Parametric and Non-parametric Tests
• Parametric tests
– for particular assumptions about the underlying
population distributions
– usually normal population is assumed
• Non-Parametric Tests
– may be used on any distribution
– with nominal ordinal data--only non-parametric
tests can be used