Download II. Inferential Statistics (4) Testing a hypothesis 1.

II. Inferential Statistics (4)
Testing a hypothesis
!  Introduction
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to Hypothesis testing
test (for a population mean)
!  More about Hypothesis Testing
!  z
Let’s see an example:
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1.- Hypothesis Testing
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Research Problem
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Null Hypothesis (H0)
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Research Hypothesis: Is the informal hypothesis that
inspires the investigation
A statistical hypothesis that usually asserts that nothing
special is happening with respect to some characteristic of
the underlying population
Common Outcomes
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!  Hypothesized
An observed sample mean where the difference between
its value and that of the hypothesized population is small
enough to be viewed as a probable outcome under the null
hypothesis
Rare Outcomes
! 
An observed sample mean where the difference between
its value and that of the hypothesized population is too
large to be viewed as a probable outcome under the null
hypothesis
Sampling Distribution
!  A
distribution, centered about the population
mean, which plays a key role in testing H0 (used
to generate the decision rule)
!  Properties:
•  The standard error of the mean applies to it
•  Its shape approximates a normal curve (when sample
size satisfies the requirements of the central limit
theorem)
A statistical hypothesis that negates H0. It is often
identified with the research hypothesis
Hypothesis Testing
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Hypothesis Testing
Alternative Hypothesis (H1)
! 
Research Problem: Does the mean SAT verbal
score for all local freshmen differ from the
national average of 500 (with the population
standard deviation (σ) equal to 110)?
Let’s say that with a random sample of 100
local freshmen we obtain a mean score of 533.
How can we solve the research question?
2.- z test (for a population mean)
!  Sampling
distribution of z
!  The
distribution of z values that would be
obtained if a value of z were calculated for each
sample mean for all possible random samples of a
given size from some population
!  z
test (for a population mean)
!  A
hypothesis test that evaluates how far the
observed sample mean deviates (in standard error
units) from the hypothesized population mean
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z test (for a population mean)
! 
Assumptions of z test
1. 
2. 
z test (for a population mean)
Decision Rule
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The population is normally distributed (or the
sample is large enough to satisfy the
requirements of the central limit theorem)
The population standard deviation is known
! 
Specifies precisely when H0 should be rejected
(because the observed z qualifies as a rare
outcome)
Critical Value (cutoffs)
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Test statistic value (e.g., z score) beyond which
we will reject H0
z test (for a population mean)
z test (for a population mean)
!  Common
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and rare outcomes
!  One has to choose a level of significance (α)
Let’s see an example:
Research Problem: Does the mean SAT verbal
score for all local freshmen differ from the
national average of 500 (with the population
standard deviation (σ) equal to 110)?
Let’s say that with a random sample of 100
local freshmen we obtain a mean score of 533.
How can we solve the research question?
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-1.96
Rare outcomes
Reject H0
1.96
Common Outcomes
Retain H0
Rare outcomes
Hypothesis Testing: 6 steps
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Reject H0
State the research problem
Identify the statistical hypotheses
Specify a decision rule
Calculate the value of the observed test statistic
Make a decision
Interpret the decision
3. More about Hypothesis Testing
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Strong and Weak decisions
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Retaining H0 is a weak decision
Rejecting H0 is a strong decision
The decision to retain H0 implies (only) that
H0 could be true, whereas the decision to
reject H0 implies that H0 is probably false
(and that H1 is probably true)
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3. More about Hypothesis Testing
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One-tailed test (or directional test)
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Rejection region is located in just one tail of the
distribution
It has extra-sensitivity
Two-tailed tests (or nondirectional test)
! 
II. Inferential Statistics (5)
!  Errors
in Statistical Tests
!  Type
!  Type
I Error
II Error
!  Power
Rejection regions are located in both sides of the
distribution
1.- Errors in Statistical
Tests
Errors in Statistical Tests
!  Type
!  Probability
I error
error in statistical decision making that
occurs if the null hypothesis (H0) is rejected
when actually it is true in the population
associated with:
!  The
!  Type
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Type I error: α
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Type II error: β
II error
!  The
error in statistical decision making that
occurs if the null hypothesis is not rejected when
actually it is false in the population, and the
alternative hypothesis H1 is true
Errors in Statistical Tests
Decision by the Experimenter
H0 true
Fails to reject H0
Rejects H0,
Accepts H1
Correct decision:
Type I error:
p=α
p=1-α
True situation in
The Population
H1 true
Type II error:
p=β
Correct decision:
p=1-β
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