Transcript z-Scores and the Normal Curve

Hypothesis testing and
Decision Making
Formal aspects of hypothesis testing
Null and Alternative Hypotheses
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Null hypothesis (H0)
sets the ‘what if’ for
calculating probabilities
Alternative hypothesis
(Ha) sets the rejection
region. Oddly enough,
the alternative is
usually what you have
in mind to prove during
the study.
H0 :   50
H 0 : 1  2  0
H a :   50
H a : 1  2  0
Tails
The rejection region can fit into 1 or 2 tails of the
sampling distribution of means. The RR is
determined by the alternative hypothesis.
Two Tails
H0 :   value
One Tail
H0 :   value
H a :   value Ha :   value or
H a :   value
Tails illustrated
Two tails.
-1.96 Lik ely Outc ome 1.96
If N ull is True
H a :   value
Note 1.96 vs. 1.65
R ejec t
D on't rejec t
R ejec t
Lik ely Outc ome
If N ull is True
-3
-2
-1
0
Z
1
2
1.65
3
R ejec t
D on't rejec t
One tail.
H a :   value
-3
-2
-1
0
Z
1
2
3
Example of 2 tails

Lik ely Outc ome
If N ull is True
Suppose:
H0 :   75; H a :   75
  10, N  25

Then:
R ejec t
10
75  1.96
 71.08  78.92
25
71.08
Note 5 percent is split into two tails.
D on't rejec t
75
X
R ejec t
78.92
Example of 1 tail

Suppose:
H0 :   75; H a :   75
  10, N  25
Sampling Distribution of Means
0 .2 0
Likely Outcome if Null is True
0 .1 6
0 .1 2
10
75  1.65
 78.3
25
0 .0 8
D on't R ejec t
R ejec t
0 .0 4
0 .0 0
70 71 72 73 74 75 76 77 78
X
Note all 5 percent is at the top tail.
79 80
78.3
Review
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Suppose H0 :   650;   100; N  100
Sampling Distribution of Means
Draw sampling
distribution of means.
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What is the shape of this
distribution?
What is the mean of this
distribution?
What is the standard
deviation of this
distribution?
Draw RR if
H a :   650
H a :   650
620
630
640
650
660
670
680
Statistical Decision Making
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Must decide how to act even if uncertain
(developed by Fisher for agriculture)
Make decisions like bets in gambling (null is
false if null is improbable)
If we do this, some times we will be right;
sometimes wrong.
We can calculate probabilities of mistakes
and correct decisions. Some have names.
Decisions
Population Condition
Fire alarm
No fire
Fire
Alarm
silent
Right,
but…
Beta
Alarm
sounds
Alpha
Correct
rejection
Null
true
Accept Right,
Null
but…
Sample Decision
Three named probabilities:
Alpha, beta, and power.
Null
False
Beta
(type II
error)
Reject Alpha Correct
Null
(type I rejection
error) (power)
Researcher’s Choice
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We can pick alpha (but usually .05)
We can improve power by
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Good alternative hypothesis
Good design (minimize error)
Large samples
Review
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Define the following
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Alpha
Beta
Power
Why is it good to have alpha be a small
number?
Why is it good to have power be a large
number?
Definition
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The alternative hypothesis sets the
placement of the _____.
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1 alpha
2 omega
3 rejection region
4 standard error
Definition
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Alpha refers to what kind of error?
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1 descriptive
2 primary
3 type I
4 type II
Application
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A researcher can increase the _____ to
increase the power of a study.
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1 number of outcomes
2 sample size
3 standard error
4 study duration