HOW MUCH SLEEP DID YOU GET LAST
NIGHT?
1.
2.
3.
4.
5.
6.
<6
6
7
8
9
>9
17%
17%
17%
17%
17%
17%
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1
2
3
4
5
6
CHAPTERS 17
Testing Hypotheses and Confidence
Intervals
ISU ACT MATH SCORES - 2012
New students on campus
 Averages

ISU Math Score = 23.5
 State of Illinois = 21.0
 National = 21.1



SD =5.3
New students at ISU=5,147
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HYPOTHESIS TESTING USING P-VALUE
P-value: Given the null hypothesis, the
probability that we observe the sample we
collected.
 If p-value is small

1) Ho is probably wrong and Ha is probably right
 2) we really do have an odd sample


Conclusion
P-value small => reject null
 P-value large => fail to reject null

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APPROACHES TO HYPOTHESIS TESTING

P-value vs. Alpha Level


One-sided tests (using our z-tables)
Z-score vs. Critical Z*
One-sided tests
 Two-sided tests


Confidence Intervals

Two-sided tests
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HYPOTHESIS TEST USING
Z-SCORE VS. P-VALUE

| Z-score| > Critical Z*


| Z-score| < Critical Z*


Fail to reject the null hypothesis
P-value < alpha


Reject the null hypothesis
Reject the null hypothesis
P-value > alpha

Fail to reject the null hypothesis
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CRITICAL VALUES AGAIN (CONT.)

Here are the traditional critical values from the
Normal model:

1-sided
2-sided
0.05
1.645
1.96
0.01
2.28
2.575
0.001
3.09
3.29
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THE TRUE PROPORTION WHO PASS THE AP
EXAM IS 18%.
IN THE AP INCENTIVE PROGRAM, 20% OF THE
STUDENTS PASS THE AP EXAM, WITH 225,000
STUDENTS THAT TOOK THE PROGRAM.
The researchers believes that those in the AP Incentive
Program are better than most students at passing the
AP exam.
 Does the program work? That is, does the program
increase the likelihood of passing the AP exam.

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WHAT IS THE APPROPRIATE HYPOTHESIS
TEST?
1.
2.
3.
4.
5.
6.
Ho: pProgram=0.18
Ha: pProgram>0.18
Ho: pProgram=0.18
Ha: pProgram<0.18
Ho: pProgram=0.18
Ha: pProgram≠0.18
Ho: pProgram=0.20
Ha: pProgram>0.20
Ho: pProgram=0.20
Ha: pProgram<0.20
Ho: pProgram=0.20
Ha: pProgram≠0.20
17%
17%
17%
17%
17%
17%
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1
2
3
4
5
6
DOES THE PROGRAM WORK? TEST
SIGNIFICANCE LEVEL AT 0.05
Yes, there is enough evidence to suggest that
25%
the program works.
2.
Yes, there is NOT enough evidence to suggest
25% that the program works.
3.
No, there is enough evidence to suggest that the
25% program works.
4.
No, there is NOT enough evidence to suggest
that the program works.
1.
25%
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RADIOACTIVE FALLOUT FROM TESTING ATOMIC
BOMBS DRIFTED ACROSS A REGION. THERE WERE
240 PEOPLE IN THE REGION AT THE TIME. 46
DIED OF CANCER. CANCER EXPERTS ESTIMATE
ABOUT 28 CANCER DEATHS IS NORMAL FOR A
GROUP THIS SIZE.
IS THE DEATH RATE FOR THIS GROUP UNUSUALLY
HIGH? WHAT IS OUR HYPOTHESES?
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WHAT IS OUR HYPOTHESES?
1.
2.
3.
4.
5.
6.
H0: pregion = 0.1167
HA: pregion > 0.1167
H0: pregion = 0.1167
HA: pregion < 0.1167
H0: pregion = 0.1167
HA: pregion ≠ 0.1167
H0: pregion = 0.1917
HA: pregion > 0.1917
H0: pregion = 0.1917
HA: pregion < 0.1917
H0: pregion = 0.1917
HA: pregion ≠ 0.1917
17%
17%
17%
17%
17%
17%
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1
2
3
4
5
6
WHAT ALPHA SHOULD WE CHOOSE?
1.
2.
3.
4.
α= 0.10
α= 0.05
α= 0.01
α= 0.001
25%
25%
25%
25%
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1
2
3
4
WHAT IS OUR P-VALUE?
1.
2.
3.
4.
0.9999
0.0001
3.622
0.1917
25%
25%
25%
25%
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1.
2.
3.
4.
IS THE DEATH RATE FOR THIS GROUP UNUSUALLY
HIGH?
25%
25%
25%
25%
1.
2.
3.
4.
P-value is low enough to conclude that the death
rate is unusually high
P-value is too low to conclude that the death rate
is unusually high
P-value is too high to conclude that the death
rate is unusually high
P-value is high enough to conclude that the
death rate is unusually high
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WE CONCLUDED THAT THE DEATH RATE WAS
UNUSUALLY HIGH. BUT DOES IT PROVE THAT
EXPOSURE TO RADIATION INCREASES
33%
33%THE RISK
33%
OF CANCER?
1.
2.
3.
No, there is not enough evidence.
Yes, there is enough evidence.
Whether the death rate by cancer is unusually high
or not, the CAUSE cannot be determined.
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1
2
3
P-VALUES AND SIGNIFICANCE LEVELS

Significance at 0.01


the test is also significant at 0.05 and 0.10
Significance at 0.05

the test is also significant at 0.10
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A MARKET RESEARCHER CONCLUDED THAT A
MORE PEOPLE LIKE 4LOCO SIGNIFICANTLY
BETTER THAN SPARKS. HIS DECISION WAS
BASED ON ALPHA=0.025. WOULD HIS
DECISION HAVE BEEN DIFFERENT UNDER
ALPHA=0.20 ?
1.
2.
3.
Yes, he still would have rejected the null.
No, he would not have rejected the null.
Maybe, we would need to know the p-value.
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HIS DECISION WAS BASED ON ALPHA=0.025.
WOULD HIS DECISION HAVE BEEN DIFFERENT
UNDER ΑLPHA=0.001 ?
1.
2.
3.
Yes, he still would have
rejected the null.
No, he would not have
rejected the null.
Maybe, we would need to
know the p-value.
33%
33%
33%
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1
2
3
CONFIDENCE INTERVALS AND HYPOTHESIS
TESTS

Confidence intervals and hypothesis tests are
built from the same calculations.


They have the same assumptions and conditions.
You can approximate a hypothesis test by
examining a confidence interval.
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ONE-PROPORTION Z-INTERVAL
When the conditions are met, we are ready to find the
confidence interval for the population proportion, p.
 The confidence interval is

pˆ  z  SE  pˆ 

where

ˆˆ
SE( pˆ )  pq
n
The critical value, z*, depends on the particular
confidence level, C, that you specify.
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CONFIDENCE INTERVALS AND
HYPOTHESIS TESTS

Construct CI based on sample

If hypothesized value falls in CI,


Fail to Reject the null hypothesis
If hypothesized value is outside of the CI,

Reject the null hypothesis
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CRITICAL Z* AND SIGNIFICANCE LEVEL
FOR TWO-SIDED TEST
α= .20  CI = 80%  z*=1.282
 α= .10  CI = 90% z*=1.645
 α= .05  CI = 95% z*=1.96
 α= .02  CI = 98%z*=2.326
 α= .01  CI = 99% z*=2.576
 α= .001  CI = 99.9% z*=3.29

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EXAMPLE
A magazine reported the results of a random telephone
poll to examine what they use to measure their idea of
success.
 The poll survey 1085 men.
 28 said their measure of success was through work

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1.
2.
SUPPOSE WE WISH TO SEE IF THE FRACTION
HAS FALLEN BELOW THE 5% MARK. WHAT
DOES YOUR 99.9% CI INDICATE
50%?
50%
5% is in the interval
5% is NOT in the interval
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1
2
SUPPOSE WE WISH TO SEE IF THE FRACTION HAS FALLEN
BELOW THE 5% MARK. WHAT DOES YOUR CI INDICATE?
1. 5% is in the interval, there is strong evidence that
25% MORE than 5% of men use work as their measure of
success
2. 5% is in the interval, there is strong evidence that
25% FEWER than 5% of men use work as their measure of
success
3. 5% is NOT in the interval, there is strong evidence that
25%
MORE than 5% of men use work as their measure of
success
25%
4. 5% is NOT in the interval, there is strong evidence that
FEWER than 5% of men use work as their measure of
success
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MAKING ERRORS
When we perform a hypothesis test, we can
make mistakes in two ways:

I.
The null hypothesis is true, but we mistakenly
reject it. (Type I error)
II.
The null hypothesis is false, but we fail to reject
it. (Type II error)
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MAKING ERRORS (CONT.)
Which type of error is more serious depends on
the situation at hand. In other words, the gravity
of the error is context dependent.
 Here’s an illustration of the four situations in a
hypothesis test:

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POWER OF THE TEST

The probability that it correctly rejects a false
null hypothesis.
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A BASKETBALL PLAYER WITH A POOR FOULSHOT RECORD PRACTICES INTENSIVELY DURING
THE OFF-SEASON.



He claims he improved his proficiency from 40% to
50%.
Skeptical, the coach ask him to take 10 shots, and is
surprised that he makes 9 out of 10.
Assume
Ho: p=0.4
 Ha: p>0.4

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IF THE SHOOTER HAS NOT IMPROVED FROM
40%, BUT STILL MANAGES TO MAKE 9 OF 10,
THEN THE COACH WILL THINK
HE HAS 50%
50%
IMPROVED.
WHAT TYPE OF ERROR IS THE COACH MAKING?
1.
2.
Type I – reject null even though it is true
Type II – fail to reject null even though it is false
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1
2
IF THE PLAYER REALLY CAN HIT 50%, AND IT
TAKES AT LEAST 9 OUT OF 10 SUCCESSFUL
SHOTS TO CONVINCE THE COACH.

What’s the power of the test?
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UPCOMING WORK

HW #9 due Sunday

Part 3 of Data Project due Monday

Quiz #5 in class next Wednesday