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WARM – UP
Are twin births on the rise in the United States? In 2001 3% of
all births produced twins. In 2009 Data from an SRS of 469
mothers in a large city found that 4.05% resulted in sets of
twins being born. Test the appropriate hypothesis and state
your conclusions and assumptions.
p = The true proportion of twins births in the United States.
0.0405 0.03
pˆ p z
One
z
H0: p = 0.03
0.03 1 0.03
Proportion
p1 p
Ha: p > 0.03 z – Test
469
n
1.3345
1. SRS – The data was collected randomly
2. Appr. Normal: 469 · (0.03) ≥ 10 AND 469 · (1 – 0.03) ≥ 10
3. Population of US births is ≥ 10 · (469)
P Value
P Z
1.3345
normalcdf 1.3345, E 99
0.0910
Since the P-Value is NOT less than α = 0.05 we Fail to REJECT H0 .
There is no evidence to suggest that Twin births are on the rise in the
US.
ONE-Tailed Test –
Ha: p > # or
TWO-Tailed Test :
Ha: p ≠ #
Ha: p < #
H0 = NO CHANGE in Population, No effect, THE NORM, or
THE EXPECTATION.
Chapter 21
STATISTICAL ERRORS
If we Reject H0 when in fact H0 is TRUE this is a
TYPE I ERROR = (false positive).
(α = probability of committing a Type I Error.)
If we Fail to Reject H0 when in fact H0 is FALSE (Ha is True)
this is a TYPE II ERROR = (false negative).
(β = probability of committing a Type II Error.)
Decision
based on
sample
H0 is True
H0 is False
Reject H0
Type I
Error
Correct
Decision
Fail to
Reject H0
Correct
Decision
Type II
Error
In each example List the Hypothesis and then
Describe the Type I and Type II Error. Page 492
11. In 2003 the Dept. of Commerce reported that 68.2% of
American families own homes. In order to encourage home
ownership the city offers a plan of tax breaks for first time
home buyers. Since the plan cost the city tax revenues,
they will continue using it only if there is strong evidence
that the rate of home ownership is increasing.
H0: Home ownership remains Unchanged Not rising.
Ha: Home ownership is Increasing.
TYPE I ERROR = The city feels that home ownership is
on the rise but in fact the tax breaks are not helping. The
city forgoes tax revenues for nothing.
TYPE II ERROR = The city feels homeownership remains
unchanged (68.2%), but it is increasing. They may retract the
tax breaks feeling they are useless.
In each example List the Hypothesis and then
Describe the Type I and Type II Error.
12. Recently a group of doctors devised a quick test to test for
the Alzheimer in the population of senior citizens. A patient
that tested positive would then go through a more expensive
and time consuming battery of tests and medical diagnosis.
H0: The Patient does not have Alzheimer (Healthy)
Ha: The Patient has Alzheimer
TYPE I ERROR = The Patient is diagnosed with
Alzheimer but does NOT have the disease. The Patient
waste time and money… and stress. FALSE POSITIVE
TYPE II ERROR = The patient is diagnosed as healthy
when they do have Alzheimer. The patient does not
receive the beneficial treatment. FALSE NEGATIVE
Wednesday’s Quiz
1. Definition P-Value:
Probability of obtaining statistics or ones more
extreme, given H0 is true.
2. Definition of Confidence Level:
In Repeating Sampling C% of
the constructed intervals will contain the True Parameter.
3. Definition of Central Limit Theorem: A large random sample
will produce an approximately normal distribution.
4. Proportion Assumptions/Conditions:
5. Decision/Conclusions based on p-value:
6. Definition of Type I and II Errors:
7. Construct and Interpret Confidence Interval
8. Perform Significance Test.
Homework- Page 492: 12a-d, 13
In each example List the Hypothesis and then
Describe the Type I and Type II Error.
13. Clean Air standards require that vehicle exhausts not
exceed specified limits. Many states require annual test.
State Regulators sample shops all the time. They will
revoke the shop’s license if they find that they are certifying
vehicles that do not meet the standards.
H0: The Shop is certifying cars that are meeting the standards.
Ha: The Shop is certifying cars that DO NOT meet the standards.
TYPE I ERROR = The state regulator decides that the
shop is not meeting the standards when they really are.
The shop is fined and license revoked.
TYPE II ERROR = The state regulators decide that the
shop is following standards when it is NOT.
Consequently the air is being polluted.
Power – The Probability that a significance test will correctly
reject a false H0. “Doing the right thing.”
Power = 1 – β (Type II Prob.)
Increasing sample sizes decrease Type II errors and
consequently increase the Power of the test
H0 is True
Decision
based on
sample
Reject H0
Type I
Error
H0 is False
Correct
Decision
POWER
Fail to
Reject H0
Correct
Decision
Type II
Error
Chapter 21 – More about Tests
• The alpha level,
level.
α =0.05,is also called the significance
• Common alpha levels are 0.05, 0.10, and 0.01.
– When we reject the null hypothesis, we say that the
test is “significant at that level.”
Back to the Basics…
Are twin births on the rise in the United States? In 2001 3% of all births
produced twins. In 2005 Data from an SRS of 469 mothers in a large city
found that 19 sets of twins were born to 469 mothers. Test the appropriate
hypothesis and state your conclusions and assumptions.
Because our sample size is 469 we get a Sampling
Distribution of…
S .D.
.03(1 .03)
469
.0079
P-value = 0.0910
N(0.03, 0.0079)
.03 .038 .046 .054
pˆ 19 / 469 0.0405
Definintion of P-value: The Probability of obtaining this
statistic, p = 19/469 = 0.0405, given that the Null hypothesis
of μ = 0.03 is TRUE.
.006
.014
.022
Back to the Basics…
What happens if you increase the sample size? Lets say that
we obtain a sample of 2000 births of which 81 result in twins.
Because our sample size is 2000 we get a Sampling
Distribution of…
S .D.
.03(1 .03)
2000
.0038
P-value = 0.0030
N(0.03, 0.0038)
.019
.022
.026
.03
.034
.038
.041
pˆ 81/ 2000 0.0405
Definintion of P-value: The Probability of obtaining this
statistic, p = 81/2000 = 0.0405, given that the Null hypothesis
of μ = 0.03 is TRUE.
Decreasing α means that you would need more evidence to
Reject Ho. But α has an INVERSE relationship with β. This
means that if you increase α then you decrease β. Since
Power is 1 – β then an increase in α makes for an increase in
Power (Failing to reject a TRUE hypothesis.)