Download Chapter 7 Hypothesis Testing

Overview
Definition
Hypothesis
in statistics, is a claim or statement about
a property of a population
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Null Hypothesis: H0
 Statement about value of population
parameter
 Must contain condition of equality
 =, , or 
 Test the Null Hypothesis directly
 Reject H0 or fail to reject H0
2
Alternative Hypothesis: Ha
 Must be true if H0 is false
 , <, >
 ‘opposite’ of Null
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Note about Forming Your Own Claims
(Hypotheses)
If you are conducting a study and want
to use a hypothesis test to support your
claim, the claim must be worded so that
it becomes the alternative hypothesis.
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Legal Trial
Hypothesis Test
HO
The defendant
is not guilty
Claim about a
population parameter
HA
The defendant
is guilty
Opposing claim about a
population parameter
The evidence
convinces the
jury to reject
the assumption
of innocence. The
verdict is guilty
The statistic indicates a
rejection of HO, and the
alternate hypothesis is
accepted.
Result
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Test Statistic
a value computed from the sample data that is
used in making the decision about the
rejection of the null hypothesis
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Test Statistic
a value computed from the sample data that is
used in making the decision about the
rejection of the null hypothesis
For large samples, testing claims about
population means
z=
x - µx

n
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Critical Region
Set of all values of the test statistic that
would cause a rejection of the
null hypothesis
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Critical Region
Set of all values of the test statistic that
would cause a rejection of the
null hypothesis
Critical
Region
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Critical Region
Set of all values of the test statistic that
would cause a rejection of the
null hypothesis
Critical
Region
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Critical Region
Set of all values of the test statistic that
would cause a rejection of the
null hypothesis
Critical
Regions
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Significance Level
 denoted by 
 the probability that the
test statistic will fall in the
critical region when the null
hypothesis is actually true.
 common choices are 0.05,
0.01, and 0.10
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Critical Value
Value or values that separate the critical region
(where we reject the null hypothesis) from the
values of the test statistics that do not lead
to a rejection of the null hypothesis
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Critical Value
Value or values that separate the critical region
(where we reject the null hypothesis) from the
values of the test statistics that do not lead
to a rejection of the null hypothesis
Critical Value
( z score )
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Critical Value
Value or values that separate the critical region
(where we reject the null hypothesis) from the
values of the test statistics that do not lead
to a rejection of the null hypothesis
Reject H0
Fail to reject H0
Critical Value
( z score )
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Two-tailed,Right-tailed,
Left-tailed Tests
The tails in a distribution are the
extreme regions bounded
by critical values.
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Two-tailed Test
H0: µ = 100
Ha: µ  100
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Two-tailed Test
H0: µ = 100
 is divided equally between
Ha: µ  100
the two tails of the critical
region
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Two-tailed Test
H0: µ = 100
 is divided equally between
Ha: µ  100
the two tails of the critical
region
Means less than or greater than
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Two-tailed Test
H0: µ = 100
 is divided equally between
Ha: µ  100
the two tails of the critical
region
Means less than or greater than
Reject H0
Fail to reject H0
Reject H0
100
Values that differ significantly from 100
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Right-tailed Test
H0: µ  100
Ha: µ > 100
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Right-tailed Test
H0: µ  100
Ha: µ > 100
Points Right
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Right-tailed Test
H0: µ  100
Ha: µ > 100
Points Right
Fail to reject H0
100
Reject H0
Values that
differ significantly
from 100
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Left-tailed Test
H0: µ  100
Ha: µ < 100
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Left-tailed Test
H0: µ  100
Ha: µ < 100
Points Left
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Left-tailed Test
H0: µ  100
Ha: µ < 100
Points Left
Reject H0
Values that
differ significantly
from 100
Fail to reject H0
100
26
Conclusions
in Hypothesis Testing
always test the null hypothesis
1. Reject the H0
2. Fail to reject the H0
need to formulate correct wording of final
conclusion
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Wording of Final Conclusion
Start
Does the
original claim contain
the condition of
equality
Yes
(Original claim
contains equality
and becomes H0)
No
Do
you reject
H0?.
“There is sufficient
evidence to warrant
(Reject H0) rejection of the claim
that. . . (original claim).”
Yes
No
(Fail to
reject H0)
(Original claim
does not contain
equality and
becomes Ha)
Do
you reject
H0?
Yes
(Reject H0)
“There is not sufficient
evidence to warrant
rejection of the claim
that. . . (original claim).”
“The sample data
supports the claim that
. . . (original claim).”
No
(Fail to
reject H0)
(This is the
only case in
which the
original claim
is rejected).
(This is the
only case in
which the
original claim
is supported).
“There is not sufficient
evidence to support
the claim
that. . . (original claim).”
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Accept versus Fail to Reject
some texts use “accept the null
hypothesis
we are not proving the null hypothesis
sample evidence is not strong enough
to warrant rejection (such as not
enough evidence to convict a suspect)
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Definition
Power of a Hypothesis Test
is the probability (1 - ) of rejecting a
false null hypothesis, which is
computed by using a particular
significance level  and a particular
value of the mean that is an alternative
to the value assumed true in the null
hypothesis.
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Assumptions
for testing claims about population means
1) The sample is a simple random
sample.
2) The sample is large (n > 30).
a) Central limit theorem applies
b) Can use normal distribution
3) If  is unknown, we can use sample
standard deviation s as estimate for .
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Traditional (or Classical) Method of
Testing Hypotheses
Goal
Identify a sample result that is significantly
different from the claimed value
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The traditional (or classical) method
of hypothesis testing converts the
relevant sample statistic into a test
statistic which we compare to the
critical value.
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Hypotheses Testing
5 Step Process
1. State the hypotheses
2. Decide on a model.
3. Determine the endpoints of the rejection region
and state the decision rule.
4. Compute the test statistic
5. State the conclusion
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Test Statistic for Claims about µ when n > 30
z=
x - µx

n
Test Statistic for Claims about µ when n < 30
t=
x - µx

n
35
Decision Criterion
Reject the null hypothesis if the test
statistic is in the critical region
Fail to reject the null hypothesis if the test
statistic is not in the critical region
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Wording of Final Conclusion
FIGURE 7-4
Start
Does the
original claim contain
the condition of
equality
Yes
(Original claim
contains equality
and becomes H0)
No
Do
you reject
H0?.
“There is sufficient
evidence to warrant
(Reject H0) rejection of the claim
that. . . (original claim).”
Yes
No
(Fail to
reject H0)
(Original claim
does not contain
equality and
becomes Ha)
Do
you reject
H0?
Yes
(Reject H0)
“There is not sufficient
evidence to warrant
rejection of the claim
that. . . (original claim).”
“The sample data
supports the claim that
. . . (original claim).”
No
(Fail to
reject H0)
(This is the
only case in
which the
original claim
is rejected).
(This is the
only case in
which the
original claim
is supported).
“There is not sufficient
evidence to support
the claim
that. . . (original claim).”
37
Example: Given a
data set of 106 healthy body temperatures,
where the mean was 98.2o and s = 0.62o , at the 0.05 significance level,
test the claim that the mean body temperature of all healthy adults is
equal to 98.6o.
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Example: Given a
data set of 106 healthy body temperatures,
where the mean was 98.2o and s = 0.62o , at the 0.05 significance level,
test the claim that the mean body temperature of all healthy adults is
equal to 98.6o.
Steps:
1)
2)
State the hypotheses
H0 :  = 98.6o
Ha :   98.6o
Determine the model
Two tail Z test, n > 30
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3) Determine the Rejection Region
 = 0.05
/2 = 0.025 (two tailed test)
0.4750
0.025
z = - 1.96
0.4750
0.025
1.96
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4) Compute the test statistic
z = x-µ =
 n
98.2 - 98.6
0.62 106
= - 6.64
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5) State the Conclusion
Sample data:
x = 98.2o
or
z = - 6.64
Reject
H0: µ = 98.6
Fail to Reject
H0: µ = 98.6
z = - 1.96
µ = 98.6
Reject
H0: µ = 98.6
z = 1.96
or z = 0
z = - 6.64
REJECT H0
There is sufficient evidence to
warrant rejection of claim that
the mean body temperatures of
healthy adults is equal to 98.6o.
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Assumptions
for testing claims about population means
1) The sample is a simple random sample.
2) The sample is small (n  30).
3) The value of the population
standard deviation  is unknown.
4) The sample values come from a
population with a distribution that is
approximately normal.
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Test Statistic
for a Student t-distribution
x -µx
t= s
n
Critical Values
Found in Table A-3
Degrees of freedom (df) = n -1
Critical t values to the left of the mean are
negative
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Important Properties of the
Student t Distribution
1. The Student t distribution is different for different sample sizes (see
Figure 6-5 in Section 6-3).
2. The Student t distribution has the same general bell shape as the
normal distribution; its wider shape reflects the greater variability that
is expected with small samples.
3. The Student t distribution has a mean of t = 0 (just as the standard
normal distribution has a mean of z = 0).
4. The standard deviation of the Student t distribution varies with the
sample size and is greater than 1 (unlike the standard normal
distribution, which has a  = 1).
5. As the sample size n gets larger, the Student t distribution get closer to
the normal distribution. For values of n > 30, the differences are so
small that we can use the critical z values instead of developing a
much larger table of critical t values. (The values in the bottom row of
Table A-3 are equal to the corresponding critical z values from the
normal distributions.)
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Figure 7-11
Choosing between the Normal and Student
t-Distributions when Testing a Claim about a Population Mean µ
Start
Use normal distribution with
Is
n > 30
?
Yes
Z
(If  is unknown use s instead.)
No
Is the
distribution of
the population essentially
normal ? (Use a
histogram.)
x - µx
/ n
No
Yes
Is 
known
?
No
Use nonparametric methods,
which don’t require a normal
distribution.
Use normal distribution with
Z
x - µx
/ n
(This case is rare.)
Use the Student t distribution
with
x - µx
t  s/
n
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The larger Student t critical value
shows that with a small sample,
the sample evidence must be more
extreme before we consider the
difference is significant.
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A company manufacturing rockets claims to use an
average of 5500 lbs of rocket fuel for the first 15 seconds of
operation. A sample of 6 engines are fired and the mean fuel
consumption is 5690 lbs with a sample standard deviation of
250 lbs. Is the claim justified at the 5% level of significance?
1. HO: µ = 5500 HA: µ  5500
2. Two tail t test, n < 30, unknown
population standard deviation
-2.571
1.862
3. t critical for 5% for a two tail test with 5 d.f. is 2.571
4. t 
2.571
x -μ
5690  5500

 1.862
s n
250 6
5. Fail to reject HO, there is no evidence at the .05 level
that the average fuel consumption is different from µ = 5500 lbs
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P-Value Method
Table A-3 includes only selected values
of 
Specific P-values usually cannot be
found
Use Table to identify limits that contain
the P-value
Some calculators and computer
programs will find exact P-values
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P-Value Method
of Testing Hypotheses
very similar to traditional method
key difference is the way in which we
decide to reject the null hypothesis
approach finds the probability (P-value) of
getting a result and rejects the null
hypothesis if that probability is very low
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P-Value Method
of Testing Hypotheses
Definition
P-Value (or probability value)
the probability that the test statistic is as far or
farther from  if the null hypothesis is true
The attained significance level of a hypothesis
test is the P value of its test statistic
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P-Value Method
of Testing Hypotheses
Guidelines for rejecting HO based
on the P value:
If P < , then reject HO and
accept HA
If P > , then reserve judgement
about HO
52
P-value
Interpretation
Small P-values
(such as 0.05 or
lower)
Unusual sample results.
Significant difference from the
null hypothesis
Large P-values
(such as above
0.05)
Sample results are not unusual.
Not a significant difference from
the null hypothesis
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Figure 7-8
Finding P-Values
Start
What
type of test
?
Left-tailed
Right-tailed
Two-tailed
Left
P-value = area
to the left of the
test statistic
P-value
µ
Test statistic
Is
the test statistic
to the right or left of
center
?
P-value = twice
the area to the left
of the test statistic
P-value is twice
this area
µ
Test statistic
Right
P-value = twice
the area to the right
of the test statistic
P-value = area
to the right of the
test statistic
P-value is twice
this area
P-value
µ
µ
Test statistic
Test statistic
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Testing Claims with
Confidence Intervals
A confidence interval estimate of a
population parameter contains the likely
values of that parameter. We should
therefore reject a claim that the population
parameter has a value that is not included in
the confidence interval.
55
Testing Claims
with Confidence Intervals
Claim: mean body temperature = 98.6º,
where n = 106, x = 98.2º and s = 0.62º
 95% confidence interval of 106 body temperature
data (that is, 95% of samples would contain true
value µ )
 98.08º < µ < 98.32º
 98.6º is not in this interval
 Therefore it is very unlikely that µ = 98.6º
 Thus we reject claim µ = 98.6º
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Underlying Rationale of
Hypotheses Testing
 If, under a given observed assumption, the
probability of getting the sample is exceptionally
small, we conclude that the assumption is
probably not correct.
 When testing a claim, we make an assumption
(null hypothesis) that contains equality. We then
compare the assumption and the sample results
and we form one of the following conclusions:
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Underlying Rationale of
Hypotheses Testing
If the sample results can easily occur when the
assumption (null hypothesis) is true, we
attribute the relatively small discrepancy
between the assumption and the sample results
to chance.
If the sample results cannot easily occur when
that assumption (null hypothesis) is true, we
explain the relatively large discrepancy between
the assumption and the sample by concluding
that the assumption is not true.
58
Type I Error
The mistake of rejecting the null hypothesis
when it is true.
 (alpha) is used to represent the probability
of a type I error
Example: Rejecting a claim that the mean
body temperature is 98.6 degrees when the
mean really does equal 98.6
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Type II Error
the mistake of failing to reject the null
hypothesis when it is false.
ß (beta) is used to represent the probability of
a type II error
Example: Failing to reject the claim that the
mean body temperature is 98.6 degrees when
the mean is really different from 98.6
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Type I and Type II Errors
True State of Nature
We decide to
reject the
null hypothesis
The null
hypothesis is
true
The null
hypothesis is
false
Type I error
(rejecting a true
null hypothesis)

Correct
decision
Correct
decision
Type II error
(rejecting a false
null hypothesis)

Decision
We fail to
reject the
null hypothesis
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Controlling Type I and Type II Errors
For any fixed , an increase in the sample
size n will cause a decrease in 
For any fixed sample size n , a decrease in 
will cause an increase in . Conversely, an
increase in  will cause a decrease in  .
To decrease both  and , increase the
sample size.
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