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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
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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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HO The defendant Claim about a is not guilty population parameter
HA The defendant Opposing claim about ais guilty population parameter
Result The evidence The statistic indicates aconvinces the rejection of HO, and thejury to reject alternate hypothesis isthe assumption accepted.of innocence. Theverdict is guilty
Legal Trial Hypothesis Test
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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 RegionSet of all values of the test statistic that
would cause a rejection of the null hypothesis
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Critical RegionSet of all values of the test statistic that
would cause a rejection of thenull hypothesis
CriticalRegion
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Critical RegionSet of all values of the test statistic that
would cause a rejection of the null hypothesis
CriticalRegion
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Critical RegionSet of all values of the test statistic that
would cause a rejection of the null hypothesis
CriticalRegions
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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 ValueValue 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
Critical Value( z score )
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
Critical Value( z score )
Fail to reject H0Reject H0
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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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 TestH0: µ = 100
Ha: µ 100
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Two-tailed TestH0: µ = 100
Ha: µ 100 is divided equally between
the two tails of the critical region
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Two-tailed TestH0: µ = 100
Ha: µ 100
Means less than or greater than
is divided equally between the two tails of the critical
region
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Two-tailed TestH0: µ = 100
Ha: µ 100
Means less than or greater than
100
Values that differ significantly from 100
is divided equally between the two tails of the critical
region
Fail to reject H0Reject H0 Reject H0
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Right-tailed TestH0: µ 100
Ha: µ > 100
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Right-tailed TestH0: µ 100
Ha: µ > 100
Points Right
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Right-tailed TestH0: µ 100
Ha: µ > 100
Values that differ significantly
from 100100
Points Right
Fail to reject H0 Reject H0
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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 < 100Points Left
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Left-tailed Test
H0: µ 100
Ha: µ < 100
100
Values that differ significantly
from 100
Points Left
Fail to reject H0Reject H0
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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
Does theoriginal claim contain
the condition ofequality
Doyou reject
H0?.
Yes
(Original claim
contains equality
and becomes H0)
No(Original claimdoes not containequality and
becomes Ha)
Yes
(Reject H0)
“There is sufficientevidence to warrantrejection of the claimthat. . . (original claim).”
“There is not sufficientevidence to warrantrejection of the claimthat. . . (original claim).”
“The sample datasupports the claim that . . . (original claim).”
“There is not sufficientevidence to support the claimthat. . . (original claim).”
Doyou reject
H0?
Yes
(Reject H0)
No(Fail to
reject H0)
No(Fail to
reject H0)
(This is theonly case inwhich theoriginal claimis rejected).
(This is theonly case inwhich theoriginal claimis supported).
Start
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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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Assumptionsfor 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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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
Hypotheses Testing5 Step Process
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Test Statistic for Claims about µ when n > 30
n
x - µxz =
n
x - µxt =
Test Statistic for Claims about µ when n < 30
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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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FIGURE 7-4 Wording of Final Conclusion
Does theoriginal claim contain
the condition ofequality
Doyou reject
H0?.
Yes
(Original claim
contains equality
and becomes H0)
No(Original claimdoes not containequality and
becomes Ha)
Yes
(Reject H0)
“There is sufficientevidence to warrantrejection of the claimthat. . . (original claim).”
“There is not sufficientevidence to warrantrejection of the claimthat. . . (original claim).”
“The sample datasupports the claim that . . . (original claim).”
“There is not sufficientevidence to support the claimthat. . . (original claim).”
Doyou reject
H0?
Yes
(Reject H0)
No(Fail to
reject H0)
No(Fail to
reject H0)
(This is theonly case inwhich theoriginal claimis rejected).
(This is theonly case inwhich theoriginal claimis supported).
Start
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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.
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H0 : = 98.6o
Ha : 98.6o
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) State the hypotheses
2) Determine the model
Two tail Z test, n > 30
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3) Determine the Rejection Region
z = - 1.96 1.96
0.0250.025
0.47500.4750
= 0.05/2 = 0.025 (two tailed test)
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4) Compute the test statistic
z = = = - 6.64x - µ 98.2 - 98.6
n0.62 106
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RejectH0: µ = 98.6
RejectH0: µ = 98.6
Sample data:
x = 98.2o
or
z = - 6.64
z = - 1.96 µ = 98.6or z = 0
z = 1.96
z = - 6.64
Fail to RejectH0: µ = 98.6
5) State the Conclusion
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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Assumptionsfor 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
Critical ValuesFound in Table A-3
Degrees of freedom (df) = n -1
Critical t values to the left of the mean are negative
t = x -µxsn
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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
Is n > 30
?
Is thedistribution of
the population essentiallynormal ? (Use a
histogram.)
No
Yes
Yes
No
No
Is known
?
Use normal distribution with
x - µx
/ nZ
(If is unknown use s instead.)
Use nonparametric methods, which don’t require a normal distribution.
Use normal distribution with
x - µx
/ nZ
(This case is rare.)
Use the Student t distributionwith x - µx
s/ nt
Start
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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 anaverage of 5500 lbs of rocket fuel for the first 15 seconds ofoperation. A sample of 6 engines are fired and the mean fuelconsumption is 5690 lbs with a sample standard deviation of250 lbs. Is the claim justified at the 5% level of significance?
2. Two tail t test, n < 30, unknown population standard deviation
3. t critical for 5% for a two tail test with 5 d.f. is 2.571
86216250
550056904 .
ns
-μx. t
5. Fail to reject HO, there is no evidence at the .05 level that the average fuel consumption is different from µ = 5500 lbs
2.571-2.571 1.862
1. HO: µ = 5500 HA: µ 5500
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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 Methodof 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 Methodof Testing Hypotheses
DefinitionP-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 Methodof 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
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Unusual sample results. Significant difference from the null hypothesis
Small P-values (such as 0.05 or lower)
P-value Interpretation
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
Isthe test statistic
to the right or left ofcenter
?
P-value = areato the left of the test statistic
P-value = twice the area to the left of the test statistic
P-value = areato the right of the test statistic
Left-tailed Right-tailed
RightLeft
Two-tailed
P-value = twice the area to the right of the test statistic
Whattype of test
?
µ µ µ µ
P-value P-value is twicethis area
P-value is twicethis area
P-value
Test statistic Test statistic Test statistic Test statistic
Start
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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.
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Testing Claims with Confidence Intervals
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
Claim: mean body temperature = 98.6º, where n = 106, x = 98.2º and s = 0.62º
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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.
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Type I ErrorThe 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 Errorthe 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
We fail to
reject the
null hypothesis
The null
hypothesis is
true
The null
hypothesis is
false
Type I error
(rejecting a true
null hypothesis)
Type II error
(rejecting a false
null hypothesis)
Correct
decision
Correct
decision
Decision
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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.
