Transcript Chapter 4

Chapter 8
Inferences Based on a Single
Sample: Tests of Hypothesis
The Elements of a Test of
Hypothesis
7 elements
1.
2.
3.
4.
5.
6.
7.
The Null hypothesis
The alternate, or research hypothesis
The test statistic
The rejection region
The assumptions
The Experiment and test statistic calculation
The Conclusion
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The Elements of a Test of
Hypothesis
Does a manufacturer’s pipe meet building
code?
Null hypothesis – Pipe does not meet code
(H0): < 2400
Alternate hypothesis – Pipe meets
specifications
(Ha): > 2400
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The Elements of a Test of
Hypothesis
Test statistic to be used z  x  2400  x  2400
x

n
Rejection region
Determined by Type I error, which is the probability of
rejecting the null hypothesis when it is true, which is .
Here, we set =.05
Region is z>1.645, from
z value table
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The Elements of a Test of
Hypothesis
Assume that s is a good approximation of 
Sample of 60 taken, x  2460 , s=200
Test statistic is z  x  2400  2460 2400  60  2.12
s
n
200 50
28.28
Test statistic lies in rejection region,
therefore we reject H0 and accept Ha that the
pipe meets building code
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The Elements of a Test of
Hypothesis
Type I vs Type II Error
Conclusions and Consequences for a Test of Hypothesis
True State of Nature
Conclusion
H0 True
Ha True
Accept H0
(Assume H0 True)
Correct decision
Type II error
(probability )
Reject H0
(Assume Ha True)
Type I error
(probability )
Correct decision
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The Elements of a Test of
Hypothesis
1. The Null hypothesis – the status quo. What
we will accept unless proven otherwise.
Stated as H0: parameter = value
2. The Alternative (research) hypothesis (Ha) –
theory that contradicts H0. Will be accepted if
there is evidence to establish its truth
3. Test Statistic – sample statistic used to
determine whether or not to reject Ho and
accept Ha
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The Elements of a Test of
Hypothesis
4. The rejection region – the region that will lead to
H0 being rejected and Ha accepted. Set to
minimize the likelihood of a Type I error
5. The assumptions – clear statements about the
population being sampled
6. The Experiment and test statistic calculation –
performance of sampling and calculation of
value of test statistic
7. The Conclusion – decision to (not) reject H0,
based on a comparison of test statistic to
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rejection region
Large-Sample Test of Hypothesis
about a Population Mean
Null hypothesis is the status quo, expressed in one
of three forms
H0:  = 2400
H0:  ≤ 2400
H0:  ≥ 2400
It represents what must be accepted if the
alternative hypothesis is not accepted as a result
of the hypothesis test
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Large-Sample Test of Hypothesis
about a Population Mean
Alternative hypothesis can take one of 3 forms:
One-tailed, upper tail
Ha: <2400
One-tailed, upper tail
Ha: >2400
Two-tailed
Ha: 2400
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Large-Sample Test of Hypothesis
about a Population Mean
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Large-Sample Test of Hypothesis
about a Population Mean
If we have: n=100, x = 11.85, s = .5, and we
want to test if   12 with a 99% confidence
level, our setup would be as follows:
H0:  = 12
Ha:   12
x  12
Test statistic z 
x
Rejection region z < -2.575 or z > 2.575
(two-tailed)
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Large-Sample Test of Hypothesis
about a Population Mean
CLT applies, therefore no assumptions
about population are needed
Solve
x  12 x 12 11.85 12 11.85 12  .15
z
x


n

 100

s 10

.5 10
 .3
Since z falls in the rejection region, we
conclude that at .01 level of significance the
observed mean differs significantly from 12
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Observed Significance Levels: pValues
The p-value, or observed significance level,
is the smallest  that can be set that will
result in the research hypothesis being
accepted.
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Observed Significance Levels: pValues
Steps:
Determine value of test statistic z
The p-value is the area to the right of z if Ha
is one-tailed, upper tailed
The p-value is the area to the left of z if Ha is
one-tailed, lower tailed
The p-valued is twice the tail area beyond z
if Ha is two-tailed.
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Observed Significance Levels: pValues
When p-values are used, results are
reported by setting the maximum  you are
willing to tolerate, and comparing p-value to
that to reject or not reject H0
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Small-Sample Test of Hypothesis
about a Population Mean
When sample size is small (<30) we use a different
sampling distribution for determining the rejection
region and we calculate a different test statistic
The t-statistic and t distribution are used in cases
of a small sample test of hypothesis about 
All steps of the test are the same, and an
assumption about the population distribution is
now necessary, since CLT does not apply
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Small-Sample Test of Hypothesis
about a Population Mean
Small-Sample Test of Hypothesis about 
One-Tailed Test
H 0:
H a:
  0
  0
Test Statistic:
(or Ha:
t  t
  0
)
x  0
t
s n
Rejection region:
(or
Two-Tailed Test
  0
H a:
  0
Test Statistic:
t  t
when Ha:
H 0:
x  0
t
s n
Rejection region:
t  t 2
  0
where t and t/2 are based on (n-1) degrees of freedom
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Large-Sample Test of Hypothesis
about a Population Proportion
Large-Sample Test of Hypothesis about p
One-Tailed Test
H0:
p  p0
Ha:
p  p0
Two-Tailed Test
H0:
(or Ha:
Test Statistic:
p  p0
)
pˆ  p0
z
 pˆ
where, according to H0,
Rejection region: z   z

(or z  z when p  p0

p  p0
Ha: p 
p0
Test Statistic:
pˆ  p0
z
p
 pˆ  p0 q0 n
Rejection region:
and
q0  1  p0
z  z 2
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Large-Sample Test of Hypothesis
about a Population Proportion
Assumptions needed for a Valid Large-Sample
Test of Hypothesis for p
•A random sample is selected from a binomial
population
•The sample size n is large (condition satisfied if
p0  3 pˆ falls between 0 and 1
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Calculating Type II Error
Probabilities: More about 
Type II error is associated with , which is
the probability that we will accept H0 when
Ha is true
Calculating a value for  can only be done if
we assume a true value for 
There is a different value of  for every value
of 
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Calculating Type II Error
Probabilities: More about 
Steps for calculating  for a Large-Sample Test about 
1. Calculate the value(s) of x corresponding to the
borders of the rejection region using one of the
following:
Upper-tailed test:
Lower-tailed test:
Two-tailed test:
s
n
s
x 0  0  z  x  0  z
n
s
x 0 L  0  z  x  0  z
n
s
x 0U  0  z  x  0  z
n
x 0  0  z  x  0  z
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Calculating Type II Error
Probabilities: More about 
2.
3.
Specify the value of a in Ha for
which  is to be calculated.
Convert border values of x 0 to
z values using the mean a, and
the formula
z
4.
x 0  a
x
Sketch the alternate distribution,
shade the area in the acceptance
region and use the z statistics and
table to find the shaded area, 
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Calculating Type II Error
Probabilities: More about 
The Power of a test – the probability that the
test will correctly lead to the rejection of H0
for a particular value of  in Ha. Power is
calculated as 1- .
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Tests of Hypothesis about a
Population Variance
Hypotheses about the variance use the ChiSquare distribution and statistic
n  1s
The quantity 
has a sampling
distribution that follows the
chi-square distribution
assuming the population the
sample is drawn from is
normally distributed.
2
2
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Tests of Hypothesis about a
Population Variance
Small-Sample Test of Hypothesis about  2
One-Tailed Test
H 0:
Ha:
Two-Tailed Test
 2   02
 2   02(or Ha:
Test Statistic:
 2  ) 02
2


n

1
s
2 
 02
2  2 
2
2
when Ha:    0
Rejection region:
(or
 2   2
 2   02
2
2
Ha:   0
H0 :
1
Test Statistic:
2


n

1
s
2 
 02
2
2



Rejection region:
1 2 
2
2



 2 
Or
where  02 is the hypothesized variance and the distribution of  is based
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on (n-1) degrees of freedom
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