Download Case Ⅱ : Large

Probability and Statistics
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Case Ⅱ : Large-Sample Test
When the sample size is large, we can use the test statistic:
X  μo
Z
S/ n
which has approximately a standard normal distribution when Ho is true.
The use of rejection region given previously for case Ⅰ then the results in the test
procedures for which the significance level is approximately (rather than exactly) α .
The rule of thumb n>40 will again be used to characterize a large sample size.
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β and Sample Size Determination
Determination of β and necessary sample size for
these large-sample tests can be based either on
specifying a plausible value of σ and using the
case Ⅰ formulas or on using the curves to be
introduced shortly in connection with case Ⅲ .
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Case Ⅲ : A Normal Population Distribution
When the sample size n is small, the CIT can no longer be
invoked to justify the use the large-sample test.
But when x̅ is the mean of a random sample of size n from
a normal distribution with the mean μ , the rv :
T
X μ
S/ n
has a t distribution with n-1 degrees of freedom.
So we can use the test statistic:
X  μo
T
S/ n
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The One-Sample t Test
H0 : μ=μo
Null hypothesis:
Test statistic value:
Alternative Hypothesis
Ha : μ>μo
Ha : μ<μo
Ha : μ≠μo
x  μo
t
S/ n
Rejection Region for Level α Test
t≥tα,n-1 (upper-tailed test)
t≤-tα,n-1 (lower-tailed test)
either t≥tα/2,n-1 or t≤-tα/2,n-1 (two-tailed test)
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β and Sample Size Determination
The calculation of β(μ') for the t test is much less
straightforward.
This is because the distribution of the test statistic is
quite complicated when Ho is false and Ha is true.
This must be done numerically, but fortunately it
has been done by research statisticians for both oneand two-tailed t test.
The results are summarized in the graphs of β that
appears in Appendix Table A.17 [Probability and
statistics for engineering and sciences, Jay L.
Devore].
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3. Test Concerning a Population Proportion
Large-Sample Tests
Large-sample tests concerning p are a special case
of the more general large-sample procedures for a
parameter θ.
Let θ̂ be an estimator of θ that is unbiased and
approximately a normal distribution. Ho : θ=θo .
Suppose that when Ho is true, the standard
deviation θ̂ of σ θˆ , involves no unknown
parameter.
ˆ θ
θ
o
Then the test statistic is : Z 
σ θˆ
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The estimator pˆ  X n is unbiased, has σ  p1  p n
approximately a normal distribution, and the
standard deviation is
.
So the test statistic is:
pˆ
pˆ  p o
Z
p o 1  p o  n
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Null hypothesis:
Ho : p=po
Test statistic value:
Alternative Hypothesis
Ha : p>po
Ha : p<po
Ha : p≠po
pˆ  p o
z
p o 1  p o  n
Rejection Region for Level α Test
z>zα (upper-tailed test)
z<zα (lower-tailed test)
either z≥zα/2 or z≤-zα/2 (two-tailed test)
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β and Sample Size Determination
Alternative Hypothesis
Ha : p >po
Ha : p <po
Ha : p ≠ po
β(μ')
 pˆ  p o  z a p o 1  p o  n 


p1  p n


 pˆ  p o  z a p o 1  p o  n 


p1  p n


 pˆ  p o  z a/2 p o 1  p o  n 
 pˆ  p o  z a/2 p o 1  p o  n 

  

p1  p n
p1  p n




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The sample size n for which the level α test also
satisfies β(μ') =β is
  z p 1  p   z p1  p  2
o

 a o

 
p  p o

n
2
 z a/2 p o 1  p o   z p1  p 


p  p o


one - tailed test
two - tailed test
(an approxomat e solution)
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Small-Sample Tests
Test procedures when the sample size n is small are
based on the binominal distribution rather than the
normal distribution.
Consider the alternative hypothesis Ha : p>po and again
let X be the number of successes in the sample. Then the
X is the test statistic.
When Ho is true, P( typeⅠ errors) =1-B(c-1; n, po).
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Because X has a discrete probability distribution, it is
usually not possible to find a value c for which
P(
typeⅠ errors) is exactly the desired significance level α .
Instead, the largest rejection region of the form {c,c+1,…,n}
satisfying 1-B(c-1; n, po) ≤ α is used.
The procedures for Ha : p<po and Ha : p≠po are constructed in
a similar manner.
And β(μ') is the result of a straightforward binominal
probability calculation.
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4. P-Values
A P-value conveys much information about the strength of
evidence against Ho and allow an individual decision maker
to draw a conclusion at any specified level α .
The P-value is the smallest level of significance at which Ho
would be rejected when a specified test procedure is used on
a given data set.
Once the P-value has been determined, the conclusion at any
particular level α results from comparing the P-value to α :
1. P-value ≤ α → rejected Ho at level α .
2. P-value ＞ α → do not rejected Ho at level α .
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P-value is customary to call the data significant
when Ho is rejected and not significant otherwise.
The P-value is then the smallest level at which the
data is significant. An easy way to visualize the
comparison of the P-value with the chosen α is to
draw a picture like that of Figure 8.6(page 347).
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DEFINITION
The P-value is the probability, calculated
assuming Ho is true, of obtaining a test statistic
value at least as contradictory to Ho as the value
that actually resulted.
The smaller the P-value, the more contradictory is
the data to Ho .
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P-values for z Test
Let z = test statistic value for z test.
Then
P-value:
 1  z 

P   z 
21    z 

for an upper - tailed test
for an lower - tailed test
for a two - tailed test
Each of these is the probability of getting a value at least as extreme as what was obtained
(assuming Ho is true).
The three cases are illustrated in Figure 8.7 .
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P-values for t Tests
Let t = test statistic value for t test
Then P-value for a t test will be a t curve area. (Figure
8.8)
 area in upper tail

P   area in lower tail
sum of area in two tails

upper - tailed test
lower - tailed test
two - tailed test
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5. Some Comments on Selecting a Test Procedure
Once the experimenter has decided on the
question of interest and method for
gathering data, construction of an
appropriate test procedure consists of three
distinct steps:
1. Specify a test statistic.
2. Decide on the general form of the rejection
region.
3. Select the specific numerical critical value or
values that will separate the rejection region from
the acceptance region.
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Issues to be considered in carrying out Steps 1-3
encompass the following questions:
1. What are the practical implications and consequences
of choosing a particular level of significance once the
other aspects of test procedure have been determined ?
2. Does there exits a general principle, not dependent
just on intuition, that can be used to obtain best or good
test procedures ?
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3. When two or more tests are appropriate in given
situation, how can the test be compared to decided
which should be used ?
4. If a test is derived under specific assumptions
about the distribution or population be sampled,
how well will the test procedure work when the
assumptions are violated ?
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