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Chapter 8.
Inferences
on a Population Mean
8.1 Confidence Intervals
8.2 Hypothesis Testing
8.3 Summary
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8.1 Confidence Intervals
8.1.1 Confidence Interval Construction(1/8)
•
Confidence Intervals
– A confidence interval for an unknown parameter θ is an interval that
contains a set of plausible values of the parameter.
– It is associated with a confidence level 1-, which measures the
probability that the confidence interval actually contains the
unknown parameter value.
– Confidence levels of 90%, 95%, and 99% are typically used.
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8.1.1 Confidence Interval Construction(2/8)
• Inferences on a Population Mean
– Inference methods on a population mean based upon the tprocedure are appropriate for large sample sizes n ≥ 30 and
also for small sample sizes as long as the data can
reasonably be taken to be approximately normally
distributed.
– Nonparametric techniques (Chapter 15) can be employed
for small sample sizes with data that are clearly not
normally distributed.
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8.1.1 Confidence Interval Construction(3/8)
•
Two-Sided t-Interval
– A confidence interval with confidence level 1- for a population
mean  based upon a sample of n continuous data observations
with a sample mean x and a sample standard deviation s is
t / 2,n 1s
t / 2, n 1s 

  x 
,x 

n
n


– The interval is known as a two-sided t-interval or variance unknown
confidence interval.
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8.1.1 Confidence Interval Construction(4/8)
• A two-sided t-interval
x
t / 2,n 1s
s
t / 2,n 1 
n
x
n
x
̂
critical point  s.e.( ˆ )
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t / 2,n 1s
n
8.1.1 Confidence Interval Construction(5/8)
• The length of two-sided t-interval is
L
2t / 2,n 1s
n
 2  critical point  s.e.( ˆ )
• As the standard error of ̂ decreases, so that ˆ  x becomes a
more “accurate” estimate of .
• The length of a confidence interval also depends upon the
confidence level. As the confidence level increases, the length
of the confidence interval also increase.
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8.1.1 Confidence Interval Construction(6/8)
• We know that
n X  
S
tn 1
and so the definition of the critical point of the t-distribution
ensures that


n  X  
P  t / 2,n1 
 t / 2,n1   1  


S


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8.1.1 Confidence Interval Construction(7/8)
• And
t
S
t
S

P  X   / 2,n 1    X   / 2,n 1   1  
n
n 

• This probability statement should be interpreted as saying that
there is a probability of 1- that the random confidence interval
limits take values that “straddle” the fixed value .
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8.1.1 Confidence Interval Construction(8/8)
• Technically speaking,
n  X    S has a t-distribution only
when the random variables X i are normally distributed.
• Nevertheless, the central limit theorem ensures that the
distribution of
X is approximately normal for reasonably large
sample sizes, and in such cases it is sensible to construct tintervals regardless of the actual distribution of the data
observations.
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Example 14 : Metal Cylinder Production (p.365)
•
Data : 60 metal cylinder diameters (page 290, Figure 6.5).
•
Summary statistics:
•
n = 60
Median = 50.01
Max. = 50.36
x = 49.999
Upper quartile = 50.07 Min. = 49.74
s = 0.134
Lower quartile = 49.91
Critical points:
Sample size n = 60
Confidence level 90%: t0.05,59 = 1.671
Confidence level 95%: t0.025,59 = 2.001
Confidence level 99%: t0.005,59 = 2.662
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Example 14 : Metal Cylinder Production(2/4)
• Confidence interval with confidence level 90%:
1.671 0.134
1.671 0.134 

, 49.999 
 49.999 
  (49.970,50.028)
60
60


• Confidence interval with confidence level 95%:
2.001 0.134
2.001 0.134 

49.999

,
49.999


  (49.964,50.033)
60
60


• Confidence interval with confidence level 99%:
2.662  0.134
2.662  0.134 

, 49.999 
 49.999 
  (49.953,50.045)
60
60


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Example 14 : Metal Cylinder Production(3/4)
• Confidence intervals for mean metal cylinder diameter
49.970
x =49.999
50.028
90%
49.964
x =49.999
50.033
95%
49.953
x =49.999
99%
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50.045
Example 14 : Metal Cylinder Production(4/4)
• Conclusion with confidence interval:
With over 99% certainty, the average cylinder diameter lies
within 0.05 mm of 50.00mm, that is, within the interval
(49.95, 50.05).
• Comment:
It is important to remember that this confidence interval is for
the mean cylinder diameter, and not for the actual diameter
of a randomly selected cylinder.
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8.1.2 Effect of the Sample Size
on Confidence Intervals(1/4)
• Recall:
L
2t / 2,n 1s
n
• For a fixed critical point, a confidence interval length L is
inversely proportional to the square root of the sample size n.
• Notice that this dependence of the critical point on the sample
size also serves to produce smaller confidence intervals with
large sample sizes.
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8.1.2 Effect of the Sample Size
on Confidence Intervals(2/4)
• If a confidence interval with a length no large than L0 is required,
then sample size
 t / 2, n 1s 
n  4

L
0


2
must be used. This inequality can be used to find a suitable
sample size n if approximate values or upper bounds are used for
t/2, n-1 and s.
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8.1.2 Effect of the Sample Size
on Confidence Intervals(3/4)
• Example (p. 364): For the mean thickness of plastic sheets, an
experimenter wishes to construct a 95% confidence interval with
a length no larger than L0 = 2.0 mm.
– Known fact: the standard deviation (s)  4.0 mm
– Assumption: t0.025, n-1  2.1, for a large enough sample size
– Then a sample size is
 t / 2, n 1s 
 2.1 4.0 
2
  2
  L0
n 
n 



 2.1 4.0 
n  4
  70.56
 2.0 
2
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8.1.2 Effect of the Sample Size
on Confidence Intervals(4/4)
•
Additional sampling:
– For a sample size n1 and a sample standard deviation s, the length of
L
the confidence interval is
2t / 2,n1 1s
n1
– To reduce the confidence interval length to L0 < L, the size of the
additional sample required is
n  n1 ,
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 t / 2,n1 1s 
where n  4  

L
0


2
Example 14 : Metal Cylinder Production (p. 365)
•
n = 60, 99% confidence interval (49.953, 50.045) and confidence
length = 0.092 mm
•
Question: How much additional sampling is required to provide the
increased precision of a confidence interval with a length of 0.08
mm at the same confidence level?
•
Answer: A total sample size required is
2
 t0.005,59 s 
 2.662  0.134 
n  4
  4
  79.53
0.08


 L0 
2
– Therefore, an additional sample of at least 80-60 = 20 cylinders
is needed.
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8.1.4 Simulation Experiment(1/2)
•
Figure 8.8(page 369) shows the 500 simulation results (a mean of  =
10, a sample size of n = 30) and 95% confidence intervals.
•
Notice that, in simulations 24 and 37, the confidence intervals do not
include the correct value  = 10.
•
Each simulation provides a 95% confidence interval which has a
probability of 0.05 of not containing the value  = 10.
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8.1.4 Simulation Experiment(2/2)
• Since the simulations are independent of each other, the
number of simulations out of 500 for which the confidence
interval does not contain  = 10 has a binomial distribution with
n = 500 and p = 0.05.
• In practice, an experimenter observes just one data set, and it
has a probability of 0.95 of providing a 95% confidence interval
that does indeed straddle the true value .
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8.1.5 One-Sided Confidence Intervals(1/4)
• One-Sided t-Interval: One-sided confidence intervals with
confidence levels 1- for a population mean  based on a
sample of n continuous data observations with a sample mean x
and a sample standard deviation s are
t ,n 1s 

   , x 

n


which provides an upper bound on the population mean , and

  x 


,
n

t ,n 1s
which provides a lower bound on the population mean .
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8.1.5 One-Sided Confidence Intervals(2/4)
• c.f.
Since
n X  
S
tn 1
the definition of the critical point t,n-1 implies that

n  X   
  1
P  t ,n1 


S


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8.1.5 One-Sided Confidence Intervals(3/4)
This may be rewritten
t
S

P    X   ,n 1   1  
n 

so that
t ,n 1s 

   , x 

n


is a one-sided confidence interval for  with a confidence level
of 1-.
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8.1.5 One-Sided Confidence Intervals(4/4)
• Figure 8.10: Comparison of two-sided and one-sided
confidence intervals
x
One-sided(upper bound)
x
t / 2,n 1s
Two-sided
One-sided(lower bound)
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t ,n 1s
x
n
x
x
x
t ,n 1s
n
x
n
t / 2,n 1s
n
Example 45 : Hospital Worker Radiation Exposures
• n = 28,
sample mean x = 5.145,
• sample standard deviation s = 0.7524,
• critical point t0.01, 27 = 2.473.
• 99% one-sided confidence interval for  is
t ,n 1s  

2.473  0.7524 
   , x 
   ,5,145 
   ,5.496 
n  
28


• Consequently, with a confidence level of 0.99 the experimenter
can conclude that the average radiation level at a 50cm
distance from a patient is no more than about 5.5.
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8.1.6 z-Intervals(1/2)
• Two-Sided z-Interval
If an experimenter wishes to construct a confidence interval for
a population mean  based on a sample of size n with a sample
mean x and using an assumed known value for the population
standard deviation , then the appropriate confidence interval is
z / 2
z / 2 

  x 
, x

n
n


which is known as a two-sided z-interval or variance known
confidence interval.
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8.1.6 z-Intervals(2/2)
• One-Sided z-Interval
One-sided 1- level confidence intervals for a population
mean  based on a sample of n observations with a sample x
mean and using a known value of the population standard
deviation  are

   , x 

z  

n 
and

  x 

z 

,
n

These confidence intervals are known as one-sided z-intervals.
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8.2 Hypothesis Testing
8.2.1 Hypotheses(1/2)
• Hypothesis Tests of a Population Mean
– A null hypothesis H0 for a population mean  is a statement that
designates possible values for the population mean.
– It is associated with an alternative hypothesis HA, which is the
“opposite” of the null hypothesis.
– A two-sided set of hypotheses is
H0 :  = 0
versus
for specified value of .
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HA :  ≠ 0
8.2.1 Hypotheses(2/2)
– A one-sided set of hypotheses is either
H0 :   0
versus
HA :  > 0
versus
HA :  < 0
or
H0 :  ≥ 0
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Example 14 : Metal Cylinder Production
• The machine that produces metal cylinders is set to make
cylinders with a diameter of 50 mm.
• The two-sided hypotheses of interest are
H0 :  = 50
versus
HA :  ≠ 50
where the null hypothesis states that the machine is calibrated
correctly.
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Example 47 : Car Fuel Efficiency
• A manufacturer claim : its cars achieve an average of at least 35
miles per gallon in highway driving.
• The one-sided hypotheses of interest are
H0 :  ≥ 35
versus
HA :  < 35
• The null hypothesis states that the manufacturer’s claim
regarding the fuel efficiency of its cars is correct.
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8.2.2 Interpretation of p-values(1/4)
• Types of error
– Type I error: An error committed by rejecting the null
hypothesis when it is true.
– Type II error: An error committed by accepting the null
hypothesis when it is false.
• Significance level
– is specified as the upper bound of the probability of type I
error.
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8.2.2 Interpretation of p-values(2/4)
• p-value of a test (or observed level of significance)
– Definition: The p-value of a test is the probability of
obtaining a given data set or worse when the null hypothesis
is true.
– A data set can be used to measure the plausibility of null
hypothesis H0 through the construction of a p-value.
– The smaller the p-value, the less plausible is the null
hypothesis. (why?)
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8.2.2 Interpretation of p-values(3/4)
• Rejection of the Null Hypothesis
– If a p-value is smaller than the significance level, then the
hypothesis H0 is rejected in favor of the alternative
hypothesis HA.
•
Acceptance of the Null Hypothesis
– A p-value larger than 0.10 is generally taken to indicate that the null
hypothesis H0 is a plausible statement. The null hypothesis H0 is
therefore accepted.
– However, this does not mean that the null hypothesis H0 has been
proven to be true.
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8.2.2 Interpretation of p-values(4/4)
• Intermediate p-values
– A p-value in the range 1% ~ 10% is generally taken to
indicate that the data analysis is inconclusive. There is some
evidence that the null hypothesis is not plausible, but the
evidence is not overwhelming.
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Example 14 : Metal Cylinder Production
• Q : whether the machine can be shown to be calibrated
incorrectly.
• H0 :  = 50, HA :  ≠ 50
• With a small p-value,
– The null hypothesis is rejected and the machine is
demonstrated to be miscalibrated.
• With a large p-value,
– The null hypothesis is accepted and the experimenter
concludes that there is no evidence that the machine is
calibrated incorrectly.
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8.2.3 Calculation of p-values
• Two-sided t-test
– Consider testing
–
H 0 :   0 vs H A :   0
p  value  2  P( X  | t |)
where
 X 
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x  0
s/ n
• One-sided t-test
– Consider testing
– Then
p  value  P( X  t )
– Consider testing
– Then
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H 0 :   0 vs H A :   0
H 0 :   0 vs H A :   0
p  value  P( X  t )
Example 14 : Metal Cylinder Production
H0:  = 0
versus
HA:  ≠ 0
• The data set of metal cylinder diameters: n = 60,
s = 0.1334
• 0 = 50.0 
=x 49.99856,
49.99856  50.0 

t
 0.0836
0.1334 60
• p-value = 2 x P(X≥0.0836), X ~ t-distribution with n-1=59 d.f.
• p-value = 2 x 0.467 = 0.934
• With such a large p-value,
t59 distribution
0.467
the null hypothesis is accepted.
0 ltl=0.0836
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Example 47 : Car Fuel Efficiency (1/2)
H 0 :   35 versus H A :   35
•
n=20,
•
the fuel efficiency with a sample mean x =34.271 miles/gallon
•
sample standard deviation s=2.915 miles/gallon.
•
0 = 35.0
t
•
20  34.271  35.0 
2.915
 1.119
The alternative hypothesis is HA:  < 35, so that
p  value  P  X  1.119
where X has a t-distribution with n-1=19 d.f.
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Example 47 : Car Fuel Efficiency (2/2)
• The value can be shown to
t19 distribution
be p-value = 0.1386.
• This p-value is larger than 0.10
and so the null hypothesis
should be accepted.
-1.119
p-value
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0
8.2.4 Significance Levels(1/14)
•
Significance Level of a Hypothesis Test
– A hypothesis test with a significance level or size 
rejects the null hypothesis H0 if a p-value smaller than  is obtained
and
accepts the null hypothesis H0 if a p-value larger than  is obtained.
•
P-values are more informative than knowing whether a size  test
accepts or rejects the null hypothesis. (Why?)
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8.2.4 Significance Levels(2/14)
•
Two-Sided Problems
– Two-Sided Hypothesis Test for a Population Mean
• A size  test for the two-sided hypotheses
H0:  = 0
versus
HA:  ≠ 0
rejects the null hypothesis H0 if the test statistic ltl falls in
the rejection region
t  t / 2, n1
and accepts the null hypothesis H0 if the test statistic ltl falls in
the acceptance region
•
What is the role of
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
here?
t  t / 2, n1
Example 14 : Metal Cylinder Production
H0:  = 0
versus
HA:  ≠ 0
• The data set of metal cylinder diameters gives a test statistic of
ltl = 0.0836
• =0.10  t0.05,59=1.671
• =0.05  t0.025,59=2.001
• =0.01  t0.005,59=2.662
• The test statistic is smaller than each of these critical points,
and so the hypothesis tests all accept the null hypothesis.
• The p-value is therefore known to be larger than 0.10, and in fact
the previous analysis found the p-value to be 0.934.
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8.2.4 Significance Levels(3/14)
•
Relationship Between Confidence Intervals and Hypothesis Tests
– The value 0 is contained within a 1- level two-sided confidence
interval
t / 2,n 1s
t / 2,n 1s 

, x
x 

n
n


if the p-value for the two-sided hypothesis test
H0:  = 0
is larger than .
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versus
HA:  ≠ 0
8.2.4 Significance Levels(4/14)
• If 0 is contained within the 1- level confidence interval,
the hypothesis test with size  accepts the null hypothesis,
and
• if 0 is not contained within the 1- level confidence interval,
the hypothesis test with size  rejects the null hypothesis.
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8.2.4 Significance Levels(5/14)
• (FIGURE 8.36, Page 399)
H 0 :   0
p  value  
x
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versus
H A :   0
p  value  
t / 2,n 1s
x
p  value  
x
t / 2,n 1s
n
n
1- level two-sided confidence interval
Example 14 : Metal Cylinder Production (1/2)
• A 90% two-sided t-interval for the mean cylinder diameter was
found to be (49.970, 50.028).
• This contains the value 0=50.0
and so is consistent with
• the hypothesis testing problem
H0:  = 50.0
versus
HA:  ≠ 50.0
having a p-value of 0.934, so that the null hypothesis is accepted at
size =0.10.
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Example 14 : Metal Cylinder Production (2/2)
• The 90% confidence interval implies that the hypothesis testing
problem
H0:  = 0
versus
HA:  ≠ 0
has a p-value larger than 0.10 for 49.970 050.028 and a p-value
smaller than 0.10 otherwise. (Why ?)
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8.2.4 Significance Levels(6/14)
• One-Sided Inferences on a Population Mean(H0:   0) (p.401)
– A size  test for the one-sided hypothesis
H0:   0
versus
HA:  > 0
rejects the null hypothesis when
t>t
,n-1
and accepts the null hypothesis when
t t
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,n-1
8.2.4 Significance Levels(7/14)
• One-sided Inferences on a Population Mean(H0:   0)
– The 1- level one-sided upper confidence interval

  x 


,
n

t ,n 1s
consists of the values 0 for which this hypothesis testing
problem has a p-value larger than .
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8.2.4 Significance Levels(8/14)
•
Relationship between hypothesis testing and confidence intervals for
one-sided problems
(FIGURE 8.40, Page 403)
H 0 :   0
p  value  
x
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versus
H A :   0
p  value  
t ,n 1s
x
n
1- level one-sided confidence interval
8.2.4 Significance Levels(9/14)
• One-sided Inferences on a Population Mean (H0:  ≥ 0)
– A size  test for the one-sided hypothesis
H0:  ≥ 0
versus
HA:  < 0
rejects the null hypothesis when
t < -t
,n-1
and accepts the null hypothesis when
t ≥ -t
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,n-1
8.2.4 Significance Levels(10/14)
• One-sided Inferences on a Population Mean(H0:  ≥ 0)
– The 1- level one-sided lower confidence interval

   , x 

t ,n 1s 

n 
consists of the values 0 for which this hypothesis testing
problem has a p-value larger than .
NIPRL
8.2.4 Significance Levels(11/14)
• Relationship between hypothesis testing and confidence intervals
for one-sided problems (FIGURE 8.39, Page 402)
H 0 :   0
versus
H A :   0
p  value  
p  value  
x
x
t ,n 1s
1- level one-sided confidence interval
NIPRL
n
Example 47 : Car Fuel Efficiency (1/2)
• The one-sided hypotheses of interest here are
– H0:  ≥ 35.0
versus
HA:  < 35.0
• t = -1.119 > –t0.10,19 = -1.328.
• So
a size  = 0.10 hypothesis test accepts the null hypothesis.
• This conclusion is consistent with the previous analysis where
the p-value was found to be 0.1386, which is larger than 
=0.10.
NIPRL
Example 47 : Car Fuel Efficiency (2/2)
• Furthermore, the one-sided 90% lower t-interval

   , x 

t ,n 1s  
1.328  2.915 


,34.271

 
   ,35.14 
n  
20

contains the value 0 = 35.0, as expected.
• In fact, this confidence interval indicates that the hypothesis
testing problem
– H0:  ≥ 0
versus
HA:  < 0
has a p-value larger than 0.10 for any value of 0  35.14.
NIPRL
8.2.4 Significance Levels(12/14)
• Power of a hypothesis Test
– The power of a hypothesis test is defined to be
• power = 1 – (probability of Type II error | H A )
which is the probability that the null hypothesis is rejected
when it is false.
• Larger power levels and shorter confidence intervals are both
indications of an increase in the “precision” of an experiment
( why? ).
NIPRL
8.2.4 Significance Levels(13/14)
FIGURE 8.42 The specification of two quantities determines
the third quantity
Hypothesis Testing
Sample size n
Significance level
Power of hypothesis test

Confidence Intervals
Confidence level 1-

NIPRL
Sample size n
Length of confidence interval
8.2.4 Significance Levels(14/14)
• Relationship Between Power and Sample Size
– For a fixed significance level , the power of a hypothesis
test increases as the sample size n increases. (Why? )
NIPRL
8.2.5 z-Tests (1/7)
(t-test, s, t , n-1 )  (z-test,  , z  )
• Two-Sided z-test
– The p-value for the two-sided hypothesis testing problem
H0:  = 0
versus
HA:  ≠ 0
based upon a data set of n observations with a sample
mean
x and an assumed “known” population standard
deviation , is
p  value  2     z 
where   x  is the standard normal cumulative distribution
function.
NIPRL
8.2.5 z-Tests (2/7)
•
Two-Sided z-test
z
n  x  0 

which is known as the z-statistic.
•
A size  test rejects the null hypothesis H0 if the test statistic lzl falls in
the rejection regionz  z / 2
and accepts the null hypothesis H0 if the test statistic lzl falls in the
acceptance region
NIPRL
z  z / 2
8.2.5 z-Tests (3/7)
– The 1-  level two-sided confidence interval
•
z / 2
z / 2 

  x 
,x 

n
n


consists of the value 0 for which this hypothesis testing
problem has a p-value larger than .
NIPRL
8.2.5 z-Tests (4/7)
• One-Sided z-Test (H0:   0)
– The p-value for the one-sided hypothesis testing problem
H0:   0
versus
HA:  > 0
based upon a data set of n observations with a sample mean
x and an assumed known population standard deviation ,
is
p  value  1    z 
This testing procedure is called a one-sided z-test.
NIPRL
8.2.5 z-Tests (5/7)
• One-Sided z-Test (H0:   0)
– A size  test rejects the null hypothesis when
z  z
and accepts the null hypothesis when
z  z
– The 1-  level upper one-sided confidence interval
z


  x   ,
n


NIPRL
8.2.5 z-Tests (6/7)
• One-Sided z-Test (H0:  ≥ 0)
– The p-value for the one-sided hypothesis testing problem
• H0:  ≥ 0
versus
HA:  < 0
based upon a data set of n observations with a sample mean
x and an assumed known population standard deviation ,
is
•
p  value    z 
This testing procedure is called a one-sided z-test.
NIPRL
8.2.5 z-Tests (7/7)
• One-Sided z-Test (H0:  ≥0)
– A size  test rejects the null hypothesis when
z   z
and accepts the null hypothesis when
z   z
– The 1-  level lower one-sided confidence interval
z

   , x   
n 

NIPRL