Transcript σ 2

Chapter 9: One- and Two- Sample
Estimation
• Statistical Inference
– Estimation
– Tests of hypotheses
– “Even the most efficient unbiased estimator is unlikely to
estimate the population parameter exactly.” (Walpole et
al, pg. 272)
• Interval estimation: (1 – α)100% confidence
interval for the unknown parameter.
– Example: if α = 0.01, we develop a _______ confidence
interval.
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Single Sample: Estimating the Mean
• Given:
– σ is known and X is the mean of a random sample of
size n,
• Then,
– the (1 – α)100% confidence interval for μ is given by
X  z / 2 (

n
)    X  z / 2 (

n
)
Z
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Example
A traffic engineer is concerned about the delays at an
intersection near a local school. The intersection is
equipped with a fully actuated (“demand”) traffic light and
there have been complaints that traffic on the main street
is subject to unacceptable delays.
To develop a benchmark, the traffic engineer randomly
samples 25 stop times (in seconds) on a weekend day.
The average of these times is found to be 13.2 seconds,
and the variance is known to be 4 seconds2.
Based on this data, what is the 95% confidence interval
(C.I.) around the mean stop time during a weekend day?
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Example (cont.)
X = ______________
σ = _______________
α = ________________
α/2 = _____________
Z0.025 = _____________
Z0.975 = ____________
__________________ < μ < ___________________
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Your turn …
• What is the 90% C.I.? What does it mean?
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How big a sample do we need?
• If we want to be sure that our error in estimating
µ is less than a specified amount, e, the required
sample size is given by,
 z / 2 
n

 e 
2
• In our example, if the traffic engineer wants to be
95% confident that the mean stop time is off by
less than 0.1, then he should take
n = ___________________________________
samples.
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What if σ2 is unknown?
For example, what if the traffic engineer doesn’t know
the variance of this population?
1. If n is sufficiently large (> _______), then the large
sample confidence interval is:
s
X  z / 2 (
)
n
2. Otherwise, must use the t-statistic …
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Single Sample: Estimating the Mean
(σ unknown, n not large)
• Given:
– σ is unknown and X is the mean of a random sample
of size n (where n is not large),
• Then,
– the (1 – α)100% confidence interval for μ is given by
X  t / 2,n 1(
-5
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-4
-3
-2
s
s
)    X  t / 2,n 1(
)
n
n
-1
0
1
2
3
4
5
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Recall Our Example
A traffic engineer is concerned about the delays at an
intersection near a local school. The intersection is
equipped with a fully actuated (“demand”) traffic light
and there have been complaints that traffic on the main
street is subject to unacceptable delays.
To develop a benchmark, the traffic engineer randomly
samples 25 stop times (in seconds) on a weekend day.
The average of these times is found to be 13.2 seconds,
and the sample variance, s2, is found to be 4 seconds2.
Based on this data, what is the 95% confidence interval
(C.I.) around the mean stop time during a weekend day?
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Example (cont.)
X = ______________
s = _______________
α = ________________
α/2 = _____________
t0.025,24 = _____________
__________________ < μ < ___________________
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Your turn
A thermodynamics professor gave a physics
pretest to a random sample of 15 students who
enrolled in his course at a large state university.
The sample mean was found to be 59.81 and
the sample standard deviation was 4.94.
Find a 99% confidence interval for the mean on
this pretest.
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Solution
X = ______________
s = _______________
α = ________________
α/2 = _____________
(draw the picture)
T___ , ____ = _____________
__________________ < μ < ___________________
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Standard Error of a Point Estimate
• Case 1: σ known
– The standard deviation, or standard error of X is

n
• Case 2: σ unknown, sampling from a normal
distribution
– The standard deviation, or (usually) estimated
standard error of X is
______
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9.6: Prediction Interval
• For a normal distribution of unknown mean μ, and
standard deviation σ, a 100(1-α)% prediction
interval of a future observation, x0 is
1
1
X  z / 2 1   x0  X  z / 2 1 
n
n
if σ is known, and
1
1
X  t / 2,n 1s 1   x0  X  t / 2,n 1s 1 
n
n
if σ is unknown
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9.7: Tolerance Limits
• For a normal distribution of unknown mean μ,
and unknown standard deviation σ, tolerance
limits are given by
x + ks
where k is determined so that one can assert
with 100(1-γ)% confidence that the given limits
contain at least the proportion 1-α of the
measurements.
• Table A.7 gives values of k for (1-α) = 0.9, 0.95,
0.99; γ = 0.05, 0.01; and for selected values of
n.
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Summary
• Confidence interval  population mean μ
• Prediction interval 
a new observation x0
• Tolerance interval 
a (1-α) proportion of
the measurements
can be estimated with
a 100(1-γ)%
confidence
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Estimating the Difference Between Two
Means
• Given two independent random samples, a point
estimate the difference between μ1 and μ2 is
given by the statistic
x1  x 2
We can build a confidence interval for μ1 - μ2
(given σ12 and σ22 known) as follows:
( x 1  x 2 )  z / 2
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 12
n1

 22
n2
 1  2  ( x 1  x 2 )  z / 2
 12
n1

 22
n2
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An example
• Look at example 9.9, pg. 289
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Differences Between Two Means:
Variances Unknown
• Case 1: σ12 and σ22 unknown but equal
( x 1  x 2 )  t / 2,n1 n2 2Sp
Where,
1 1
1 1

 1  2  ( x1  x 2 )  t / 2,n1 n2 2Sp

n1 n2
n1 n2
(n1  1)S12  (n2  1)S22
S 
n1  n2  2
2
p
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Differences Between Two Means:
Variances Unknown
• Case 2: σ12 and σ22 unknown and not equal
( x 1  x 2 )  t / 2,
Where,
s12 s22
s12 s22

 1  2  ( x 1  x 2 )  t / 2,

n1 n2
n1 n2
(S12 / n1  S22 / n2 )2

 S2 / n 2   S2 / n 2 
 1 1  2 2 
 n1  1   n2  1 

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


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Estimating μ1 – μ2
• Example (σ12 and σ22 known) :
A farm equipment manufacturer wants to compare the
average daily downtime of two sheet-metal stamping
machines located in two different factories.
Investigation of company records for 100 randomly
selected days on each of the two machines gave the
following results:
x1 = 12 minutes
x2 = 10 minutes
12 = 12
 22 = 8
n1 = n2 = 100
Construct a 95% C.I. for μ1 – μ2
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Solution
α/2 = _____________
Picture
z_____ = ____________
( x 1  x 2 )  z / 2
 12
n1

 22
n2
 1  2  ( x 1  x 2 )  z / 2
 12
n1

 22
n2
__________________ < μ1 – μ2 < _________________
Interpretation:
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μ1 – μ2 : σi2 Unknown
• Example (σ12 and σ22 unknown but equal):
Suppose the farm equipment manufacturer was
unable to gather data for 100 days. Using the data
they were able to gather, they would still like to
compare the downtime for the two machines. The
data they gathered is as follows:
x1 = 12 minutes
s12 = 12
n1 = 18
x2 = 10 minutes
s22 = 8
n2 = 14
Construct a 95% C.I. for μ1 – μ2
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Solution
α/2 = _____________
Picture
t____ , ________= ____________
(n1  1)S12  (n2  1)S22
S 
 ______________
n1  n2  2
2
p
( x 1  x 2 )  t / 2,n1 n2 2Sp
1 1
1 1

 1  2  ( x1  x 2 )  t / 2,n1 n2 2Sp

n1 n2
n1 n2
__________________ < μ1 – μ2 < _________________
Interpretation:
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Paired Observations
• Suppose we are evaluating observations that
are not independent …
For example, suppose a teacher wants to compare
results of a pretest and posttest administered to the
same group of students.
• Paired-observation or Paired-sample test …
Example: murder rates in two consecutive years for
several US cities (see attached.) Construct a 90%
confidence interval around the difference in
consecutive years.
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Solution
D = ____________
Picture
tα/2, n-1 = _____________
sd 
2
(
d

d
)
 i
n 1
 _________
a (1-α)100% CI for μ1 – μ2 is: d  t / 2,n1(
sd
)
n
__________________ < μ1 – μ2 < _________________
Interpretation:
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C.I. for Proportions
• The proportion, P, in a binomial experiment may be

estimated by
X
P
n
where X is the number of successes in n trials.
• For a sample, the point estimate of the parameter is

x
p
n
• The mean for the sample proportion is    p
p
pq
and the sample variance  2 
n
p
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C.I. for Proportions
• An approximate (1-α)100% confidence interval
for p is:

p  z / 2
 
pq
n
• Large-sample C.I. for p1 – p2 is:


( p1  p2 )  z / 2




p1 q1 p2 q2

n1
n2
Interpretation: _______________________________
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Example 9.16
1. C.I. = (-0.0017, 0.0217), therefore no reason to
believe there is a significant decrease in the
proportion defectives using the new process.
2. What if the interval were (+0.0017, 0.0217)?
3. What if the interval were (-0.9, -0.7)?
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