Download LESSON 18: CONFIDENCE INTERVAL ESTIMATION ESTIMATION

LESSON 18: CONFIDENCE INTERVAL ESTIMATION
Outline
• Confidence interval: mean
– Known s
– Selecting sample size
– Unknown s
– Small population
• Confidence interval: proportion
• Confidence interval: variance
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ESTIMATION
• Point estimator: A point estimator draws inferences about a
population by estimating the value of an unknown
parameter using a single value or point.
• Interval estimator: An interval estimator draws inferences
about a population by estimating the value of an unknown
parameter using an interval.
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1
ESTIMATION
• Example: A manager of a plant making cellular phones
wants to estimate the time to assemble a phone. A sample
of 30 assemblies show a mean time of 400 seconds. The
sample mean time of 400 seconds is a point estimate. An
alternate estimate is a range e.g., 390 to 410. Such a range
is an interval estimate. The computation method of interval
estimate is discussed in Chapter 10.
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ESTIMATION
• Interval estimates are reported with the end points e.g.,
[390, 410]
or, equivalently, with a central value and its difference from
each end point e.g.,
400±10
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2
ESTIMATION
• Precision of an interval estimate: The limits indicate the
degree of precision. A more precise estimate is the one
with less spread between limits e.g., [395,405] or 400±5
• Reliability of an interval estimate: The reliability of an
interval estimate is the probability that it is correct.
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ESTIMATION
• Unbiased estimator: an unbiased estimator of a population
parameter is an estimator whose expected value is equal to
that parameter.
• In Chapter 2, the sample variance is defined as follows:
n
s2 =
∑(X
i
− X )2
i =1
n −1
• The use of n-1 in the denominator is necessary to get an
unbiased estimator of variance. The use of n in the
denominator produces a smaller value of variance.
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3
ESTIMATION
• Consistent estimators: An estimator is consistent if the
precision and reliability improves as the sample size is
increased. The estimators X and P are consistent.
• Efficient estimators: An estimator is more efficient than
another if for the same sample size it will provide a greater
sampling precision and reliability.
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INTERVAL ESTIMATOR OF MEAN (KNOWN s )
• For some confidence level 1 -α, sample size n, sample
mean, X and the population standard deviation, σ the
confidence interval estimator of mean, µ is as follows:
σ

σ
σ 
X ± zα / 2
also written as  X − zα / 2
, X + zα / 2
n
n
n 

• Recall that zα / 2 is that value of z for which area in the upper
tail is α/2
• Lower confidence limit (LCL)
X − zα / 2
• Upper confidence limit (UCL)
X + zα / 2
σ
n
σ
n
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Area=1-α
σX =
f(x)
CONFIDENCE
INTERVAL
σ
n
Area
Area
=0.5-α/2 =0.5-α/2
Area=α/2
Area=α/2
-zα/2
X − zα / 2
n
X
zα/2
X + zα / 2
σ
n
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Area=0.82
σX =
f(x)
AREAS FOR
THE 82%
CONFIDENCE
INTERVAL
σ
z=0
Area
=0.41
Area
=0.41
Area=0.09
Area=0.09
-z0.09 z=0
X − z0.09
σ
n
σ
n
X
z0.09
X + z0.09
σ
n
10
5
Area=0.82
σX =
f(x)
AREAS AND z
AND X
VALUES FOR
THE 82%
CONFIDENCE
INTERVAL
Area
=0.41
Area
=0.41
Area=0.09
Area=0.09
-1.34 z=0
X −1.34
σ
n
σ
n
X
1.34
X + 1.34
σ
n
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INTERVAL ESTIMATOR OF MEAN (KNOWN s )
• Interpretation:
– There is (1-α) probability that the sample mean will be
equal to a value such that the interval (LCL, UCL) will
include the population mean
– If the same procedure is used to obtain a confidence
interval estimate of the population mean for a sufficiently
large number of k times, the interval (LCL, UCL) is
expected to include the population mean (1-α)k times
• Wrong interpretation: It’s wrong to interpret that there is
(1-α) probability that the population mean lies between LCL
and UCL. Population mean is fixed, not uncertain /
probabilistic.
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6
INTERVAL ESTIMATOR OF MEAN (KNOWN s )
• Interpretation of the 95% confidence interval:
– There is 0.95 probability that the sample mean will be
equal to a value such that the interval (LCL, UCL) will
include the population mean
– If the same procedure is used to obtain a confidence
interval estimate of the population mean for a sufficiently
large number of k times, the interval (LCL, UCL) is
expected to include the population mean 0.95k times –
• Wrong interpretation: It’s wrong to interpret that there is
0.95 probability that the population mean lies between LCL
and UCL. Population mean is fixed, not uncertain /
probabilistic.
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INTERVAL ESTIMATOR OF MEAN (KNOWN s )
Example 1: The following data represent a random sample of
10 observations from a normal population whose standard
deviation is 2. Estimate the population mean with 90%
confidence: 7,3,9,11,5,4,8,3,10,9
f(x)
σx =
σ
n
x
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SELECTING SAMPLE SIZE
• A narrow confidence interval is more desirable.
• For a given a confidence level, a narrow confidence interval
can be obtained by increasing the sample size.
• Desired precision or maximum error: If the confidence
interval has the form of X ± d then, d is the desired
precision or the maximum error.
• For a given confidence level (1-α), desired precision d and
the population standard deviation σ the sample size
necessary to estimate population mean, µ is
z σ
n =  α /2 
 d 
2
An approximation for σ: σ =
Q.999 − Q. 001
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SELECTING SAMPLE SIZE
Example 2: Determine the sample size that is required to
estimate a population mean to within 0.2 units with 90%
confidence when the standard deviation is 1.0.
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INTERVAL ESTIMATOR OF MEAN (UNKNOWN s )
• If the population standard deviation s is unknown, the
normal distribution is not appropriate and the mean is
estimated using Student t distribution. Recall that
X −µ
t=
s/ n
• For some confidence level 1 -α, sample size n, sample
mean, X and the sample standard deviation, s the
confidence interval estimator of mean, µ is as follows:
s

s
s 
X ± tα / 2
also written as  X − tα / 2
, X + tα / 2
n
n
n 

• Where, tα / 2 is that value of t for which area in the upper tail
is α/2 at degrees of freedom, d.f. = n-1.
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SMALL POPULATION
• For small, finite population, a correction factor is applied in
computing σ X . So, the confidence interval is computed as
follows:
µ = X ± zα / 2
σ
n
N −n
N −1
(σ
known )
µ = X ± tα / 2
s
n
N −n
N −1
(σ
unknown )
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UNKNOWN s AND SMALL POPULATION
Example 3: An inspector wishes to estimate the mean
weight of the contents in a shipment of 16-ounce cans of
corn. The shipment contains 500 cans. A sample of 25
cans is selected, and the contents of each are weighed.
The sample mean and standard deviation were compute
to be X = 16.1 ounces and s = 0.25 ounce. Construct a
90% confidence interval of the population mean.
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INTERVAL ESTIMATOR OF PROPORTION
• Confidence interval of the proportion for large population:
P(1 − P )
n
• Confidence interval of the proportion for small population:
π = P ± zα / 2
P(1 − P ) N − n
n
N −1
• Required sample size for estimating the proportion:
π = P ± zα / 2
zα2 / 2π (1 − π )
n=
d2
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INTERVAL ESTIMATOR OF PROPORTION
Example 4: The controls in a brewery need adjustment
whenever the proportion p of unfulfilled cans is 0.01 or
greater. There is no way of knowing the true proportion,
however. Periodically, a sample of 100 cans is selected
and the contents are measured.
(a) For one sample, 3 under-filled can were found.
Construct the resulting 95% confidence interval estimate
of p.
(b) What is probability of getting as many or more underfilled cans as in (a) when in fact p is only 0.01.
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INTERVAL ESTIMATOR OF VARIANCE
• The chi-square distribution is asymmetric. As a result, two
critical values are required to compute the confidence
interval of the variance.
• Confidence interval of the variance:
(n − 1)s 2 ≤ σ 2 ≤ (n − 1)s 2
χα2 / 2
χ12−α / 2
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INTERVAL ESTIMATOR OF VARIANCE
Example 5: The sample standard deviation for n = 25
observations was computed to be s = 12.2. Construct a
98% confidence interval estimate of the population
standard deviation.
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READING AND EXERCISES
Lesson 18
Reading:
Section 10-1 to 10-4, pp. 295-319
Exercises:
10-9, 10-10, 10-13, 10-21, 10-24, 10-26, 10-31, 10-32
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