Large-Sample Confidence Intervals
Proposition
If n is sufficiently large, the standardized variable
Z=
X −µ
√
S/ n
has approximately a standard normal distribution. This implies that
s
x̄ ± zα/2 · √
n
is a large-sample confidence interval for µ with confidence level
approximately 100(1 − α)%. This formula is valid regardless of the shape
of the population distribution.
Liang Zhang (UofU)
Applied Statistics I
July 17, 2008
1/7
Large-Sample Confidence Intervals
Example (a variant of Problem 16)
The charge-to-tap time (min) for a carbon steel in one type of open hearth
furnace was determined for each heat in a sample of size 46, resulting in a
sample mean time of 382.1 and a sample standard deviation of 31.5.
Calculate a 95% confidence interval for true average charge-to-tap time.
Liang Zhang (UofU)
Applied Statistics I
July 17, 2008
2/7
Large-Sample Confidence Intervals
Example (Problem 19)
The article “Limited Yield Estimation for Visual Defect Sources” (IEEE
Trans. on Semiconductor Manuf., 1997: 17-23) reported that, in a study
of a particular wafer inspection process, 356 dies were examined by an
inspection probe and 201 of these passed the probe. Assuming a stable
process, calculate a 95% confidence interval for the proportion of all dies
that pass the probe.
Liang Zhang (UofU)
Applied Statistics I
July 17, 2008
3/7
Large-Sample Confidence Intervals
Proposition
A confidence interval for a population proportion p with confidence
level approximately 100(1 − α)% has
r
p̂ +
lower confidence limit =
2
zα/2
2n
p̂ +
Liang Zhang (UofU)
+
2
zα/2
4n2
2 )/n
1 + (zα/2
and
upper confidence limit =
p̂q̂
n
− zα/2
2
zα/2
2n
Applied Statistics I
r
+ zα/2
1+
p̂q̂
n
+
2
zα/2
4n2
2 )/n
(zα/2
July 17, 2008
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Large-Sample Confidence Intervals
Example (Problem 16)
The charge-to-tap time (min) for a carbon steel in one type of open hearth
furnace was determined for each heat in a sample of size 46, resulting in a
sample mean time of 382.1 and a sample standard deviation of 31.5.
Calculate a 95% upper confidence bound for true average charge-to-tap
time.
Liang Zhang (UofU)
Applied Statistics I
July 17, 2008
5/7
Large-Sample Confidence Intervals
Example (Problem 19)
The article “Limited Yield Estimation for Visual Defect Sources” (IEEE
Trans. on Semiconductor Manuf., 1997: 17-23) reported that, in a study
of a particular wafer inspection process, 356 dies were examined by an
inspection probe and 201 of these passed the probe. Assuming a stable
process, calculate a 95% lower confidence bound for the proportion of all
dies that pass the probe.
Liang Zhang (UofU)
Applied Statistics I
July 17, 2008
6/7
Large-Sample Confidence Intervals
Proposition
A large-sample upper confidence bound for µ is
s
µ < x̄ + zα · √
n
and a large-sample lower confidence bound for µ is
s
µ > x̄ − zα · √
n
A one-sided confidence bound for p results from replacing zα/2 by zα
and ± by either + or − in the CI formula for p. In all cases the confidence
level is approximately 100(1 − α)%
Liang Zhang (UofU)
Applied Statistics I
July 17, 2008
7/7