Large-Sample Confidence Intervals
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
x¯ ± 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.
Large-Sample Confidence Intervals
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.
Large-Sample Confidence Intervals
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.
Large-Sample Confidence Intervals
Large-Sample Confidence Intervals
Proposition
A confidence interval for a population proportion p with
confidence level approximately 100(1 − α)% has
r
pˆ +
lower confidence limit =
2
zα/2
2n
p
ˆq
ˆ
n
− zα/2
+
2
zα/2
4n2
2 )/n
1 + (zα/2
and
pˆ +
upper confidence limit =
2
zα/2
2n
r
+ zα/2
p
ˆq
ˆ
n
2 )/n
1 + (zα/2
+
2
zα/2
4n2
Large-Sample Confidence Intervals
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.
Large-Sample Confidence Intervals
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.
Large-Sample Confidence Intervals
Large-Sample Confidence Intervals
Proposition
A large-sample upper confidence bound for µ is
s
µ < x¯ + zα · √
n
and a large-sample lower confidence bound for µ is
s
µ > x¯ − 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)%