Confidence Intervals
Confidence Intervals
Definition
A 100(1 − α)% confidence interval for the mean µ of a normal
population when the value of σ is known is given by
σ
σ
x − zα/2 · √ , x + zα/2 · √
n
n
or, equivalently, by x ∓ zα/2 ·
√σ
n
Confidence Intervals
Confidence Intervals
Proposition
To obtain a 100(1 − α)% confidence interval with width w for the
mean µ of a normal population when the value of σ is known, we
need a random sample of size at least
σ 2
n = 2zα/2 ·
w
Confidence Intervals
Proposition
To obtain a 100(1 − α)% confidence interval with width w for the
mean µ of a normal population when the value of σ is known, we
need a random sample of size at least
σ 2
n = 2zα/2 ·
w
Remark:
The half-width w2 of the 100(1 − α)% CI is called the bound on
the error of estimation associated with a 100(1 − α)%
confidence level.
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
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.
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
p̂ +
lower confidence limit =
2
zα/2
2n
p̂q̂
n
− zα/2
+
2
zα/2
4n2
2 )/n
1 + (zα/2
and
p̂ +
upper confidence limit =
2
zα/2
2n
r
+ zα/2
p̂q̂
n
2 )/n
1 + (zα/2
+
2
zα/2
4n2
Large-Sample Confidence Intervals
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)%
Confidence Intervals for Normal Distribution
Confidence Intervals for Normal Distribution
Example (a variant of Problem 62, Ch5)
The total time for manufacturing a certain component is known to
have a normal distribution. However, the mean µ and variance σ 2
for the normal distribution are unknown. After an experiment in
which we manufactured 10 components, we recorded the sample
time which is given as follows:
1
2
3
4
5
time 63.8 60.5 65.3 65.7 61.9
with
6
7
8
9
10
time 68.2 68.1 64.8 65.8 65.4
X = 64.95, s = 2.42
Confidence Intervals for Normal Distribution
Example (a variant of Problem 62, Ch5)
The total time for manufacturing a certain component is known to
have a normal distribution. However, the mean µ and variance σ 2
for the normal distribution are unknown. After an experiment in
which we manufactured 10 components, we recorded the sample
time which is given as follows:
1
2
3
4
5
time 63.8 60.5 65.3 65.7 61.9
with
6
7
8
9
10
time 68.2 68.1 64.8 65.8 65.4
X = 64.95, s = 2.42
What is the 95% confidence interval for the population mean µ?
Confidence Intervals for Normal Distribution
Confidence Intervals for Normal Distribution
Theorem
Let X1 , X2 , . . . , Xn be a random sample from a normal distribution
with mean µ and variance σ 2 , where µ and σ are unknown. The
random variable
X −µ
√
T =
S/ n
has a probability distribution called a t distribution with
n − 1 degrees of freedom (df). Here X is the sample mean
and S is the sample standard deviation.
Confidence Intervals for Normal Distribution
Confidence Intervals for Normal Distribution
Confidence Intervals for Normal Distribution
Confidence Intervals for Normal Distribution
Properties of t Distributions:
Confidence Intervals for Normal Distribution
Properties of t Distributions:
Let tν denote the density function curve for ν df.
1. tν is governed by only one parameter ν, the number of
degrees of freedom.
Confidence Intervals for Normal Distribution
Properties of t Distributions:
Let tν denote the density function curve for ν df.
1. tν is governed by only one parameter ν, the number of
degrees of freedom.
2. Each tν curve is bell-shaped and centered at 0.
Confidence Intervals for Normal Distribution
Properties of t Distributions:
Let tν denote the density function curve for ν df.
1. tν is governed by only one parameter ν, the number of
degrees of freedom.
2. Each tν curve is bell-shaped and centered at 0.
3. Each tν curve is more spread out than the standard normal
(z) curve.
Confidence Intervals for Normal Distribution
Properties of t Distributions:
Let tν denote the density function curve for ν df.
1. tν is governed by only one parameter ν, the number of
degrees of freedom.
2. Each tν curve is bell-shaped and centered at 0.
3. Each tν curve is more spread out than the standard normal
(z) curve.
4. As ν increases, the spread of the corresponding tν curve
decreases.
Confidence Intervals for Normal Distribution
Properties of t Distributions:
Let tν denote the density function curve for ν df.
1. tν is governed by only one parameter ν, the number of
degrees of freedom.
2. Each tν curve is bell-shaped and centered at 0.
3. Each tν curve is more spread out than the standard normal
(z) curve.
4. As ν increases, the spread of the corresponding tν curve
decreases.
5. As ν → ∞, the sequence of tν curves approaches the standard
normal curve (so the z curve is often called the t curve with
df=∞).
Confidence Intervals for Normal Distribution
Confidence Intervals for Normal Distribution
Notation
Let tα,ν = the number on the measurement axis for which the area
under the t curve with ν df to the right of tα,ν is α; tα,ν is called a
t critical value.
Confidence Intervals for Normal Distribution
Notation
Let tα,ν = the number on the measurement axis for which the area
under the t curve with ν df to the right of tα,ν is α; tα,ν is called a
t critical value.
Confidence Intervals for Normal Distribution
Confidence Intervals for Normal Distribution
Proposition
Let x̄ and s be the sample mean and sample standard deviation
computed from the results of a random sample from a normal
population with mean µ. Then a 100(1 − α)% confidence
interval for µ is
s
s
α
α
√
√
x̄ − t 2 ,n−1 ·
, x̄ + t 2 ,n−1 ·
n
n
or, more compactly, x̄ ± t α2 ,n−1 · √sn .
An upper confidence bound for µ is
s
x̄ + tα,n−1 · √
n
and replacing + by − in this latter expression gives a lower
confidence bound for µ, both with confidence level 100(1 − α)%.
Confidence Intervals for Normal Distribution
Confidence Intervals for Normal Distribution
Example (a variant of Problem 62, Ch5)
The total time for manufacturing a certain component is known to
have a normal distribution. However, the mean µ and variance σ 2
for the normal distribution are unknown. After an experiment in
which we manufactured 10 components, we recorded the sample
time which is given as follows:
1
2
3
4
5
time 63.8 60.5 65.3 65.7 61.9
with
6
7
8
9
10
time 68.2 68.1 64.8 65.8 65.4
X = 64.95, s = 2.42
Confidence Intervals for Normal Distribution
Example (a variant of Problem 62, Ch5)
The total time for manufacturing a certain component is known to
have a normal distribution. However, the mean µ and variance σ 2
for the normal distribution are unknown. After an experiment in
which we manufactured 10 components, we recorded the sample
time which is given as follows:
1
2
3
4
5
time 63.8 60.5 65.3 65.7 61.9
with
6
7
8
9
10
time 68.2 68.1 64.8 65.8 65.4
X = 64.95, s = 2.42
What is the 95% confidence interval for the 11th component?
Confidence Intervals for Normal Distribution
Confidence Intervals for Normal Distribution
Proposition
A prediction interval (PI) for a single observation to be selected
from a normal population distribution is
r
1
x̄ ± t α2 ,n−1 · s 1 +
n
The prediction level is 100(1 − α)%.
Confidence Intervals for Normal Distribution
Confidence Intervals for Normal Distribution
Example (a variant of Problem 62, Ch5)
The total time for manufacturing a certain component is known to
have a normal distribution. However, the mean µ and variance σ 2
for the normal distribution are unknown. After an experiment in
which we manufactured 10 components, we recorded the sample
time which is given as follows:
1
2
3
4
5
time 63.8 60.5 65.3 65.7 61.9
with
6
7
8
9
10
time 68.2 68.1 64.8 65.8 65.4
X = 64.95, s = 2.42
Confidence Intervals for Normal Distribution
Example (a variant of Problem 62, Ch5)
The total time for manufacturing a certain component is known to
have a normal distribution. However, the mean µ and variance σ 2
for the normal distribution are unknown. After an experiment in
which we manufactured 10 components, we recorded the sample
time which is given as follows:
1
2
3
4
5
time 63.8 60.5 65.3 65.7 61.9
with
6
7
8
9
10
time 68.2 68.1 64.8 65.8 65.4
X = 64.95, s = 2.42
What is the 95% confidence interval such that at least 90% of the
values in the population are inside this interval?
Confidence Intervals for Normal Distribution
Confidence Intervals for Normal Distribution
Proposition
A tolerance interval for capturing at least k% of the values in a
normal population distribution with a confidence level 95%has the
form
x̄ ± (tolerance critical value) · s
Confidence Intervals for Normal Distribution
Proposition
A tolerance interval for capturing at least k% of the values in a
normal population distribution with a confidence level 95%has the
form
x̄ ± (tolerance critical value) · s
The tolerance critical values for k = 90, 95, and 99 in combination
with various sample sizes are given in Appendix Table A.6.