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.
Confidence Intervals for the Variance of a Normal
Population
Confidence Intervals for the Variance of a Normal
Population
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 the Variance of a Normal
Population
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 a 95% confidence for the population variance σ 2 ?
Confidence Intervals for the Variance of a Normal
Population
Confidence Intervals for the Variance of a Normal
Population
Theorem
Let X1 , X2 , . . . , Xn be a random sample from a distribution with
mean µ and variance σ 2 . Then the random variable
P
(Xi − X )2
(n − 1)S 2
=
σ2
σ2
has s chi-squared (χ2 ) probability distribution with n − 1 degrees
of freedom (df).
Confidence Intervals for the Variance of a Normal
Population
Confidence Intervals for the Variance of a Normal
Population
Confidence Intervals for the Variance of a Normal
Population
Confidence Intervals for the Variance of a Normal
Population
Notation
Let χ2α,ν , called a chi-squared critical value, denote the number
on the measurement axis such that α of the area under the
chi-squared curve with ν df lies to the right of χ2α,ν .
Confidence Intervals for the Variance of a Normal
Population
Notation
Let χ2α,ν , called a chi-squared critical value, denote the number
on the measurement axis such that α of the area under the
chi-squared curve with ν df lies to the right of χ2α,ν .
Confidence Intervals for the Variance of a Normal
Population
Confidence Intervals for the Variance of a Normal
Population
Proposition
A 100(1 − α)% confidence interval for the variance σ 2 of a
normal population has lower limit
(n − 1)s 2 /χ2α ,n−1
2
and upper limit
(n − 1)s 2 /χ21− α ,n−1
2
A confidence interval for σ has lower and upper limits that are
the square roots of the corresponding limits in the interval for σ 2 .