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 X = 64.95, s = 2.42
6
7
8
9
10
time 68.2 68.1 64.8 65.8 65.4
What is the 95% confidence interval for the population mean µ?
Liang Zhang (UofU)
Applied Statistics I
July 17, 2008
1 / 10
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.
Liang Zhang (UofU)
Applied Statistics I
July 17, 2008
2 / 10
Confidence Intervals for Normal Distribution
Liang Zhang (UofU)
Applied Statistics I
July 17, 2008
3 / 10
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=∞).
Liang Zhang (UofU)
Applied Statistics I
July 17, 2008
4 / 10
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.
Liang Zhang (UofU)
Applied Statistics I
July 17, 2008
5 / 10
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)%.
Liang Zhang (UofU)
Applied Statistics I
July 17, 2008
6 / 10
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 X = 64.95, s = 2.42
6
7
8
9
10
time 68.2 68.1 64.8 65.8 65.4
What is the 95% confidence interval for the 11th component?
Liang Zhang (UofU)
Applied Statistics I
July 17, 2008
7 / 10
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)%.
Liang Zhang (UofU)
Applied Statistics I
July 17, 2008
8 / 10
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 X = 64.95, s = 2.42
6
7
8
9
10
time 68.2 68.1 64.8 65.8 65.4
What is the 95% confidence interval such that at least 90% of the values
in the population are inside this interval?
Liang Zhang (UofU)
Applied Statistics I
July 17, 2008
9 / 10
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.
Liang Zhang (UofU)
Applied Statistics I
July 17, 2008
10 / 10