Confidence Intervals
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
6
7
8
9
10
time 68.2 68.1 64.8 65.8 65.4
We know that both MME and MLE for the population mean µ is the
sample mean X , i.e. µ̂ = X = 64.95. How accurate is this estimation?
Liang Zhang (UofU)
Applied Statistics I
July 14, 2008
1 / 10
Confidence Intervals
• Assume the other parameter σ is known, e.g. σ = 2.7
• X is normally distributed with mean µ and variance σ 2 /n. Therefore,
X −µ
√ is a standard normal random variable.
Z = σ/
n
• For the interval [−A, A], how large should A be such that with 95%
confidence we are sure Z falls in that interval?
P(−A < Z < A) = .95
A is the 97.5th percentle, which is 1.96.
X −µ
√ < 1.96 = .95
• P −1.96 < σ/
n
σ
σ
√
√
• P X − 1.96 · n < µ < X + 1.96 · n = .95
• The interval X − 1.96 · √σn , X + 1.96 · √σn is called the 95%
confidence interval for µ.
• In our case, 95% confidence interval for µ is (63.28, 66.62).
Liang Zhang (UofU)
Applied Statistics I
July 14, 2008
2 / 10
Confidence Intervals
Interpretation of Confidence Interval
• The 95% confidence interval for µ (63.28, 66.62) doesn’t mean
P(µ falls in the interval(63.28, 66.62)) = .95
• It is a long-run effect: if we have 1000 random samples, then for
approximately 950 of them, µ falls
in the interval
σ
σ
√
√
X − 1.96 · n , X + 1.96 · n .
Liang Zhang (UofU)
Applied Statistics I
July 14, 2008
3 / 10
Confidence Intervals
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 for the normal distribution is
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
6
7
8
9
10
time 68.2 68.1 64.8 65.8 65.4
We know that both MME and MLE for the population mean µ is the
sample mean X , i.e. µ̂ = X = 64.95. We further assume the standard
deviation is known to be σ = 2.7. What is the 99% confidence
interval for µ?
Liang Zhang (UofU)
Applied Statistics I
July 14, 2008
4 / 10
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 ·
Liang Zhang (UofU)
√σ
n
Applied Statistics I
July 14, 2008
5 / 10
Confidence Intervals
Graphically interpretation:
Liang Zhang (UofU)
Applied Statistics I
July 14, 2008
6 / 10
Confidence Intervals
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 for the normal distribution is
unknown. Thus we decide to do an experiment in which we manufacture n
components to estimate the population mean µ. We know that both
MME and MLE for the population mean µ is the sample mean X , i.e.
µ̂ = X . We further assume the standard deviation is known to be σ = 2.7.
If we want a 99% confidence interval for µ with width 3.34, how
large should n be?
Liang Zhang (UofU)
Applied Statistics I
July 14, 2008
7 / 10
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.
Liang Zhang (UofU)
Applied Statistics I
July 14, 2008
8 / 10
Confidence Intervals
Example:
Extensive experience with fans of a certain type used in diesel engines has
suggested that the exponential distribution provides a good model for time
until failure. However, the parameter λ is unknown. The following table
records the data for a size 10 sample:
1
2
3
4
5
time 1.199 0.105 0.373 0.266 0.888
6
7
8
9
10
time 0.574 0.244 0.008 0.689 0.235
What is a 95% confidence interval for λ?
Liang Zhang (UofU)
Applied Statistics I
July 14, 2008
9 / 10
Confidence Intervals
Proposition
Let X1 , X2 , . . . , Xn i.i.d random variables from an expentional
distribution
P
with parameter λ. Then the random variable Y = 2λ ni=1 Xi has the
chi-squared distribution with 2n degrees of freedom, i.e., Y ∼ χ2 (2n)
Liang Zhang (UofU)
Applied Statistics I
July 14, 2008
10 / 10