Lecture 1 Interval Estimation - fsalamri · Methods for constructing confidence intervals...
Transcript of Lecture 1 Interval Estimation - fsalamri · Methods for constructing confidence intervals...
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Lecture 1
Interval Estimation
Dr. Hoda Ragab Rezk
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The interval estimator of θ is called
a 100(1 − 𝛼)% confidence interval for θ if
P(L ≤ θ ≤ U) = 1 − 𝛼.
The L is called the lower confidence limit
The U is called the upper confidence limit.
The number (1− 𝛼) is called the confidence
coefficient or degree of confidence.
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Remark The length of confidence interval =
upper confidence limit - lower confidence limit
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Methods for constructing
confidence intervals
(1)Pivotal Quantity Method
(2)Maximum Likelihood Estimator (MLE) Method (3)Bayesian Method
(4)Invariant Methods
(5)Inversion of Test Statistic Method
(6)The Statistical or General Method.
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Pivotal Quantity Method
Definition
Let X1, X2, ..., Xn be a random sample of size n
from a population X with probability density function
f(x; θ),
where θ is an unknown parameter.
A pivotal quantity Q is a function of X1,X2, ...,Xn and θ
whose probability distribution is independent of the
parameter θ.
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Remark 1 The pivotal quantity Q(X1,X2, ...,Xn, θ) will
usually contain both the parameter θ and an
estimator (a statistic) of θ.
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Example 1
Let X1, X2, ..., X11 be a random sample of size 11
from a normal distribution with unknown mean μ and
variance 𝜎2 = 9.9. If 𝑥𝑖11𝑖=1 =132 , then what is the
95% confidence interval for μ?
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Answer 1
Since 𝑋𝑖 ∼ 𝑁(𝜇, 9.9), the confidence interval for 𝜇 is given by
X − zα2
σ
n, X + zα
2
σ
n
Since 𝑥𝑖11𝑖=1 = 132, the sample mean 𝑥 =
132
11= 12,
𝜎2
𝑛 =
9.9
11= 0.9
Since 1 − 𝛼 = 0.95 , 𝛼 = 0.05, So 𝑧𝛼2= 𝑧0.025 = 1.96
(from normal table)
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The confidence interval for μ is
12 − 1.96 0.9 , 12 + 1.96 0.9
10.141, 13.859
Example 2
Let X1, X2, ..., X11 be a random sample of size 11
from a normal distribution with unknown mean μ and
variance 𝜎2 = 9.9. If 𝑥𝑖11𝑖=1 =132, then for what value of the
constant k is 12 − 𝑘 0.9 , 12 + 𝑘 0.9
a 90% confidence interval for μ?
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Answer 2
The 90% confidence interval for μ when the variance
is given is
X − zα2
σ
n, X + zα
2
σ
n
Thus we need to find X ,σ
n and zα
2 corresponding to
1 − 𝛼 = 0.90. Hence
𝑥 =132
11= 12
𝜎2
𝑛 =
9.9
11= 0.9
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𝑧𝛼2= 𝑧0.05 = 1.64 (from normal table).
Hence, the confidence interval for μ at 90%
confidence level is
12 − 1.64 0.9 , 12 + 1.64 0.9
10.444, 13.556
Comparing this interval with the given interval, we get
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K=1.64
and the corresponding 90% confidence interval is
[10.141, 13.859]
Remark
• The confidence level is directly proportional to the length
of the confidence interval.
For example
• Notice that the length of the 90% confidence interval for μ
is 3.112. However, the length of the 95% confidence
interval is 3.718
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Thank You