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

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.

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?

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