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Unformatted text preview: interval. How big the interval
We will use a probabilistic way of doing this.
Conﬁdence Interval General Form
Suppose unknown µ (or any population parameter) is of interest. And we ﬁgured out
that there are two values, L and U such that when we calculate the probability
P (L < µ < U ) = 1 − α where α is in (0, 1). Then the interval (L, U ) is called the
100(1 − α)% conﬁdence interval.
Is this a good idea? We will see. . . .
But for now, let’s try to do what this asks for. Utku Suleymanoglu (UMich) Interval Estimation 3 / 17 Interval Estimation of Population Mean Case 1: Normally distributed population, σ known
We learned that if X ∼ N (µ, σ 2 ) then X ∼ N (µ, σ 2 /n).
¯ X −µ
So that Z = σ/√n has a standard normal distribution. Then we know that there is a
value zα/2 such that 1 − α = P (−zα/2 < Z < zα/2 )
√ < zα/2 )
= P − zα/2 (σ/ n) < X − µ < zα/2 (σ/ n)
= P (−zα/2 < √
= P − X − zα/2 (σ/ n) < −µ < −X + zα/2 (σ/ n)
= P X − zα/2 (σ/ n) < µ < X + zα/2 (σ/ n)
=P L<µ<U √
where L = X...
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This note was uploaded on 03/17/2014 for the course ECON 404 taught by Professor Staff during the Spring '08 term at University of Michigan.
- Spring '08