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Unformatted text preview: hat 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 − zα/2 σ/ n and U = X + zα/2 σ/ n
Utku Suleymanoglu (UMich) Interval Estimation 4 / 16 Interval Estimation of Population Mean We ﬁgured out the lower and upper bounds of the 100(1 − α)% conﬁdence interval.
Notice that they are in the format (as promised) x ± ME .
Margin of error in this case is zα/2 σ/ n. Or zα/2 σx
Let’s summarize this result before we discuss what it means:
CI for µ, Case 1: Normal Population, σ known
If population has a normal distribution and σ is known, a 100(1 − α)% conﬁdence
interval for µ can be constructed via:
x ± zα/2 σ/ n
where zα/2 is the z value such that F (z ) = 1 − α/2 or simply z value with upper tail
probability of α/2.
Now let’s do an example. Utku Suleymanoglu (UMich) Interval Estimation 5 / 16 Interval Estimation of Population Mean Example Suppose you have a normally distributed population with variance σ 2 = 4. You have a
sample of 400 observations, whose sample average is x = 3. Let’s build a 90% conﬁdence
interval for µ.
If (1 − α) = 0.90, then α = 0.10 and α/2 = 0.05. What is th...
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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