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Unformatted text preview: glu (UMich) Interval Estimation 7 / 16 Interval Estimation of Population Mean Case 2: Normally distributed population, σ unknown
We keep talking about unknown population parameters: µ is unknown, we are building
an interval estimate for it.
How realistic is that σ is known. Not really: we almost never know what σ is. So what
do we do?
We can use s , sample standard deviation, instead. We can do this, but need a
modiﬁcation to make it work.
It turns out the sampling distribution of x is not normal if you use s instead of σ .
Sampling Distribution of x with unknown σ
If x is the sample mean of a sample drawn from a population with N (µ, σ 2 ) distribution,
t= x −µ
s/ n has a Student’s t distribution with n-1 degrees of freedom Utku Suleymanoglu (UMich) Interval Estimation 8 / 16 Interval Estimation of Population Mean Student’s t-distribution is another continuous probability distribution.
It looks a lot like the standard normal distribution with symmetricity and mean at
It has a single parameter, called its degrees of freedom. As df increases, it converges
to N (0, 1).
We will rely on another table to calculate probabilities for it. Utku Suleymanoglu (UMich) Interval Estimation 9 / 16 Interval Estimation of Population Mean Utku Suleymanoglu (UMich) Interval Estimation 10 / 16 Interval Estimation of Population Mean CI for µ, Case 2: Normal Population, σ unknown
If population has a normal distribution and σ is unknown, but sample standard
deviation,s , is known, a 100(1 − α)% conﬁdence interval for µ can be constructed via:
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- Spring '08