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Unformatted text preview: tα/2,n−1 s / n
where tα/2,n−1 is t-value with n − 1 degrees of freedom and α/2 upper tail probability.
Notice now the margin of error is a function of both x and s
¯ Utku Suleymanoglu (UMich) Interval Estimation 11 / 16 Interval Estimation of Population Mean Example Similar example: Suppose you have a normally distributed population with variance σ 2
unknown. You have a sample of 14 observations, whose sample average is x = 7. Sample
standard deviation is measured to be 1.21. Let’s build a 95% conﬁdence interval for µ.
We need to get the margin of error. Let’s calculate tα/2,n−1 is ﬁrst. α = 0.05, so
α/2 = 0.025. n = 14, so n − 1 = 13. Let’s look up the table:
We have tα/2,n−1 = 2.160. So margin of error is tα/2,n−1 s / n = 2.160 × √21 . It is
0.6985. So the conﬁdence interval:
(7 − 0.6985, 7 + 0.6985) = (6.3015, 7.6985)
Food for thought: suppose the 1.21 was σ not s . What would happen to the width of
our interval estimate? Why? Utku Suleymanoglu (UMich) Interval Estimation 12 / 16 Interval Estimation of Population Mean Case 3: Population Not Normally Distributed, σ unknown Above results relied on the population values having a normal distribution. If this is
violated, sampling distribution of x cannot be normal or t-distribution. So the conﬁdence
intervals cannot be constructed as described.
Luckily, CLT comes to rescue. For large enough n, x is approximately normally
distributed, so we can build conﬁdence intervals.
CI for µ, Case 3: Any Population, σ unknown
If population has unknown dis...
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- Spring '08