1. CI for μ When σ is Unknown
Suppose X1, X2,…,Xn are random sample from N(μ, σ2) where both
μ and σ are unknown.
If σ2 is unknown we can estimate it using s2 and use the tn-1 distribution.
A 100(1-α)% confidence interval for μ from a small sample, nearly normal random sample, , is …
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2. Example

3. Prediction interval
In many applications, a researcher is interested in predicting a single
value of a variable to be observed in the future, rather than estimating
the mean value of that variable.
A 100(1-α)% prediction interval for a single new observation x, based
on a small sample, nearly normal random sample, is …
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4. Example

5. Summary

6. CI for Population Variance
In many case we will be interested in making inference about the population variance.
Suppose X1, X2,…,Xn are random sample from N(μ, σ2) where both
μ and σ are unknown. A CI for σ2 is …
This interval is valid even if the distribution we are sampling from is near normal.

7. CI for Population Standard Deviation
In many case we will be interested in making inference about the population standard deviation.
Suppose X1, X2,…,Xn are random sample from N(μ, σ2) where both
μ and σ are unknown. A CI for σ is …
This interval is valid even if the distribution we are sampling from is near normal.

8. Examples