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 … week 5 1

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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 … week 5 3

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Summary

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.

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.

Examples