Let X1, , Xn be iid n(, 2), where 2 is known We know that a 1 confidence interval for is z 2 n a Show that a 1 prediction interval for Xn 1 is z 2 1 1 n b Show that a 1 tolerance interval for 100p of the underlying population is given by z 2 (1 1 n) c Find a 1 prediction interval for Xn 1 if 2 is u...

a The distribution of X n1 is n0 2 1 1n so P X n1 z 2 1 1n P Z z 2 1 b p percent of the no ...

Let X1,..., Xn be iid n(, 2), where 2 is known. We know that a 1 -

Question:

Let X1,..., Xn be iid n(μ, σ2), where σ2 is known. We know that a 1 - α confidence interval for μ is ± zα/2 σ/√n.
a. Show that a 1 - α prediction interval for Xn+1 is ± zα/2 σ √1 + 1/n.
b. Show that a 1 - α tolerance interval for 100p% of the underlying population is given by ± zα/2 σ (1 + 1/√n).
c. Find a 1 - α prediction interval for Xn+1 if σ2 is unknown.