6.3.25 (One-sided confidence intervals for means) Suppose that (x1,..., xn) is a sample from an N(, ?2 0) distribution, where ? R1 is unknown and ?2 0 is known. Suppose we want to make inferences about the interval ?() = (??, ). Consider the problem of finding an interval C(x1,..., xn) = (??, u (x1,..., xn)) that covers the interval (??, ) with probability at least ? . So we want u such that for every , P ( ? u (X1,..., Xn)) ? ? . Note that (??, ) ? (??, u (x1,..., xn)) if and only if ? u (x1,..., xn), so C(x1,..., xn) is called a left-sided ? -confidence interval for . Obtain an exact leftsided ? -confidence interval for using u (x1,..., xn) = x + k # ?0/ ?n $ , i.e., find the k that gives this property.

6.3.25 (One-sided confidence intervals for means) Suppose that (x1, . . ., Xn) is a sam- ple from an N(u, ) distribution, where u E R is unknown and o is known. Sup- pose we want to make inferences about the interval y (u) = (-0, u). Consider the problem of finding an interval C(x1, . . ., Xn) = (-0o, u (x1, ..., Xn)) that covers the interval (-oo, u) with probability at least y . So we want u such that for every u, Pu (u Su (X1, . . ., Xn)) 27. Note that (-oo, u) C (-oo, u (x1, ..., Xn)) if and only if u