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Type I and Type II Errors. In Exercises 29–32, provide statements that identify the type I error and the type II error that correspond to the given claim. (Although conclusions are usually expressed in verbal form, the answers here can be expressed with statements that include symbolic expressions such as p = 0.1.)
31. The proportion of adults who use the Internet is greater than 0.87.
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- 2. Which of the following statements are (not) true? lim sup{An U Bn} 818 lim sup{A, B} 818 lim inf{An U Bn} 818 818 lim inf{A, B} An An A, Bn- A, BnB →B = = = lim sup A, U lim sup Bn; 818 818 lim sup A, lim sup Bn; 818 81U lim inf A, U lim inf Bn; 818 818 lim inf A, lim inf Bn; n→X 818 An U BRAUB as no; An OBRANB as n→∞.arrow_forwardThroughout, A, B, (An, n≥ 1), and (Bn, n≥ 1) are subsets of 2. 1. Show that AAB (ANB) U (BA) = (AUB) (AB), Α' Δ Β = Α Δ Β, {A₁ U A2} A {B₁ U B2) C (A1 A B₁}U{A2 A B2).arrow_forward16. Show that, if X and Y are independent random variables, such that E|X|< ∞, and B is an arbitrary Borel set, then EXI{Y B} = EX P(YE B).arrow_forward
- Proposition 1.1 Suppose that X1, X2,... are random variables. The following quantities are random variables: (a) max{X1, X2) and min(X1, X2); (b) sup, Xn and inf, Xn; (c) lim sup∞ X and lim inf∞ Xn- (d) If Xn(w) converges for (almost) every w as n→ ∞, then lim- random variable. → Xn is aarrow_forwardExercise 4.2 Prove that, if A and B are independent, then so are A and B, Ac and B, and A and B.arrow_forward8. Show that, if {Xn, n ≥ 1) are independent random variables, then sup X A) < ∞ for some A.arrow_forward
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