BASIC PRACTICE OF STATISTICS(REISSUE)>C
BASIC PRACTICE OF STATISTICS(REISSUE)>C
8th Edition
ISBN: 9781319341831
Author: Moore
Publisher: MAC HIGHER
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Chapter 21, Problem 21.41E

(a)

To determine

To check: Whether the sample means suggest that there is a difference in the mean trustworthy ratings between the two groups or not.

(b)

To determine

To construct: The stemplot for positive attitude and negative attitude.

To identify: There is any deviation from normality or not.

(c)

To determine

To test: Whether the students with a positive attitude rate Mitt Romney more trustworthy than those with a negative attitude or not.

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6. Let, for p = (0, 1), and xe R. X be a random variable defined as follows: P(X=-x) = P(X = x)=p. P(X=0)= 1-2p. Show that there is equality in Chebyshev's inequality for X. This means that Chebyshev's inequality, in spite of being rather crude, cannot be improved without additional assumptions.
4. Prove that, for any random variable X, the minimum of EIX-al is attained for a = med (X).
8. Recall, from Sect. 2.16.4, the likelihood ratio statistic, Ln, which was defined as a product of independent, identically distributed random variables with mean 1 (under the so-called null hypothesis), and the, sometimes more convenient, log-likelihood, log L, which was a sum of independent, identically distributed random variables, which, however, do not have mean log 1 = 0. (a) Verify that the last claim is correct, by proving the more general statement, namely that, if Y is a non-negative random variable with finite mean, then E(log Y) log(EY). (b) Prove that, in fact, there is strict inequality: E(log Y) < log(EY), unless Y is degenerate. (c) Review the proof of Jensen's inequality, Theorem 5.1. Generalize with a glimpse on (b).
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