Chapter 11: Problem 9
Use the information that, for events \(\mathrm{A}\) and \(\mathrm{B}\), we have \(P(A)=0.8, P(B)=0.4\), and \(P(A\) and \(B)=0.25.\) Find \(P(\operatorname{not} B).\)
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Curving Grades on an Exam A statistics instructor designed an exam so that the grades would be roughly normally distributed with mean \(\mu=75\) and standard deviation \(\sigma=10 .\) Unfortunately, a fire alarm with ten minutes to go in the exam made it difficult for some students to finish. When the instructor graded the exams, he found they were roughly normally distributed, but the mean grade was 62 and the standard deviation was 18\. To be fair, he decides to "curve" the scores to match the desired \(N(75,10)\) distribution. To do this, he standardizes the actual scores to \(z\) -scores using the \(N(62,18)\) distribution and then "unstandardizes" those \(z\) -scores to shift to \(N(75,10)\). What is the new grade assigned for a student whose original score was 47 ? How about a student who originally scores a \(90 ?\)
Find the specified areas for a \(N(0,1)\) density. The area above \(z=1.35\). (b) The area below \(z=-0.8\). (c) The area between \(z=-1.23\) and \(z=0.75\).
State whether the two events (A and B) described are disjoint, independent, and/or complements. (It is possible that the two events fall into more than one of the three categories, or none of them.) South Africa plays Australia for the championship in the Rugby World Cup. Let \(\mathrm{A}\) be the event that Australia wins and \(\mathrm{B}\) be the event that South Africa wins. (The game cannot end in a tie.)
Use the information that, for events \(\mathrm{A}\) and \(\mathrm{B}\), we have \(P(A)=0.8, P(B)=0.4\), and \(P(A\) and \(B)=0.25.\) Find \(P(A\) or \(B)\).
\(\mathbf{P . 1 0 8}\) Find \(P(X=2)\) if \(X\) is a binomial random variable with \(n=6\) and \(p=0.3\). if \(X\) is a binomial random variable with \(n=8\) and \(p=0.9\).
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