Chapter 5: Problem 17
Let \(Y_{1}
Chapter 5: Problem 17
Let \(Y_{1}
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Get started for freeConsider the following algorithm: (1) Generate \(U\) and \(V\) independent uniform \((-1,1)\) random variables. (2) Set \(W=U^{2}+V^{2}\). (3) If \(W>1\) goto Step (1). (4) Set \(Z=\sqrt{(-2 \log W) / W}\) and let \(X_{1}=U Z\) and \(X_{2}=V Z\).
Let \(X\) be a random variable with a continuous cdf \(F(x)\). Assume that \(F(x)\) is strictly increasing on the space of \(X\). Consider the random variable \(Z=F(X)\). Show that \(Z\) has a uniform distribution on the interval \((0,1)\).
A Weibull distribution with pdf
$$
f(x)=\left\\{\begin{array}{ll}
\frac{1}{\theta^{3}} 3 x^{2} e^{-x^{3} / \theta^{3}} & 0
Let \(Y_{1}
. Let \(p\) equal the proportion of drivers who use a seat belt in a state that does not have a mandatory seat belt law. It was claimed that \(p=0.14 .\) An advertising campaign was conducted to increase this proportion. Two months after the campaign, \(y=104\) out of a random sample of \(n=590\) drivers were wearing their seat belts. Was the campaign successful? (a) Define the null and alternative hypotheses. (b) Define a critical region with an \(\alpha=0.01\) significance level. (c) Determine the approximate \(p\) -value and state your conclusion.
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