Chapter 9: Problem 478
A family had eight children. The ages were \(9,11,8,15,14\), \(12,17,14\) (a) Find the measures of central tendency for the data. (b) Find the range of the data.
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Suppose we have a binomial distribution for which \(\mathrm{H}_{0}\) is \(\mathrm{p}=1 / 2\) where \(\mathrm{p}\) is the probability of success on a single trial. Suppose the type I error, \(\alpha=.05\) and \(\mathrm{n}=100 .\) Calculate the power of this test for each of the following alternate hypotheses, \(\mathrm{H}_{1}: \mathrm{p}=.55, \mathrm{p}=.60, \mathrm{p}=.65, \mathrm{p}=.70\), and \(\mathrm{p}=.75 .\) Do the same when \(\alpha=.01\).
Consider a simple random variable \(\mathrm{X}\) having just two possible values \(\operatorname{Pr}(\mathrm{X}=1)=\mathrm{p}\) and \(\operatorname{Pr}(\mathrm{X}=0)=1-\mathrm{p}\). Find the moment generating function of \(\mathrm{X}\) and \(\mathrm{E}\left(\mathrm{X}^{\mathrm{k}}\right)\) for all \(\mathrm{k}=1,2,3, \ldots\)
Let \(\mathrm{T}\) be distributed with density function \(f(t)=\lambda e^{-\lambda . t} \quad\) for \(t>0\) and \(=0\) otherwise If \(S\) is a new random variable defined as \(S=\) In \(\mathrm{T}\), find the density function of \(\mathrm{S}\).
My wife wanted to know whether putting cut flowers into a certain chemical solution (we'll call it 'Flower-Life') would prolong their life, so we designed the following experiment. She bought 2 fresh blooms of 25 different kinds of flowers \(-2\) roses, 2 irises, 2 carnations, and so on We then put one of each pair in a vase of water, and their partners in a vase containing 'Flower-Life'. Both vases were put side by side in the same room, and the length of life of each flower was noted. We then had 2 matched samples, so the results could be tested for significance by Wilcoxon's Signed Ranks Test. This revealed a smaller rank total of \(50 .\) Is there a statistical difference between 'Flower-Life' and plain water?
Given the probability distribution of the random variable \(\mathrm{X}\) in the table below, compute \(\mathrm{E}(\mathrm{X})\) and \(\operatorname{Var}(\mathrm{X})\). $$ \begin{array}{|c|c|} \hline \mathrm{x}_{\mathrm{i}} & \operatorname{Pr}\left(\mathrm{X}=\mathrm{x}_{i}\right) \\ \hline 0 & 8 / 27 \\ \hline 1 & 12 / 27 \\ \hline 2 & 6 / 27 \\ \hline 3 & 1 / 27 \\ \hline \end{array} $$
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