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Chapter 9: Problem 476
Given the following set of ungrouped measurements $$ 3,5,6,6,7, \text { and } 9 $$ determine the mean, median, and mode.
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The seven dwarfs challenged the Harlem Globetrotters to a basketball game. Besides the obvious difference in height, we are interested in constructing a \(95 \%\) confidence in ages between dwarfs and basketball players. The respective ages are, $$ \begin{array}{|l|l|l|l|} \hline \text { Dwardfs } & & \text { Globetrotters } & \\ \hline \text { Sneezy } & 20 & \text { Meadowlark } & 43 \\ \hline \text { Grumpy } & 39 & \text { Curley } & 37 \\ \hline \text { Dopey } & 23 & \text { Marques } & 45 \\ \hline \text { Doc } & 41 & \text { Bobby Joe } & 25 \\ \hline \text { Sleepy } & 35 & \text { Theodis } & 34 \\ \hline \text { Happy } & 29 & & \\ \hline \text { Bashful } & 31 & & \\ \hline \end{array} $$ Can you construct the interval? Assume the variance in age is the same for dwarfs and Globetrotters.
Consider the Poisson distribution \(\left(\mathrm{e}^{-\lambda} \lambda^{\mathrm{k}}\right) / \mathrm{k} !\). Prove \(\left[\left(e^{-\lambda} \lambda^{k-1}\right) /(k-1) !\right]<\left(e^{-\lambda} \lambda^{k}\right) / k ! \quad\) for \(k<\lambda\), \(\left[\left(\mathrm{e}^{-\lambda} \lambda^{\mathrm{k}-1}\right) /(\mathrm{k}-1) !\right]>\left(\mathrm{e}^{-\lambda} \lambda^{\mathrm{k}}\right) / \mathrm{k} ! \quad\) for \(\mathrm{k}>\lambda\) \(\left[\left(e^{-\lambda} \lambda^{k-1}\right) /(k-1) !\right]=\left(e^{-\lambda} \lambda^{k}\right) / k ! \quad\) if \(\lambda\) is an integer and \(\mathrm{k}=\lambda\).
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 the exponential distribution \(\mathrm{f}(\mathrm{x})=\lambda \mathrm{e}^{-\lambda \mathrm{x}}\) for \(\mathrm{x}>0\). Find the moment generating function and from it, the mean and variance of the exponential distribution.
Flapjack Computers is interested in developing a new tape drive for a proposed new computer. Flapjack does not have research personnel available to develop the new drive itself and so is going to subcontract the development to an independent research firm. Flapjack has set a fee of \(\$ 250,000\) for developing the new tape drive and has asked for bids from various research firms. The bid is to be awarded not on the basis of price (set at \(\$ 250,000\) ) but on the basis of both the technical plan shown in the bid and the firm's reputation. Dyna Research Institute is considering submitting a proposal (i.e., a bid) to Flapjack to develop the new tape drive. Dyna Research Management estimated that it would cost about \(\$ 50,000\) to prepare a proposal; further they estimated that the chances were about \(50-50\) that they would be awarded the contract. There was a major concern among Dyna Research engineers concerning exactly how they would develop the tape drive if awarded the contract. There were three alternative approaches that could be tried. One involved the use of certain electronic components. The engineers estimated that it would cost only \(\$ 50,000\) to develop a prototype of the tape drive using the electronic approach, but that there was only a 50 percent chance that the prototype would be satisfactory. A second approach involved the use of certain magnetic apparatus. The cost of developing a prototype using this approach would cost \(\$ 80,000\) with 70 percent chance of success. Finally, there was a mechanical approach with cost of \(\$ 120,000\), but the engineers were certain of success. Dyna Research could have sufficient time to try only two approaches. Thus, if either the magnetic or the electronic approach tried and failed, the second attempt would have to use the mechanical approach in order to guarantee a successful prototype. The management of Dyna Research was uncertain how to take all this information into account in making the immediate decision-whether to spend \(\$ 50,000\) to develop a proposal for Flapjack. Can you help?
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