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Chapter 9: Problem 474
Arrange the values \(7,8,5,10,3\) in ascending order and identify the value of the median.
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A research worker wishes to estimate the mean of a population using a sample large enough that the probability will be \(.95\) that the sample mean will not differ from the population mean by more than 25 percent of the standard deviation. How large a sample should he take?
Find the expected values of the random variables \(\mathrm{X}\) and \(\mathrm{Y}\) if \(\quad \operatorname{Pr}(\mathrm{X}=0)=1 / 2 \quad\) and \(\operatorname{Pr}(\mathrm{X}=1)=1 / 2\) and \(\operatorname{Pr}(\mathrm{Y}=1)=1 / 4 \quad\) and \(\operatorname{Pr}(\mathrm{Y}=2)=3 / 4\). Compare the sum of \(\mathrm{E}(\mathrm{X})+\mathrm{E}(\mathrm{Y})\) with \(\mathrm{E}(\mathrm{X}+\mathrm{Y})\) if \(\operatorname{Pr}(\mathrm{X}=\mathrm{x}, \mathrm{Y}=\mathrm{y})=\operatorname{Pr}(\mathrm{X}=\mathrm{x}) \operatorname{Pr}(\mathrm{Y}=\mathrm{y})\)
Given the probability distribution of the random variable \(\mathrm{X}\) in the table below, compute \(\mathrm{E}(\mathrm{X})\) and \(\operatorname{Var}(\mathrm{X})\). $$ \begin{array}{|c|c|} \hline \mathrm{x}_{\mathrm{i}} & \operatorname{Pr}\left(\mathrm{X}=\mathrm{x}_{i}\right) \\ \hline 0 & 8 / 27 \\ \hline 1 & 12 / 27 \\ \hline 2 & 6 / 27 \\ \hline 3 & 1 / 27 \\ \hline \end{array} $$
Consider a probability distribution for random orientations in which the probability of an observation in a region on the surface of the unit hemisphere is proportional to the area of that region. Two angles, \(u\) and \(v\), will determine the position of an observation. It can be shown that the position of an observation is jointly distributed with density function $$ \begin{array}{r} \mathrm{f}(\mathrm{u}, \mathrm{v})=[\\{\sin \mathrm{u}\\} /\\{2 \pi\\}] \quad 0<\mathrm{u}<2 \pi \\ 0<\mathrm{u}<\pi / 2 . \end{array} $$ Two new variables, \(\mathrm{X}\) and \(\mathrm{Y}\) are defined, where $$ \mathrm{X}=\sin \mathrm{u} \cos \mathrm{v} $$ $$ \mathrm{Y}=\sin \mathrm{u} \sin \mathrm{v} $$ Find the joint density function of \(\mathrm{X}\) and \(\mathrm{Y}\).
Plot the dependent variable against the Independent variable. Find the least squares line for this data. What is the Y-intercept? If \(3.0\) units of fertilizer were used what would be a good guess as to the resultant corn yield? $$ \begin{array}{l|l|l|l|l|l|l|l|l|l} \mathrm{X} & \text { Fertilizer } & .3 & .6 & .9 & 1.2 & 1.5 & 1.8 & 2.1 & 2.4 \\\ \hline \mathrm{Y} & \text { Corn Yield } & 10 & 15 & 30 & 35 & 25 & 30 & 50 & 45 \end{array} $$
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