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Chapter 3: Problem 20
Find the weighted average of a data set where 10 has a weight of \(5 ; 20\) has a weight of \(3 ; 30\) has a weight of 2
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Find the mean, median, and mode of the data set \(\begin{array}{lllll}10 & 12 & 20 & 15 & 20\end{array}\)
The Hill of Tara in Ireland is a place of great archaeological importance. This region has been occupied by people for more than 4000 years. Geomagnetic surveys detect subsurface anomalies in the earth's magnetic field. These surveys have led to many significant archaeological discoveries. After collecting data, the next step is to begin a statistical study. The following data measure magnetic susceptibility (centimeter-gram-second \(\times 10^{-6}\) ) on two of the main grids of the Hill of Tara (Reference: Tara: An Archaeological Survey by Conor Newman, Royal Irish Academy, Dublin). Grid E: \(x\) variable \begin{tabular}{lrlllll} \(13.20\) & \(5.60\) & \(19.80\) & \(15.05\) & \(21.40\) & \(17.25\) & \(27.45\) \\ \(16.95\) & \(23.90\) & \(32.40\) & \(40.75\) & \(5.10\) & \(17.75\) & \(28.35\) \\ Grid H: \(y\) variable & & & & & \\ \(11.85\) & \(15.25\) & \(21.30\) & \(17.30\) & \(27.50\) & \(10.35\) & \(14.90\) \\ \(48.70\) & \(25.40\) & \(25.95\) & \(57.60\) & \(34.35\) & \(38.80\) & \(41.00\) \\ \(31.25\) & & & & & & \\ \hline \end{tabular} (a) Compute \(\Sigma x, \Sigma x^{2}, \Sigma y\), and \(\Sigma y^{2}\). (b) Use the results of part (a) to compute the sample mean, variance, and standard deviation for \(x\) and for \(y\). (c) Compute a \(75 \%\) Chebyshev interval around the mean for \(x\) values and also for \(y\) values. Use the intervals to compare the magnetic susceptibility on the two grids. Higher numbers indicate higher magnetic susceptibility. However, extreme values, high or low, could mean an anomaly and possible archaeological treasure. (d) Compute the sample coefficient of variation for each grid. Use the \(\mathrm{CV}\) s to compare the two grids. If \(s\) represents variability in the sig- -nal (magnetic susceptibility) and \(\bar{x}\) represents the expected level of the signal, then \(s / \bar{x}\) can be thought of as a measure of the variability per unit of expected signal. Remember, a considerable variability in the signal (above or below average) might indicate buried artifacts. Why, in this case, would a large \(\mathrm{CV}\) be better, or at least more exciting? Explain.
What symbol is used for the standard deviation when it is a sample statistic? What symbol is used for the standard deviation when it is a population parameter?
What was the age distribution of prehistoric Native Americans? Extensive anthropologic studies in the southwestern United States gave the following information about a prehistoric extended family group of 80 members on what is now the Navajo Reservation in northwestern New Mexico (Source: Based on information taken from Prehistory in the Navajo Reservation District, by F. W. Eddy, Museum of New Mexico Press). \begin{tabular}{l|cccc} \hline Age range (years) & \(1-10^{*}\) & \(11-20\) & \(21-30\) & 31 and over \\ \hline Number of individuals & 34 & 18 & 17 & 11 \\ \hline \end{tabular} "Includes infants. For this community, estimate the mean age expressed in years, the sample variance, and the sample standard deviation. For the class 31 and over, use \(35.5\) as the class midpoint.
The Grand Canyon and the Colorado River are beautiful, rugged, and sometimes dangerous. Thomas Myers is a physician at the park clinic in Grand Canyon Village. Dr. Myers has recorded (for a 5 -year period) the number of visitor injuries at different landing points for commercial boat trips down the Colorado River in both the Upper and Lower Grand Canyon (Source: Fateful Journey by Myers, Becker, and Stevens). Upper Canyon: Number of Injuries per Landing Point Between North Canyon and Phantom Ranch \(\begin{array}{lllllllllll}2 & 3 & 1 & 1 & 3 & 4 & 6 & 9 & 3 & 1 & 3\end{array}\) Lower Canyon: Number of Injuries per Landing Point Between Bright Angel and Lava Falls \(\begin{array}{llllllllllllll}8 & 1 & 1 & 0 & 6 & 7 & 2 & 14 & 3 & 0 & 1 & 13 & 2 & 1\end{array}\) (a) Compute the mean, median, and mode for injuries per landing point in the Upper Canyon. (b) Compute the mean, median, and mode for injuries per landing point in the Lower Canyon. (c) Compare the results of parts (a) and (b). (d) The Lower Canyon stretch had some extreme data values. Compute a \(5 \%\) trimmed mean for this region, and compare this result to the mean for the Upper Canyon computed in part (a).
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