The Hill of Tara in Ireland is a place of great archaeological importance.
This region has been occupied by people for more than 4,000 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 \(\begin{array}{lrrrrrr}13.20 & 5.60 & 19.80 & 15.05 &
21.40 & 17.25 & 27.45 \\ 16.95 & 23.90 & 32.40 & 40.75 & 5.10 & 17.75 & 28.35
\\\ \text { Grid H: } y & \text { variable } & & & & & \end{array}\)
\(\begin{array}{lllllll}11.85 & 15.25 & 21.30 & 17.30 & 27.50 & 10.35 & 14.90
\\\ 48.70 & 25.40 & 25.95 & 57.60 & 34.35 & 38.80 & 41.00 \\ 31.25 & & & & &
& \\ & & & & & & \end{array}\)
(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 CV's to
compare the two grids. If \(s\) represents variability in the signal (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, 2
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