91Ó°ÊÓ

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 \(\mathbf{E}: x\) variable $$\begin{array}{ccccccc} 13.20 & 5.60 & 19.80 & 15.05 & 21.40 & 17.25 & 27.45 \\ 16.95 & 23.90 & 32.40 & 40.75 & 5.10 & 17.75 & 28.35 \end{array}$$ Grid H: \(y\) variable $$\begin{array}{lllllll} 11.85 & 15.25 & 21.30 & 17.30 & 27.50 & 10.35 & 14.90 \\ 48.70 & 25.40 & 25.95 & 57.60 & 34.35 & 38.80 & 41.00 \\ 31.25 & & & & & \end{array}$$ (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 \(C V\) 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 \(C V\) be better, or at least more exciting? Explain.

Step by step solution

01

Compute Sums and Squares

First, we will calculate the required sums: \( \Sigma x, \Sigma x^2, \Sigma y, \) and \( \Sigma y^2 \). For Grid \( E \) (\( x \)), sum the provided values to get \( \Sigma x = 285.90 \) and calculate the squares of individual \( x \) values, then sum them for \( \Sigma x^2 = 6735.985 \). Similarly, for Grid \( H \) (\( y \)), calculate \( \Sigma y = 386.40 \) and the square of each \( y \) value, then sum them for \( \Sigma y^2 = 18250.19 \).

02

Calculate Sample Mean, Variance, and Standard Deviation

The sample mean \( \bar{x} \) for Grid \( E \) is \( \frac{\Sigma x}{n} = \frac{285.90}{14} = 20.422857 \). The variance \( s_x^2 \) is calculated as \( \frac{\Sigma x^2 - \frac{(\Sigma x)^2}{n}}{n-1} = \frac{6735.985 - \frac{(285.90)^2}{14}}{13} = 71.740577 \). The standard deviation \( s_x \) is \( \sqrt{s_x^2} = 8.470616 \). Repeat similar calculations for Grid \( H \) (\( y \)): \( \bar{y} = 21.466667 \), \( s_y^2 = 247.372 \), and \( s_y = 15.7257 \).

03

Determine Chebyshev Interval

For a 75% Chebyshev interval, use the formula: \( \bar{x} \pm k \cdot s_x \) where \( k = \sqrt{\frac{1}{0.25}} = 2 \). For Grid \( E \), the interval is \( 20.422857 \pm 2 \times 8.470616 = [3.481625, 37.364089] \). For Grid \( H \), calculate [[\bar{y} \pm 2 \cdot s_y], resulting in [−9.984733, 52.918067]. Compare these intervals: Grid \( E \) and Grid \( H \) show variability ranges for identifying potential anomalies.

04

Coefficient of Variation

The coefficient of variation (CV) for Grid \( E \) is \( \frac{s_x}{\bar{x}} = \frac{8.470616}{20.422857} = 0.414925 \). For Grid \( H \), \( CV \) is \( \frac{15.7257}{21.4667} = 0.73285 \). Comparing \( CV \) values, Grid \( H \) shows higher relative variability, suggesting possible archaeological significance indicated by anomalies.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Mean

The mean is a measure of central tendency, which provides us with the average value of a dataset. It's calculated by summing all data points and dividing by the number of points.
In the context of the magnetic susceptibility data from the Hill of Tara, calculating the mean helps us understand the average level of magnetic susceptibility for the different grids (Grid E and Grid H). This is particularly important since large deviations from the mean can signal relevant anomalies.
For example, the mean for Grid E can be calculated by summing all the magnetic susceptibility values across the grid and dividing by the total number of readings, i.e., \(\bar{x} = \frac{\Sigma x}{n} \). For Grid E, the calculation is \(\bar{x} = \frac{285.90}{14} \approx 20.42 \).
This represents the average magnetic field influence found in Grid E, and similar calculations are done for Grid H.

Exploring Variance

Variance is a statistical measure that tells us how spread out the values in a dataset are. Essentially, it quantifies the degree of variation from the mean in the dataset.

• A high variance means that the data points are spread out over a larger range of values.
• A low variance means that the data points tend to be close to the mean.

In our exercise, calculating the variance for both Grid E and Grid H helps identify how much individual magnetic susceptibility readings deviate from their respective means. The formula used for variance is: \[s^2 = \frac{\Sigma x^2 - \frac{(\Sigma x)^2}{n}}{n-1}\]

A higher variance in one grid compared to another can be an indication of the presence of outliers or anomalies, which are of great interest in archaeological studies as they might indicate buried artifacts.

The Role of Standard Deviation

Standard deviation is directly related to variance. It is simply the square root of the variance. While variance gives a squared value which can be harder to interpret, standard deviation provides a more intuitive measure of variability because it is in the same unit as the original data.
For example, the standard deviation \( s \) for Grid E, derived from the variance, tells us on average how much each reading is expected to deviate from the mean reading.
Calculating standard deviation involves taking the square root of the variance: \(s = \sqrt{s^2} \). This calculation for Grid E gave us approximately \(8.47\), which indicates how spread out the values are around the mean of 20.42.
The significance of standard deviation in this exercise lies in its ability to help researchers compare data spreads and ascertain whether certain variations might indicate subterranean artifacts.

Chebyshev's Theorem Explanation

Chebyshev's theorem is a useful statistical tool when trying to understand how data points are dispersed around the mean, especially in non-normal distributions. It states that no less than \(1 - \frac{1}{k^2}\) of data lies within \(k\) standard deviations from the mean for any dataset.
In this exercise, a 75% Chebyshev interval is calculated using \(k = 2\) to encompass most of the data. This means that 75% of the values fall within two standard deviations from the mean.
Specifically, for Grid E, applying Chebyshev's theorem provides an interval that gives us an expectation that most magnetic susceptibility values lie within the range of \([3.48, 37.36]\) cm-g-s units.
Identifying such ranges is critical in archaeological surveys because it helps researchers focus on variability that falls outside these intervals, as such anomalies might indicate the presence of artifacts.

Understanding the Coefficient of Variation

The coefficient of variation (CV) is a standardized measure of dispersion calculated as the ratio of the standard deviation to the mean, often expressed as a percentage.
This concept is especially useful when comparing the degree of variation between datasets with different units or widely different means.
For our dataset, the CV for Grid E, which we calculated as \(0.414925\), informs us about how much variation exists in relation to the mean magnetic susceptibility reading for the grid.
Higher CV values point to a higher level of relative variation, which is intriguing in the context of archaeology. More variability can suggest a higher chance of finding unexpected or exceptional values that might correspond to underground discoveries. Thus, a higher CV can indicate rich areas for archaeological exploration and excavation.