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 10−6) on two of the main grids of the Hill of Tara.

Grid E: x variable
14.51 14.88 35.23 17.10 24.87 6.74 13.77
21.17 11.18 13.77 35.60 28.20 11.55 7.85
Grid H: y variable
16.11 40.55 23.63 51.83 20.81 17.52 45.25
46.19 12.35 10.00 46.66 26.92 13.29 24.10

(a) Compute Óx, Óx2, Óy, Óy2. (Round your answers to two decimal places.)
Óx =__
Óx2 = ___
Óy = ___
Óy2 = ___

(b) Use the results of part (a) to compute the sample mean, variance, and standard deviation for x and for y. (Round your answers to two decimal places.)
Grid E: x =
s2 =
s =
Grid H: y =
s2 =
s =

(c) Compute a 75% Chebyshev interval around the mean for x values and also for y values. (Round your answers to two decimal places.)
Grid E:
11 to 12

Grid H:
13 to 14

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b. For both x and y, add the scores and divide by n to find the mean. Subtract the mean from each score, and square each one. The sum of these squared differences = variance. The variance = s^2 = s squared, so standard deviation = square root of variance.

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(a) To compute Óx, Óx2, Óy, Óy2, we need to find the sum and sum of squares for both variables.

For Grid E:
Óx = sum of x variable = 14.51 + 14.88 + 35.23 + 17.10 + 24.87 + 6.74 + 13.77 + 21.17 + 11.18 + 13.77 + 35.60 + 28.20 + 11.55 + 7.85
= 261.32

Óx2 = sum of squares of x variable = 14.51^2 + 14.88^2 + 35.23^2 + 17.10^2 + 24.87^2 + 6.74^2 + 13.77^2 + 21.17^2 + 11.18^2 + 13.77^2 + 35.60^2 + 28.20^2 + 11.55^2 + 7.85^2
= 6997.17

For Grid H:
Óy = sum of y variable = 16.11 + 40.55 + 23.63 + 51.83 + 20.81 + 17.52 + 45.25 + 46.19 + 12.35 + 10.00 + 46.66 + 26.92 + 13.29 + 24.10
= 384.32

Óy2 = sum of squares of y variable = 16.11^2 + 40.55^2 + 23.63^2 + 51.83^2 + 20.81^2 + 17.52^2 + 45.25^2 + 46.19^2 + 12.35^2 + 10.00^2 + 46.66^2 + 26.92^2 + 13.29^2 + 24.10^2
= 36925.14

Therefore,
Óx = 261.32
Óx2 = 6997.17
Óy = 384.32
Óy2 = 36925.14

(b) Using the results from part (a), we can compute the sample mean, variance, and standard deviation for both variables.

For Grid E:
Sample mean (x) = Óx / n, where n is the number of data points
= 261.32 / 14
= 18.67 (rounded to two decimal places)

Variance (s^2) = (Óx2 / n) - (Óx / n)^2
= (6997.17 / 14) - (18.67)^2
= 333.37 (rounded to two decimal places)

Standard deviation (s) = √(s^2)
= √(333.37)
= 18.25 (rounded to two decimal places)

For Grid H:
Sample mean (y) = Óy / n
= 384.32 / 14
= 27.45 (rounded to two decimal places)

Variance (s^2) = (Óy2 / n) - (Óy / n)^2
= (36925.14 / 14) - (27.45)^2
= 306.61 (rounded to two decimal places)

Standard deviation (s) = √(s^2)
= √(306.61)
= 17.51 (rounded to two decimal places)

Therefore,
For Grid E: x = 18.67, s^2 = 333.37, s = 18.25
For Grid H: y = 27.45, s^2 = 306.61, s = 17.51

(c) To compute a 75% Chebyshev interval around the mean for x and y values, we can use the formula:

Interval = [mean - (k * standard deviation), mean + (k * standard deviation)]

where k is the number of standard deviations away from the mean. For a 75% interval, k = 2.

For Grid E:
Interval for x values = [18.67 - (2 * 18.25), 18.67 + (2 * 18.25)]
= [18.67 - 36.50, 18.67 + 36.50]
= [11.17, 26.17] (rounded to two decimal places)

For Grid H:
Interval for y values = [27.45 - (2 * 17.51), 27.45 + (2 * 17.51)]
= [27.45 - 35.02, 27.45 + 35.02]
= [13.43, 41.47] (rounded to two decimal places)

Therefore,
Grid E: 75% Chebyshev interval for x values is 11 to 26.
Grid H: 75% Chebyshev interval for y values is 13 to 41.