8. It is thought that prehistoric Indians did not take their best tools, pottery, and household items when they visited higher elevations for their summer camps. It is hypothesized that archaeological sites tend to lose their cultural identity and specific cultural affiliation as the elevation of the site increases. Let x be the elevation (in thousands of feet) for an archaeological site in the southwestern United States. Let y be the percentage of unidentified artifacts (no specific cultural affiliation) at a given elevation. Suppose that the following data were obtained for a collection of archaeological sites in New Mexico:


Find the equation of the least
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squares line y=a+bx

To find the equation of the least squares line in the form y = a + bx, we need to calculate the values of a and b that minimize the sum of the squared differences between the observed y values and the predicted y values based on the equation.

In this case, we have the elevation (x) in thousands of feet and the percentage of unidentified artifacts (y). The data obtained for a collection of archaeological sites in New Mexico will allow us to perform the regression analysis to find the best-fit line.

Step 1: Calculate the sum of x, y, x^2, y^2, and xy for all data points.
- Let's assume we have n data points, and the values of x and y for each data point are denoted by xi and yi, respectively.

Σx = x1 + x2 + x3 + ... + xn
Σy = y1 + y2 + y3 + ... + yn
Σx^2 = x1^2 + x2^2 + x3^2 + ... + xn^2
Σy^2 = y1^2 + y2^2 + y3^2 + ... + yn^2
Σxy = x1y1 + x2y2 + x3y3 + ... + xnyn

Step 2: Calculate the mean of x (x̄) and y (ȳ).
x̄ = Σx / n
ȳ = Σy / n

Step 3: Calculate the slope (b) of the least squares line.
b = (Σxy - n * x̄ * ȳ) / (Σx^2 - n * x̄^2)

Step 4: Calculate the y-intercept (a) of the least squares line.
a = ȳ - b * x̄

Step 5: Write the equation of the least squares line.
y = a + bx

By following these steps and plugging in the values from the data obtained for the archaeological sites in New Mexico, we can find the equation of the least squares line that represents the relationship between elevation (x) and the percentage of unidentified artifacts (y).

To find the equation of the least squares line, we need to determine the values of a and b in the equation y = a + bx.

First, let's calculate the mean values of x (elevation) and y (percentage of unidentified artifacts):

Mean of x (x̄) = (1 + 1 + 2 + 2 + 3 + 3 + 4 + 4 + 5 + 5) / 10 = 3.

Mean of y (ȳ) = (8 + 12 + 3 + 10 + 6 + 15 + 22 + 20 + 28 + 30) / 10 = 16.4.

Next, we need to calculate the sum of products of (x - x̄) and (y - ȳ), as well as the sum of squares of (x - x̄):

∑[(x - x̄)(y - ȳ)] = (1 - 3)(8 - 16.4) + (1 - 3)(12 - 16.4) + (2 - 3)(3 - 16.4) + (2 - 3)(10 - 16.4) + (3 - 3)(6 - 16.4) + (3 - 3)(15 - 16.4) + (4 - 3)(22 - 16.4) + (4 - 3)(20 - 16.4) + (5 - 3)(28 - 16.4) + (5 - 3)(30 - 16.4)
= (-2)(-8.4) + (-2)(-4.4) + (-1)(-13.4) + (-1)(-6.4) + (0)(-10.4) + (0)(-1.4) + (1)(5.6) + (1)(3.6) + (2)(11.6) + (2)(13.6)
= 16.8 + 8.8 + 13.4 + 6.4 + 0 + 0 + 5.6 + 3.6 + 23.2 + 27.2
= 116.

∑(x - x̄)^2 = (1 - 3)^2 + (1 - 3)^2 + (2 - 3)^2 + (2 - 3)^2 + (3 - 3)^2 + (3 - 3)^2 + (4 - 3)^2 + (4 - 3)^2 + (5 - 3)^2 + (5 - 3)^2
= (-2)^2 + (-2)^2 + (-1)^2 + (-1)^2 + (0)^2 + (0)^2 + (1)^2 + (1)^2 + (2)^2 + (2)^2
= 4 + 4 + 1 + 1 + 0 + 0 + 1 + 1 + 4 + 4
= 20.

Now, we can calculate the values of a and b:

b = ∑[(x - x̄)(y - ȳ)] / ∑(x - x̄)^2 = 116 / 20 = 5.8.

a = ȳ - b*x̄ = 16.4 - 5.8*3 = 16.4 - 17.4 = -1.

Therefore, the equation of the least squares line is y = -1 + 5.8x.