10. 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:



Calculate the sample correlation coefficient r.

To calculate the sample correlation coefficient, we need to use the formula:

r = Σ((xi - x̄)(yi - ȳ)) / √(Σ(xi - x̄)² * Σ(yi - ȳ)²)

Where:
- Σ denotes the sum
- xi and yi are the individual values in the dataset
- x̄ and ȳ are the mean values of x and y, respectively

Let's calculate the sample correlation coefficient step by step using the given data:

xi (thousands of feet): 1, 2, 3, 4, 5, 6, 7, 8
yi (percentage of unidentified artifacts): 1, 3, 4, 7, 9, 10, 11, 13

1. Calculate the mean values x̄ and ȳ:
x̄ = (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8) / 8 = 4.5
ȳ = (1 + 3 + 4 + 7 + 9 + 10 + 11 + 13) / 8 = 7.25

2. Calculate the deviations from the means (xi - x̄) and (yi - ȳ) for each data point:

(xi - x̄):
-3.5, -2.5, -1.5, -0.5, 0.5, 1.5, 2.5, 3.5

(yi - ȳ):
-6.25, -4.25, -3.25, -0.25, 1.75, 2.75, 3.75, 5.75

3. Calculate the sums of (xi - x̄)² and (yi - ȳ)²:

Σ(xi - x̄)² = (-3.5)² + (-2.5)² + (-1.5)² + (-0.5)² + 0.5² + 1.5² + 2.5² + 3.5² = 56

Σ(yi - ȳ)² = (-6.25)² + (-4.25)² + (-3.25)² + (-0.25)² + 1.75² + 2.75² + 3.75² + 5.75² = 115.5

4. Calculate the sum of ((xi - x̄)(yi - ȳ)):

Σ((xi - x̄)(yi - ȳ)) = (-3.5)(-6.25) + (-2.5)(-4.25) + (-1.5)(-3.25) + (-0.5)(-0.25) + 0.5(1.75) + 1.5(2.75) + 2.5(3.75) + 3.5(5.75) = 124.5

5. Substitute the values into the formula and calculate r:

r = 124.5 / √(56 * 115.5) ≈ 124.5 / √(6456)

Calculating the square root of 6456 gives us approximately 80.37.

r ≈ 124.5 / 80.37 ≈ 1.55

Therefore, the sample correlation coefficient (r) for the given data is approximately 1.55.

To calculate the sample correlation coefficient (r), we need to use the given data and the following formula:

r = [n∑xy - (∑x)(∑y)] / √([n∑x^2 - (∑x)^2] * [n∑y^2 - (∑y)^2])

where:
n = number of data points
∑xy = sum of the products of each x and y value
∑x = sum of the x values
∑y = sum of the y values
∑x^2 = sum of the squared x values
∑y^2 = sum of the squared y values

Using the given data, we have x and y values as follows:

x (elevation in thousands of feet): 2.0 3.0 4.0 5.0 6.0 7.0
y (% of unidentified artifacts): 20.0 23.0 18.0 15.0 10.0 8.0

First, we need to calculate the sums:

∑x = 2.0 + 3.0 + 4.0 + 5.0 + 6.0 + 7.0 = 27.0
∑y = 20.0 + 23.0 + 18.0 + 15.0 + 10.0 + 8.0 = 94.0
∑xy = (2.0 * 20.0) + (3.0 * 23.0) + (4.0 * 18.0) + (5.0 * 15.0) + (6.0 * 10.0) + (7.0 * 8.0) = 292.0
∑x^2 = (2.0)^2 + (3.0)^2 + (4.0)^2 + (5.0)^2 + (6.0)^2 + (7.0)^2 = 91.0
∑y^2 = (20.0)^2 + (23.0)^2 + (18.0)^2 + (15.0)^2 + (10.0)^2 + (8.0)^2 = 2358.0

Now we can substitute these values into the formula to calculate r:

r = [6(292.0) - (27.0)(94.0)] / √([6(91.0) - (27.0)^2] * [6(2358.0) - (94.0)^2])

Simplifying further:

r = (1752.0 - 2538.0) / √([546.0 - 729.0] * [14148.0 - 8836.0])
= -786.0 / √([-183.0] * [5312.0])
= -786.0 / √([-183.0] * [5312.0])
= -786.0 / √(-974096.0)
= -786.0 / 987.66
≈ -0.795

Therefore, the sample correlation coefficient (r) is approximately -0.795.