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:
x=5.00 5.75 6.25 7.25 8.25 y= 6 16 22 68 82 Find the value of the coefficient of determination
i have an answer of 0.55 is this correct, but a possible answr of 0.945
Using a statistics calculator, I found correlation coefficient r = 0.972; therefore, the coefficient of determination would be r^2 (note: ^2 means squared). Your rounded answer of 0.945 would then be correct.
To find the coefficient of determination, you need to calculate the correlation coefficient (r) and then square it.
1. Begin by finding the mean of both x and y. The mean of x is (5.00 + 5.75 + 6.25 + 7.25 + 8.25) / 5 = 6.5. The mean of y is (6 + 16 + 22 + 68 + 82) / 5 = 38.8.
2. Calculate the deviations from the mean for both x and y. This is done by subtracting the mean from each individual value. The deviations for x are (-1.5, -0.75, -0.25, 0.75, 1.75), and the deviations for y are (-32.8, -22.8, -16.8, 29.2, 43.2).
3. Multiply the deviations for x and y together for each pair and find the sum of these products. The sum of the products is (1.5 * 32.8) + (0.75 * 22.8) + (0.25 * 16.8) + (0.75 * 29.2) + (1.75 * 43.2) = 168.72.
4. Calculate the square of the deviations for both x and y. The squared deviations for x are (2.25, 0.5625, 0.0625, 0.5625, 3.0625), and the squared deviations for y are (1080.64, 519.84, 282.24, 852.64, 1868.64).
5. Sum the squared deviations for x and y separately. The sum of squared deviations for x is 6.5, and the sum of squared deviations for y is 6614.
6. Calculate the square root of the product of the sums of squared deviations for x and y. The square root of (6.5 * 6614) = 111.37.
7. Divide the sum of products (from step 3) by the square root of (step 6). 168.72 / 111.37 = 1.51.
8. Finally, square the correlation coefficient to find the coefficient of determination. (1.51)² = 2.28.
Based on the calculations, the coefficient of determination is 2.28, which does not match either of the possible answers you provided (0.55 and 0.945). Please double-check your calculations or refer to the data given to make sure the values are accurate.