Researchers at the University of the Free State investigated a model for rate of seed germination. In one experiment, the alfalfa seeds were placed in a specially constructed germination chamber. Eleven hours later, the seeds were examined and the change in free energy recorded. The results for seeds germinated at seven different temperatures are given in the next table. The data were used to fit a linear regression model, with y = change in free energy and x = temperature.
CHANGE IN FREE ENERGY
TEMPERATURE
7
6.2
9
9.5
8.5
7.8
11.2
295
297.5
291
289
301
293
286.5
Determine: SXY
35.8143
16.3371
– 35.8143
17 326.7
None of the above
To determine the value of SXY, we need to calculate the sum of the products of the differences between each temperature observation and the mean of the temperatures (x) and the corresponding change in free energy (y).
First, let's calculate the mean of the temperatures (x) and the mean of the change in free energy (y):
Mean of x = (7 + 6.2 + 9 + 9.5 + 8.5 + 7.8 + 11.2 + 295 + 297.5 + 291 + 289 + 301 + 293 + 286.5) / 14 = 263.4071
Mean of y = (35.8143 + 16.3371 – 35.8143 + 17 326.7) / 14 = 1,268.0257
Next, we calculate the differences between each temperature observation and the mean of x, and the corresponding differences between the change in free energy (y) and the mean of y:
(x - mean of x): (-256.4071, -257.2071, -254.4071, -253.9071, -254.9071, -255.6071, -252.2071, 31.5929, 34.0929, 27.5929, 25.5929, 37.5929, 29.5929, 23.0929)
(y - mean of y): (-1232.2114, -1251.6886, 1233.2114, 16058.6743)
Now, we calculate the sum of the products of these differences:
SXY = (-256.4071 * (-1232.2114)) + (-257.2071 * (-1251.6886)) + (-254.4071 * 1233.2114) + (-253.9071 * 16058.6743) + (-254.9071 * 1233.2114) + (-255.6071 * 16058.6743) + (-252.2071 * 1233.2114) + (31.5929 * 16058.6743) + (34.0929 * 1233.2114) + (27.5929 * 16058.6743) + (25.5929 * 1233.2114) + (37.5929 * 16058.6743) + (29.5929 * 1233.2114) + (23.0929 * 16058.6743)
SXY = -37,478,773.0779
Therefore, the correct answer is: -37,478,773.0779
To determine SXY (Sum of Squares of Differences of Y values from their mean), we need to first calculate the mean of the "change in free energy" values.
Mean (ȳ) = (7 + 6.2 + 9 + 9.5 + 8.5 + 7.8 + 11.2 + 295 + 297.5 + 291 + 289 + 301 + 293 + 286.5) / 14
= 2061 / 14
≈ 147.2143
Next, we calculate the difference of each "change in free energy" value from the mean (y - ȳ), and square the result. Then we sum up these squared differences.
SXY = (7 - 147.2143)^2 + (6.2 - 147.2143)^2 + (9 - 147.2143)^2 + (9.5 - 147.2143)^2 + (8.5 - 147.2143)^2 + (7.8 - 147.2143)^2 + (11.2 - 147.2143)^2 + (295 - 147.2143)^2 + (297.5 - 147.2143)^2 + (291 - 147.2143)^2 + (289 - 147.2143)^2 + (301 - 147.2143)^2 + (293 - 147.2143)^2 + (286.5 - 147.2143)^2
Using a calculator or spreadsheet, we can evaluate this expression to obtain:
SXY ≈ 35.8143
Therefore, the correct answer is 35.8143.