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: SYY
35.8143
16.5571
602 265.5
2 053
149.9286
To determine SYY, which represents the sum of squares of the dependent variable (change in free energy), we need to calculate the sum of the squared differences between each observed value and the mean of the dependent variable.
First, we need to calculate the mean of the dependent variable (change in free energy).
Mean (ȳ) = (7 + 6.2 + 9 + 9.5 + 8.5 + 7.8 + 11.2 + 295 + 297.5 + 291 + 289 + 301 + 293 + 286.5) / 14
= 2596.5 / 14
≈ 185.4643
Next, we calculate the squared differences between each observed value and the mean of the dependent variable:
(7 - 185.4643)^2
(6.2 - 185.4643)^2
(9 - 185.4643)^2
(9.5 - 185.4643)^2
(8.5 - 185.4643)^2
(7.8 - 185.4643)^2
(11.2 - 185.4643)^2
(295 - 185.4643)^2
(297.5 - 185.4643)^2
(291 - 185.4643)^2
(289 - 185.4643)^2
(301 - 185.4643)^2
(293 - 185.4643)^2
(286.5 - 185.4643)^2
Calculating these squared differences, we get:
17,199.5
19,840.2215
17,021.2857
17,135.9107
17,043.5357
17,147.9749
16,602.9565
9,883.6321
9,590.4449
8,097.7857
7,486.7143
11,245.5929
7,768.7249
11,004.5325
Finally, we sum up these squared differences to get SYY:
SYY = 17,199.5 + 19,840.2215 + 17,021.2857 + 17,135.9107 + 17,043.5357 + 17,147.9749 + 16,602.9565 + 9,883.6321 + 9,590.4449 + 8,097.7857 + 7,486.7143 + 11,245.5929 + 7,768.7249 + 11,004.5325
= 203,847.4573
To determine SYY, which represents the total sum of squares, we need to calculate the sum of the squared differences between each observed value and the mean value of the dependent variable (in this case, the change in free energy).
Here's how you can calculate SYY:
1. Find the mean (average) of the observed values of the dependent variable (change in free energy). To do this, add up all the observed values and divide the sum by the total number of observations:
mean = (7 + 6.2 + 9 + 9.5 + 8.5 + 7.8 + 11.2 + 295 + 297.5 + 291 + 289 + 301 + 293 + 286.5) / 14
mean = 294.2214
2. Subtract the mean from each observed value and square the result:
(7 - 294.2214)^2 = 82,586.9973
(6.2 - 294.2214)^2 = 83,801.0161
(9 - 294.2214)^2 = 72,553.9538
(9.5 - 294.2214)^2 = 71,268.3485
(8.5 - 294.2214)^2 = 77,203.4432
(7.8 - 294.2214)^2 = 80,530.5825
(11.2 - 294.2214)^2 = 64,765.7485
(295 - 294.2214)^2 = 0.6089
(297.5 - 294.2214)^2 = 10.6449
(291 - 294.2214)^2 = 10.2785
(289 - 294.2214)^2 = 27.1785
(301 - 294.2214)^2 = 46.3085
(293 - 294.2214)^2 = 1.4885
(286.5 - 294.2214)^2 = 59.8729
3. Add up all the squared differences:
82,586.9973 + 83,801.0161 + 72,553.9538 + 71,268.3485 + 77,203.4432 + 80,530.5825 + 64,765.7485 + 0.6089 + 10.6449 + 10.2785 + 27.1785 + 46.3085 + 1.4885 + 59.8729 = 603,619.8171
Therefore, SYY is approximately 603,619.8171.
The closest option is 602,265.5.
To find SYY (the total sum of squares), we need to calculate the sum of the squared differences between each observed y value and the mean y value.
First, calculate the mean of the y values:
mean_y = (7 + 6.2 + 9 + 9.5 + 8.5 + 7.8 + 11.2 + 295 + 297.5 + 291 + 289 + 301 + 293 + 286.5) / 14
mean_y = 2119 / 14
mean_y = 151.3571
Next, calculate the squared differences for each y value:
(7 - 151.3571)^2 = (-144.3571)^2 = 20860.682653
(6.2 - 151.3571)^2 = (-145.1571)^2 = 21086.193735
(9 - 151.3571)^2 = (-142.3571)^2 = 20243.180653
(9.5 - 151.3571)^2 = (-141.8571)^2 = 20114.577633
(8.5 - 151.3571)^2 = (-142.8571)^2 = 20408.774143
(7.8 - 151.3571)^2 = (-143.5571)^2 = 20596.411051
(11.2 - 151.3571)^2 = (-140.1571)^2 = 19641.386293
(295 - 151.3571)^2 = (143.6429)^2 = 20614.258943
(297.5 - 151.3571)^2 = (146.1429)^2 = 21376.558183
(291 - 151.3571)^2 = (139.6429)^2 = 19471.814623
(289 - 151.3571)^2 = (137.6429)^2 = 18936.470943
(301 - 151.3571)^2 = (149.6429)^2 = 22385.245943
(293 - 151.3571)^2 = (141.6429)^2 = 20058.024943
(286.5 - 151.3571)^2 = (135.1429)^2 = 18249.032943
Summing these squared differences:
SYY = 20860.682653 + 21086.193735 + 20243.180653 + 20114.577633 + 20408.774143 + 20596.411051 + 19641.386293 + 20614.258943 + 21376.558183 + 19471.814623 + 18936.470943 + 22385.245943 + 20058.024943 + 18249.032943
SYY = 276377.636963
Therefore, SYY is approximately 276377.637.