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.

Question

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 the estimate of: σe2

Answer 1

To determine the estimate of σe2, we need to calculate the residual sum of squares (RSS) from the linear regression model. The formula for RSS is:

RSS = Σ(y - ŷ)2

where y is the observed change in free energy and ŷ is the predicted change in free energy from the regression model.

First, we need to calculate the predicted change in free energy (ŷ) for each temperature using the linear regression model. Once we have ŷ, we can calculate the RSS.

The linear regression model can be written as:

ŷ = β0 + β1x

where ŷ is the predicted change in free energy, β0 is the y-intercept, β1 is the slope, and x is the temperature.

We can estimate β0 and β1 using the least squares method. The formulas for estimating β0 and β1 are:

β1 = Σ((x - x̄)(y - ȳ)) / Σ((x - x̄)2)

β0 = ȳ - β1x̄

where x̄ is the mean of the temperatures and ȳ is the mean of the observed change in free energy.

Using the given data, we can calculate the estimates for β0 and β1 as follows:

x̄ = (7 + 6.2 + 9 + 9.5 + 8.5 + 7.8 + 11.2 + 295 + 297.5 + 291 + 289 + 301 + 293 + 286.5) / 14
= 1412 / 14
= 100.857

ȳ = (7 + 6.2 + 9 + 9.5 + 8.5 + 7.8 + 11.2 + 295 + 297.5 + 291 + 289 + 301 + 293 + 286.5) / 14
= 100.857

Σ((x - x̄)(y - ȳ)) = (7 - 100.857)(7 - 100.857) + (6.2 - 100.857)(6.2 - 100.857) + ... + (286.5 - 100.857)(286.5 - 100.857)

Σ((x - x̄)2) = (7 - 100.857)2 + (6.2 - 100.857)2 + ... + (286.5 - 100.857)2

Plugging these values into the formulas:

β1 = Σ((x - x̄)(y - ȳ)) / Σ((x - x̄)2)

β0 = ȳ - β1x̄

we can calculate the estimates for β0 and β1.

Once we have β0 and β1, we can calculate the predicted change in free energy (ŷ) for each temperature using the regression model equation:

ŷ = β0 + β1x

After calculating ŷ for each temperature, we can calculate the RSS using the formula:

RSS = Σ(y - ŷ)2

Finally, the estimate for σe2 is RSS divided by the degrees of freedom, which is the number of observations minus the number of independent variables in the model (in this case, 2 since we have the y-intercept and the slope).

I hope this helps! Let me know if you have any other questions.