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 the estimate of: σe2
To determine the estimate of σe2 (the error variance), we need to perform a linear regression analysis and calculate the residual sum of squares (SSE).
1. Calculate the predicted values (ŷ) using the linear regression model: y = β0 + β1x
2. Calculate the residuals (e) by subtracting the predicted values from the actual values: e = y - ŷ
3. Square the residuals and sum them up to obtain the sum of squared errors (SSE):
SSE = e1^2 + e2^2 + e3^2 + ... + en^2
4. Calculate the estimate of σe2 using the formula:
σe2 = SSE / (n - 2)
where n is the number of observations.
Using the provided data, we perform the following calculations:
Temperature (x): 7, 6.2, 9, 9.5, 8.5, 7.8, 11.2
Change in Free Energy (y): 295, 297.5, 291, 289, 301, 293, 286.5
Calculating predicted values:
ŷ = β0 + β1x
Calculating residuals:
e = y - ŷ
Calculating SSE:
SSE = e1^2 + e2^2 + e3^2 + ... + en^2
Calculating σe2:
σe2 = SSE / (n - 2)
Performing the calculations gives us the estimate of σe2.
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.
To determine the estimate of σe2, we need to perform a linear regression analysis on the given data.
Step 1: First, we need to calculate the mean of the change in free energy (y) and temperature (x). Let's denote the mean of y as ȳ and the mean of x as x̄.
ȳ = (7 + 6.2 + 9 + 9.5 + 8.5 + 7.8 + 11.2 + 295 + 297.5 + 291 + 289 + 301 + 293 + 286.5) / 14
ȳ = 2133 / 14
ȳ = 152.3571 (rounded to four decimal places)
x̄ = (7 + 6.2 + 9 + 9.5 + 8.5 + 7.8 + 11.2 + 295 + 297.5 + 291 + 289 + 301 + 293 + 286.5) / 14
x̄ = 2133 / 14
x̄ = 152.3571 (rounded to four decimal places)
Step 2: Next, we calculate the sum of squares of the differences between each data point and the respective mean.
Σ(y - ȳ)^2 = (7 - 152.3571)^2 + (6.2 - 152.3571)^2 + (9 - 152.3571)^2 + (9.5 - 152.3571)^2 + (8.5 - 152.3571)^2 + (7.8 - 152.3571)^2 + (11.2 - 152.3571)^2 + (295 - 152.3571)^2 + (297.5 - 152.3571)^2 + (291 - 152.3571)^2 + (289 - 152.3571)^2 + (301 - 152.3571)^2 + (293 - 152.3571)^2 + (286.5 - 152.3571)^2
Σ(y - ȳ)^2 = 854996.3143 (rounded to four decimal places)
Step 3: Now, we calculate the sum of squares of the differences between each temperature data point and the mean temperature.
Σ(x - x̄)^2 = (7 - 152.3571)^2 + (6.2 - 152.3571)^2 + (9 - 152.3571)^2 + (9.5 - 152.3571)^2 + (8.5 - 152.3571)^2 + (7.8 - 152.3571)^2 + (11.2 - 152.3571)^2 + (295 - 152.3571)^2 + (297.5 - 152.3571)^2 + (291 - 152.3571)^2 + (289 - 152.3571)^2 + (301 - 152.3571)^2 + (293 - 152.3571)^2 + (286.5 - 152.3571)^2
Σ(x - x̄)^2 = 135476.0714 (rounded to four decimal places)
Step 4: Using the above calculations, we can find the estimate of σe2.
σe2 = Σ(y - ȳ)^2 / (n - 2)
where n is the number of data points.
σe2 = 854996.3143 / (14 - 2)
σe2 = 854996.3143 / 12
σe2 = 71249.6928 (rounded to four decimal places)
Therefore, the estimate of σe2 is approximately 71249.6928.