Solution:
Part a. Estimate of k using the Daniels and Jonhson data.
We will fit a simple lineal regression model to the equation established to estimate k with the data given in Table 2.
Table 2. Natural logarithm of the ratio of partial pressures as a function of time for the
first order homogeneous reaction of N2O5 in a lumped tank.
and we will use α=0.05. From the results obtained in the prior data analysis we have
The following data has been collected:
Table 1. Original Data
*Which of the following two models best represents the relationship between Y and x? Determine the values of all estimated parameters in both models.
Solution:
1. The scatter diagram of the data given in table one was made using minitab.
Figure 1. Scatter diagram of table 1
The relation deviates from a straight line most markedly in that X increases at a much faster rate at large Y than at small y. This suggests that we can try to linearize the relation by plotting ln(y) or some other logarithmic function, where η= y.
First we take the first model η=e^(β0+ β1X) and try the transformed data ln(y) given in table two, where η= y. The scatter diagram for this data, exhibits approximate linear relations.
Table 2. Data transformed using Ln(y)
Figure 2. Scatter diagram of the transformed data given in table 2
With the aid of a standard computer program for regression analysis the following results are obtained by transforming the original data.
Regression Analysis: ln(y) versus X
Analysis of Variance
Thus, the equation of the fitted line is
The proportion of the ln(y) variation that is explained by the straight line model is
which means that the model used to fit the data of table one is accurate, because the result of R^2 and a standard deviation of 0.192228, using a 95% confidence interval.
In comparison with the past model we transformed the data with the power function using the ln(y) and the ln(x) which gives the following results:
Table 3. Transformed Data ln(x) and ln(y)
Figure 3. Scattered plot of the transformed data in Table 3.
Again, with the aid of a standard computer program for regression analysis, like Minitab, the following results are obtained, where X'=ln(x) and Y'=ln(y).
