Final answer:
To determine if a simple linear regression model is appropriate, a scatter diagram of the data should be created and visually assessed for linearity.
Step-by-step explanation:
The correct answer is option (a).
To test whether a simple linear regression model is appropriate, we should first create a scatter diagram of the data. This will allow us to visually assess the relationship between the independent variable (temperature) and the dependent variable (microstructure).
If there is a clear linear relationship between the two variables, then a simple linear regression model may be appropriate. However, if the scatter plot shows a curved or non-linear pattern, then a more complex regression model may be needed.
In this case, without seeing the data or the scatter plot, we cannot definitively determine whether a simple linear regression model is appropriate. The decision should be based on visual inspection of the scatter plot.
The student's question relates to the exploration of a material's microstructure specifically focusing on partially stabilized zirconia and how its properties change with temperature.
Since the student has provided data which likely includes temperature and some measured property (e.g. porosity or grain size), a key task is to visualize this relationship through a scatter diagram and then perform a linear regression analysis to determine if there is a significant linear relationship (hypothesis test for the slope, β1).
In part c, a Type I error would occur if we incorrectly reject a true null hypothesis, meaning we would conclude that temperature has an effect on the microstructure when, in fact, it does not. For part d, the equation from fitting the simple linear regression model describes how the measured property changes with temperature.
Lastly, for part e, an estimate of the mean porosity at 1400 °C could be made using the regression equation provided the data range includes this temperature, and the model is an appropriate fit.