Question

Statistical concepts for engineering management:

Consider the data in the table on the bottom
a. Generate a model for y as a function of x
b. Is this model useful? Justify your conclusion (based on i) R2 adjusted, ii) Hypothesis test for model coefficient, iii) overall model adequacy test and iv) regression assumptions)
c. If needed, modify model as appropriate and generate the new model.

x y
5 6
6 9
7 11
8 13
9 14
10    15
11   15
12   13

Answer 1

For precise computations and efficiency, calculations are performed in MS excel.

R^2 value is not near to 1.

Summary of linear regression.

Since the model is not looking like it is following a straight line a quadratic equation is assumed.

After obtaining the equations. Mentioned in the below image. For compacted computations, they are arranged into a matrix so that an inverse and multiplication are the only operations to be carried out, to get the coefficients of the quadratic equation.

R^2 = =Σ(yest-yavg)2Σ(y-yavg)2 value should be closer to one.

It ensures the fit is perfect.

Here R^2 is 0.986, Hence going for cubic polynomial is not essential.

The fitting process can also be done in MS excel. Plot the (x,y) data on the scatter plot.

Right-click on any of the points and select format trend line

You can observe polynomial, power, linear, exponential, etc.,

choose polynomial and increase the degree to (2,3,4 up to 6)

you can select the R^2 value as well as the equation. (Highlighted in the image).

Assumptions

1. Normality of the residuals

2. Data is variation is quadratic.

3. The regression model is specified as quadratic

4. No Autocorrelation of the residuals exist

etc

3) New model with cubic polynomial

I hope you have understood.

All the Best.

Please do mention your questions in the comments section.