Residuals ° 90 Fitted values Im hardness-.) 100 110 Residuals vs Fitted Figure 1: 15 1.0 05 Standardized residuals -15 -10 -0.5 .0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 Theoretical Quantles Im(hardness -.) Normal Q-Q Statistical Modelling Exam 25/01/2024 Exercise 1 The data contained in the cement dataset represent the hardness (hardness variable) of 13 types of cement with different chemical compositions. Specifically, each type is obtained with varying proportions of aluminium (aluminium variable), silicate (silicate variable), calcium aluminoferrite (aluminium ferrite), and silicate bic (silicate_bic). The interest is explaining how the hardness of cement depends on the proportions of chemicals. A regression model was fitted for this purpose and produced the following result: Estimate Std. Error t statistic Pr(>|t|) (Intercept) 124.4809 26.7557 4.653 0.0016 aluminium 0.9739 ?? 3.435 0.0089 silicate -0.1405 0.2891 -0.486 0.6400 aluminium ferrite -0.4974 0.2751 ?? ?? silicate_bic ?? 0.3214 -2.481 0.0381 Error sum of squares 49.378 Total sum of squares R² coefficient 2715.763 ?? a) Write the model formulation and assumptions. b) Complete the missing values in the table. For "Pr(> |t)" of aluminium ferrite provide an approximate value. What variables have a statistically significant effect? c) Test the statistical hypothesis corresponding to the statement "the covariates do not have an effect on the hardness of cement". d) On a reduced model ("model B") that includes only the variables aluminium and silicate_bic the error sum of squares is equal to SSEB = 74.762. Perform an F test to compare this model with the complete model ("model A") that includes all the covariates. Interpret the result: which model would you prefer? e) Obtain the coefficient R² of model B. Instead of performing the test in point (d), could you have simply compared the coefficient R² of the two models? Why? f) Figure 1 shows two plots regarding the complete model (model A). Explain what they represent and interpret them.
Residuals ° 90 Fitted values Im hardness-.) 100 110 Residuals vs Fitted Figure 1: 15 1.0 05 Standardized residuals -15 -10 -0.5 .0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 Theoretical Quantles Im(hardness -.) Normal Q-Q Statistical Modelling Exam 25/01/2024 Exercise 1 The data contained in the cement dataset represent the hardness (hardness variable) of 13 types of cement with different chemical compositions. Specifically, each type is obtained with varying proportions of aluminium (aluminium variable), silicate (silicate variable), calcium aluminoferrite (aluminium ferrite), and silicate bic (silicate_bic). The interest is explaining how the hardness of cement depends on the proportions of chemicals. A regression model was fitted for this purpose and produced the following result: Estimate Std. Error t statistic Pr(>|t|) (Intercept) 124.4809 26.7557 4.653 0.0016 aluminium 0.9739 ?? 3.435 0.0089 silicate -0.1405 0.2891 -0.486 0.6400 aluminium ferrite -0.4974 0.2751 ?? ?? silicate_bic ?? 0.3214 -2.481 0.0381 Error sum of squares 49.378 Total sum of squares R² coefficient 2715.763 ?? a) Write the model formulation and assumptions. b) Complete the missing values in the table. For "Pr(> |t)" of aluminium ferrite provide an approximate value. What variables have a statistically significant effect? c) Test the statistical hypothesis corresponding to the statement "the covariates do not have an effect on the hardness of cement". d) On a reduced model ("model B") that includes only the variables aluminium and silicate_bic the error sum of squares is equal to SSEB = 74.762. Perform an F test to compare this model with the complete model ("model A") that includes all the covariates. Interpret the result: which model would you prefer? e) Obtain the coefficient R² of model B. Instead of performing the test in point (d), could you have simply compared the coefficient R² of the two models? Why? f) Figure 1 shows two plots regarding the complete model (model A). Explain what they represent and interpret them.
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
Related questions
Question
For context, the images attached below (the question and the related figure) is from a january 2024 past paper
Transcribed Image Text:Residuals
°
90
Fitted values
Im hardness-.)
100
110
Residuals vs Fitted
Figure 1:
15
1.0
05
Standardized residuals
-15
-10
-0.5
.0
-1.5
-1.0
-0.5 0.0
0.5
1.0
1.5
Theoretical Quantles
Im(hardness -.)
Normal Q-Q
Transcribed Image Text:Statistical Modelling
Exam 25/01/2024
Exercise 1
The data contained in the cement dataset represent the hardness (hardness variable) of 13
types of cement with different chemical compositions. Specifically, each type is obtained with
varying proportions of aluminium (aluminium variable), silicate (silicate variable), calcium
aluminoferrite (aluminium ferrite), and silicate bic (silicate_bic). The interest is explaining
how the hardness of cement depends on the proportions of chemicals.
A regression model was fitted for this purpose and produced the following result:
Estimate Std. Error t statistic Pr(>|t|)
(Intercept)
124.4809
26.7557
4.653
0.0016
aluminium
0.9739
??
3.435
0.0089
silicate
-0.1405
0.2891
-0.486
0.6400
aluminium ferrite
-0.4974
0.2751
??
??
silicate_bic
??
0.3214
-2.481
0.0381
Error sum of squares
49.378
Total sum of squares
R² coefficient
2715.763
??
a) Write the model formulation and assumptions.
b) Complete the missing values in the table. For "Pr(> |t)" of aluminium ferrite provide
an approximate value. What variables have a statistically significant effect?
c) Test the statistical hypothesis corresponding to the statement "the covariates do not have
an effect on the hardness of cement".
d) On a reduced model ("model B") that includes only the variables aluminium and silicate_bic
the error sum of squares is equal to SSEB = 74.762. Perform an F test to compare this
model with the complete model ("model A") that includes all the covariates. Interpret the
result: which model would you prefer?
e) Obtain the coefficient R² of model B. Instead of performing the test in point (d), could
you have simply compared the coefficient R² of the two models? Why?
f) Figure 1 shows two plots regarding the complete model (model A). Explain what they
represent and interpret them.
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