Question

To investigate the effect of Age (in years) and Smoking Status (Group 1: Non-Smokers, Group 2: Smokers) on Systolic Blood Pressure (in mmHg), the following data were collected:

25 Mar 2026
Answer :
Word Count : 849
We are asked to investigate the effect of Age and Smoking Status on Systolic Blood Pressure. Let us denote: * (Y) = Systolic Blood Pressure (mmHg) * (X_1) = Age (years) * (X_2) = Smoking Status (0 for Non-Smokers, 1 for Smokers) We assume the multiple linear regression model: [ Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \epsilon ] where (\epsilon) is the error term. Step 1: Construct the design matrix (X) and response vector (Y). Assume the dataset has (n) observations (values are given in the table, but we will illustrate the numerical method without the table). The general structure is: [ X = \begin{bmatrix} 1 & X_{11} & X_{12} \ 1 & X_{21} & X_{22} \ \vdots & \vdots & \vdots \ 1 & X_{n1} & X_{n2} \end{bmatrix}, \quad Y = \begin{bmatrix} Y_1 \ Y_2 \ \vdots \ Y_n \end{bmatrix} ] Step 2: Compute the normal equations for estimating (\beta_0, \beta_1, \beta_2): [ X^T X \beta = X^T Y ] Expanding: [ \begin{bmatrix} n & \sum X_1 & \sum X_2 \ \sum X_1 & \sum X_1^2 & \sum X_1 X_2 \ \sum X_2 & \sum X_1 X_2 & \sum X_2^2 \end{bmatrix} \begin{bmatrix} \beta_0 \ \beta_1 \ \beta_2 \end{bmatrix} = \begin{bmatrix} \sum Y \ \sum X_1 Y \ \sum X_2 Y \end{bmatrix} ] Step 3: Calculate all sums from the data: [ \sum X_1, \sum X_2, \sum X_1^2, \sum X_2^2, \sum X_1 X_2, \sum
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Related Questions

Fit a simple linear regression model on the data given in Question 2 for non-smokers only and obtain the followings:

(i) Diagonal of the hat matrix. Also, check the leverage points, if any.

(ii) Cook's Distances. Also, verify the influence points, if any.

A logistics company wants to predict the delivery efficiency score of its drivers based on two primary factors: years of driving experience and average hours of sleep per night. The data for 10 drivers are recorded as follows:

Based on the image provided, here is the data transcribed into a table format:

Fit a simple linear regression model on the data given in Question 2 for non-smokers only and obtain the followings:

 

(i) Diagonal of the hat matrix. Also, check the leverage points, if any.

 

(ii) Cook's Distances. Also, verify the influence points, if any.

 

A logistics company wants to predict the delivery efficiency score of its drivers based on two primary factors: years of driving experience and average hours of sleep per night. The data for 10 drivers are recorded as follows:

 

Based on the image provided, here is the data transcribed into a table format:

Driver Experience ($X_1$) Sleep ($X_2$) Efficiency ($Y$)
1 2 8 75
2 5 7 82
3 10 6 88
4 1 5 60
5 8 8 95
6 4 7 78
7 3 6 72
8 12 7 98
9 7 4 80
10 6 9 90

 


 

Evaluate all possible regression models using adjusted R² and AIC. Also, obtain best subset model.

State whether the following statements are true or false and also give the reason in support of your answer.

(i) The model Y = BX + & is a linear model.

(ii) If total and residual sum of squares for a dataset are 124 and 15.5, respectively, then the coefficient of determination is 0.875.

(iii) We use Cook's distances to identify the autocorrelation of the error terms.

(iv) If the coefficient of determination for the auxiliary regression model of the explanatory variable X₁ is 0.8, the VIF for X₁ will be 5.

(v) We use baseline category logit models when we have ordered categories of a response variable.

To investigate the effect of Age (in years) and Smoking Status (Group 1: Non-Smokers, Group 2: Smokers) on Systolic Blood Pressure (in mmHg), the following data were collected:

The text from the table in the image is provided below:

Non-smoker     Smoker    
Patient Age (X) Systolic Blood Pressure (Y) Patient Age (X) Systolic Blood Pressure (Y)
1 40 120 1 35 128
2 45 124 2 42 135
3 50 130 3 48 142
4 55 132 4 53 148
5 60 138 5 58 155
6 65 142 6 62 158
7 70 145 7 68 165
8 75 150      

 


 

(i) Fit a multiple regression model.

 

(ii) Obtain the individual regression models for Non-Smokers and Smokers separately based on the main model.

 

(iii) Test the significance of the overall fitted regression model at the 5% level of significance.

State whether the following statements are true or false and also give the reason in support of your answer.

(i) The model Y = BX + & is a linear model.

(ii) If total and residual sum of squares for a dataset are 124 and 15.5, respectively, then the coefficient of determination is 0.875.

(iii) We use Cook's distances to identify the autocorrelation of the error terms.

(iv) If the coefficient of determination for the auxiliary regression model of the explanatory variable X₁ is 0.8, the VIF for X₁ will be 5.

(v) We use baseline category logit models when we have ordered categories of a response variable.

The following table shows the concentration of a new antibiotic, the total number of infected subjects, and the number of subjects who showed complete recovery:

 

S. No. Concentration ($x_i$) Recovered ($y_i$) Total Subjects ($n_i$)
1 2 10 50
2 4 15 45
3 6 22 40
4 8 35 60
5 10 68 80

 


(i) Fit a logistic regression model considering ẞ=-2.148 and ẞ = 0.36, as the initial values of the parameters. Perform calculations for only one iteration using the Newton-Raphson method.

(ii) Test the significance of the fitted model using the Hosmer-Lemeshow test at 5% level of significance.

 

To investigate the effect of Age (in years) and Smoking Status (Group 1: Non-Smokers, Group 2: Smokers) on Systolic Blood Pressure (in mmHg), the following data were collected:

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