VIPSolutionsGive step-by-step solution with explanation and final answer:QU: Use “VIT2005" data which is in “PASWRY" package Fit multiple lincr regression of TolPicc on Age and Heating Use $% vel of significance 0 (Wie the Fed model (9)Estimat Pie ofthe house for a 15 years od house which has ceil heating Find pot timate and 90% confidence mera (timate Price of th hous or 15 years kd house which has vo hating. Find point cima and 90% confidence merval
Question:
Give step-by-step solution with explanation and final answer:
QU: Use “VIT2005" data which is in “PASWRY" package
Fit multiple lincr regression of TolPicc on Age and Heating Use $% vel of
significance 0
(Wie the Fed model
(9)Estimat Pie ofthe house for a 15 years od house which has ceil heating
Find pot timate and 90% confidence mera
(timate Price of th hous or 15 years kd house which has vo hating. Find
point cima and 90% confidence merval
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Asked by: Venky
Created at: 2025-09-21 04:44:00
Asked by: Venky
Created at: 2025-09-21 04:44:00
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Let's break down the problem step by step:
---
# Q#2: Multiple Linear Regression with "VIT2005" Data
**Dataset:** `VIT2005` from the `PASWR2` package
**Task:** Regress `TotalPrice` on `Age` and `Heating`
**Significance Level:** 5%
---
## Step 1: **Fit the Multiple Linear Regression Model**
### **(a) Write the Fitted Model**
Assume the data is loaded and the variables are:
- `TotalPrice`: Price of the house (response)
- `Age`: Age of the house (years)
- `Heating`: Type of heating (`central` or `none`, categorical)
**Model Equation:**
\[
\text{TotalPrice} = \beta_0 + \beta_1 \cdot \text{Age} + \beta_2 \cdot \text{Heating}_{\text{central}} + \varepsilon
\]
Where:
- \(\text{Heating}_{\text{central}}\) is an indicator variable (1 if central, 0 if none).
**In R:**
```r
library(PASWR2)
data("VIT2005")
model <- lm(TotalPrice ~ Age + Heating, data = VIT2005)
summary(model)
```
Suppose the output gives coefficients:
- \(\beta_0 = 210000\)
- \(\beta_1 = -2000\)
- \(\beta_2 = 15000\)
**Fitted Model:**
\[
\boxed{
\text{TotalPrice} = 210000 - 2000 \times \text{Age} + 15000 \times \text{Heating}_{\text{central}}
}
\]
---
## Step 2: **Estimate Price and 90% CI for Given Cases**
### **(b) House: 15 Years Old, Central Heating**
- Age = 15
- Heating = central (\(\text{Heating}_{\text{central}} = 1\))
#### **Point Estimate:**
\[
\text{TotalPrice} = 210000 - 2000 \times 15 + 15000 \times 1 = 210000 - 30000 + 15000 = 195000
\]
#### **90% Confidence Interval:**
In R:
```r
newdata <- data.frame(Age = 15, Heating = "central")
predict(model, newdata, interval = "confidence", level = 0.90)
```
Suppose it returns:
- fit = 195000
- lwr = 185000
- upr = 205000
**Final Answer:**
- **Point Estimate:** \$195,000
- **90% CI:** (\$185,000, \$205,000)
---
### **(c) House: 15 Years Old, No Heating**
- Age = 15
- Heating = none (\(\text{Heating}_{\text{central}} = 0\))
#### **Point Estimate:**
\[
\text{TotalPrice} = 210000 - 2000 \times 15 + 15000 \times 0 = 210000 - 30000 = 180000
\]
#### **90% Confidence Interval:**
In R:
```r
newdata <- data.frame(Age = 15, Heating = "none")
predict(model, newdata, interval = "confidence", level = 0.90)
```
Suppose it returns:
- fit = 180000
- lwr = 170000
- upr = 190000
**Final Answer:**
- **Point Estimate:** \$180,000
- **90% CI:** (\$170,000, \$190,000)
---
## **Summary Table**
| Scenario | Point Estimate | 90% CI |
|----------------------------------|---------------|--------------------|
| 15 yrs, Central Heating | \$195,000 | (\$185k, \$205k) |
| 15 yrs, No Heating | \$180,000 | (\$170k, \$190k) |
---
> **Note:** The coefficient values are hypothetical for illustration. Use actual output from R for exact answers.
> **R Code Template:**
```r
library(PASWR2)
data("VIT2005")
model <- lm(TotalPrice ~ Age + Heating, data = VIT2005)
summary(model)
predict(model, data.frame(Age=15, Heating="central"), interval="confidence", level=0.90)
predict(model, data.frame(Age=15, Heating="none"), interval="confidence", level=0.90)
```
Let me know if you need the real output with the actual dataset!
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