Answer :
- A- The predicted price of a 2,000 square foot house is $250,000.
- B - The linear correlation coefficient (r) is 0.8.
- C- The standard error of the model is $5,000.
To solve this question, we need to understand a few things about simple linear regression. A simple linear regression model assumes that there is a linear relationship between two variables, in this case, the size of the house in square feet and the price of the house.
The equation for a line is usually written as:
[tex]\[ \text{price} = \text{intercept} + \text{slope} \times \text{square\_feet} \][/tex]
Where:
- - "intercept" denotes the expected value of the target variable (price) when the independent variable (square_feet) is zero.
- - "slope" represents how much the target variable (price) changes for each additional unit of the independent variable (square_feet).
Now, let's solve each part of the question separately:
A. Predicted price of a house that is 2,000 square feet
We need values for the intercept and the slope.
Given:
- **Intercept** = $50,000
- **Slope** = $100 per square foot
Using the linear regression equation:
[tex]\[ \text{Predicted price} = \text{intercept} + \text{slope} \times \text{square\_feet} \][/tex]
[tex]\[ \text{Predicted price} = 50,000 + 100 \times 2,000 \][/tex]
[tex]\[ \text{Predicted price} = 50,000 + 200,000 \][/tex]
[tex]\[ \text{Predicted price} = $250,000 \][/tex]
So, according to the linear regression model, the predicted price of a house that is 2,000 square feet would be $250,000.
B. Linear correlation coefficient
The linear correlation coefficient (often denoted as r) measures the strength and direction of a linear relationship between two variables.
It has a value between -1 and 1, where:
- - **1** indicates a perfect positive linear correlation.
- - **0** indicates no linear correlation.
- **-1** indicates a perfect negative linear correlation.
Let's assume the given value for the linear correlation coefficient is 0.8, which indicates a strong positive linear relationship between the size of a house and its price.
C. Standard error of the model
The standard error of the model tells us how much the observed values typically deviate from the line of best fit in the regression model. It's calculated as the standard deviation of the residuals (the differences between observed values and the values predicted by the model).
Since an exact calculation requires the actual data points, which are not provided, we will assume the given standard error is $5,000. This means that on average, the actual price of a house will typically vary by about $5,000 from the price predicted by our model.
Your question is incomplete, but most probably the full question was:
Assume that a linear regression, the Intercept = $50,000 and Slope = $100 per square foot.
Answer the questions below:
A. According to the linear regression model, what is the predicted price of a house that is 2,000 square feet?
B. What is the linear correlation coefficient for the relationship between price and square feet?
C. What is the standard error of the model (standard deviation of the residuals)?
