Interpreting the values of slope and intercept coefficients using the CLRM

In summary, the given population regression equation estimates the price of a house based on its square footage. The slope coefficient, β(1), suggests that for every unit increase in square footage, the price of the house will increase by an estimated $108.78. Meanwhile, the intercept coefficient, β(0), represents the estimated price of a house without factoring in square footage. It may not accurately reflect the true price due to the limitations of the model. It would be more accurate to refer to it as the overhead cost.
  • #1
1daj
4
0
Hello there,
I'm given the following population regression equation:
PRICE(i) = β(0) + β(1)SQFT(i) + u(i) where the things in brackets are subscripts and SQFT represents the square footage.
A sample of houses is then given with their cooresponding prices and square footage.
I have solved such that β(1) = 108.7832 and β(0) = 11984.83
My question is how exactly to word the interpretation of the slope coefficient B(1) and the intercept coefficient β(0).
What I believe to be the answer is:
β1 means that a unit change in the sample living area of a house, measured in square feet, will result in an estimated $108.78 change in the price of the house and β0 represents the estimated price of the house that is not attributed to the living area size.

Is it accurate to be refer to an individual house in interpretation, as I have done, or should I be referring to houses collectively and refer to the mean sample living area and mean price.

Any help would be greatly appreciated. Thanks
 
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  • #2
What you're doing is calculating a rough estimate of how the price of a house depends on the square footage, in a population.

The equation you're using is supposing that it is reasonable to have a non-zero price for a non existing (zero square footage) house, whether this is reasonable or not is up to you to tell - if all prices include the terrain they are standing on it should be ok.

β(1) is the average price per square foot (measured in $/ft²),
β(0) corresponds to the average price of an empty plot. It may however be quite inaccurate, since it is an extrapolation using a model that might not be very accurate.
 
  • #3
winterfors said:
β(0) corresponds to the average price of an empty plot.

In my opinion it sounds better to describe β(0) as the overhead cost. The term avarage is not applicable for both the eatimates of β(0) and β(1). Think that two different regression lines possible (shall that imply two avarages of the same variable from the same sample?)

These are the values which minimizes the error of the assumed model.
 

1. What is the CLRM and why is it important in interpreting slope and intercept coefficients?

The CLRM, or Classical Linear Regression Model, is a statistical model that is commonly used to analyze the relationship between a dependent variable and one or more independent variables. It is important in interpreting slope and intercept coefficients because it provides a framework for understanding the relationship between these variables and allows us to make predictions and draw conclusions based on the data.

2. How do I interpret the slope coefficient in the CLRM?

The slope coefficient, also known as the regression coefficient, represents the change in the dependent variable for every one-unit increase in the independent variable. In other words, it tells us the amount of change in the dependent variable that is associated with a unit change in the independent variable. A positive slope coefficient indicates a positive relationship between the variables, while a negative coefficient indicates a negative relationship.

3. What does the intercept coefficient represent in the CLRM?

The intercept coefficient, also known as the constant term, represents the value of the dependent variable when all independent variables are equal to zero. In other words, it is the predicted value of the dependent variable when there is no relationship with the independent variable. It is important to note that the intercept may not always have a meaningful interpretation, especially if the independent variable is not meaningful at zero.

4. Can the slope and intercept coefficients have a p-value of zero?

No, it is not possible for the slope and intercept coefficients to have a p-value of zero. The p-value represents the probability of obtaining the observed results if the null hypothesis (i.e. no relationship between the variables) is true. In the CLRM, the p-value for the slope and intercept coefficients is used to determine if there is a statistically significant relationship between the variables. A p-value of zero would mean that there is a 100% chance of obtaining the observed results under the null hypothesis, which is not possible.

5. How do I know if the slope and intercept coefficients are statistically significant?

The statistical significance of the slope and intercept coefficients is typically determined by their corresponding p-values. A p-value less than 0.05 is considered statistically significant, indicating that there is a low probability of obtaining the results if the null hypothesis is true. However, it is also important to consider the magnitude and direction of the coefficients when interpreting their significance.

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