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4. A study was conducted to investigate the relationship between the size of a house (in square feet) and the selling price of a house (in dollars). The response variable is price in dollars, and we want to study if the covariate of the square footage helps explain the response. A random sample of 522 houses was used, and the linear regression output from R is below. 1m (formula = price sqft, data = house)

4. A study was conducted to investigate the relationship between the size of a house (in square feet) and the selling price of a house (in dollars). The response variable is price in dollars, and we want to study if the covariate of the square footage helps explain the response. A random sample of 522 houses was used, and the linear regression output from R is below. 1m (formula = price sqft, data = house)

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
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**Title: Investigating the Relationship Between House Size and Selling Price** **Introduction:** A study was conducted to examine the correlation between the size of a house (measured in square feet) and its selling price (in dollars). The study's response variable is the price of the house, and the primary objective is to determine if the square footage can help explain variations in the selling price. **Methodology:** To explore this relationship, a random sample of 522 houses was selected. The analysis utilized linear regression, with the square footage (sqft) serving as the predictor variable and the price as the response variable. Below is the output from the R statistical software used in the analysis. **Regression Model:** ```R lm(formula = price ~ sqft, data = house) ``` **Regression Output:** | Coefficient | Estimate | Std. Error | t value | Pr(>|t|) | |-------------|-------------|------------|---------|------------| | (Intercept) | -81432.946 | 11551.846 | -7.049 | 5.74e-12 *** | | sqft | 158.950 | 4.875 | 32.605 | < 2e-16 *** | - **Intercept:** The estimate for the intercept is -81432.946 with a standard error of 11551.846. The t-value is -7.049, and the p-value is extremely small (5.74e-12), which is highly significant (denoted by ***). - **sqft (Square Feet):** The estimate for the coefficient for square footage is 158.950 with a standard error of 4.875. This coefficient has a very high t-value of 32.605, and the p-value is less than 2e-16, indicating a highly significant result (also denoted by ***). **Additional Statistics:** - **Residual Standard Error:** 79120 on 520 degrees of freedom. - **Multiple R-squared:** 0.6715, which suggests that approximately 67.15% of the variability in house prices is explained by the square footage. - **Adjusted R-squared:** 0.6709, providing a slightly adjusted measure of fit that accounts for the number of predictors in the model (in this case, only one predictor). **Interpretation:** The regression analysis shows
Transcribed Image Text:**Title: Investigating the Relationship Between House Size and Selling Price** **Introduction:** A study was conducted to examine the correlation between the size of a house (measured in square feet) and its selling price (in dollars). The study's response variable is the price of the house, and the primary objective is to determine if the square footage can help explain variations in the selling price. **Methodology:** To explore this relationship, a random sample of 522 houses was selected. The analysis utilized linear regression, with the square footage (sqft) serving as the predictor variable and the price as the response variable. Below is the output from the R statistical software used in the analysis. **Regression Model:** ```R lm(formula = price ~ sqft, data = house) ``` **Regression Output:** | Coefficient | Estimate | Std. Error | t value | Pr(>|t|) | |-------------|-------------|------------|---------|------------| | (Intercept) | -81432.946 | 11551.846 | -7.049 | 5.74e-12 *** | | sqft | 158.950 | 4.875 | 32.605 | < 2e-16 *** | - **Intercept:** The estimate for the intercept is -81432.946 with a standard error of 11551.846. The t-value is -7.049, and the p-value is extremely small (5.74e-12), which is highly significant (denoted by ***). - **sqft (Square Feet):** The estimate for the coefficient for square footage is 158.950 with a standard error of 4.875. This coefficient has a very high t-value of 32.605, and the p-value is less than 2e-16, indicating a highly significant result (also denoted by ***). **Additional Statistics:** - **Residual Standard Error:** 79120 on 520 degrees of freedom. - **Multiple R-squared:** 0.6715, which suggests that approximately 67.15% of the variability in house prices is explained by the square footage. - **Adjusted R-squared:** 0.6709, providing a slightly adjusted measure of fit that accounts for the number of predictors in the model (in this case, only one predictor). **Interpretation:** The regression analysis shows
**d. The p-value to test the null hypothesis that the slope on sqft is 0 (H₀: β₁ = 0), is approximately 0. What can you say about sqft being a significant explanatory variable or covariate when explaining price?** In this context, the null hypothesis (H₀) is that the slope (β₁) of the relationship between square footage (sqft) and price is zero, indicating no relationship between these variables. A p-value of approximately 0 implies that the null hypothesis can be rejected with strong evidence. Hence, square footage (sqft) is a statistically significant explanatory variable or covariate when explaining the price. This significant relationship suggests that as square footage changes, there is a predictable change in the price.
Transcribed Image Text:**d. The p-value to test the null hypothesis that the slope on sqft is 0 (H₀: β₁ = 0), is approximately 0. What can you say about sqft being a significant explanatory variable or covariate when explaining price?** In this context, the null hypothesis (H₀) is that the slope (β₁) of the relationship between square footage (sqft) and price is zero, indicating no relationship between these variables. A p-value of approximately 0 implies that the null hypothesis can be rejected with strong evidence. Hence, square footage (sqft) is a statistically significant explanatory variable or covariate when explaining the price. This significant relationship suggests that as square footage changes, there is a predictable change in the price.
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