The following model was fitted to explain the selling prices...

The following model was fitted to explain the selling prices of condominiums in a sample of 815 sales: $$ \begin{aligned} \hat{y}= & \underset{(0.91)}{-1264}+\underset{(515)}{48.18 x_1}+\underset{(488)}{3382 x_2}-\underset{(947)}{1859 x_3}+\underset{(768)}{3219 x_4} \\ & +2005 x_5 \quad \bar{R}^2=0.86 \end{aligned} $$ (768) where $\hat{y}=$ selling price of condo, in dollars $x_1=$ square footage of living area $x_2=$ size of garage, in number of cars $x_3=$ age of condo, in years $x_4=$ dummy variable taking the value 1 if the condo has a fireplace and 0 otherwise $x_5=$ dummy variable taking the value 1 if the condo has hardwood floors and 0 if it has vinyl floors a. Interpret the estimated coefficient of $x_4$. b. Interpret the estimated coefficient of $x_5$. c. Find a $95 \%$ confidence interval for the impact of a fireplace on selling price, all other things being equal. d. Test the null hypothesis that type of flooring has no impact on selling price against the alternative that, all other things equal, condos with hardwood floors have a higher selling price than those with vinyl flooring.

The following model was fitted to explain the selling prices of condominiums in a sample of 815 sales:
$$
\begin{aligned}
\hat{y}= & \underset{(0.91)}{-1264}+\underset{(515)}{48.18 x_1}+\underset{(488)}{3382 x_2}-\underset{(947)}{1859 x_3}+\underset{(768)}{3219 x_4} \\
& +2005 x_5 \quad \bar{R}^2=0.86
\end{aligned}
$$
(768)
where
$\hat{y}=$ selling price of condo, in dollars
$x_1=$ square footage of living area
$x_2=$ size of garage, in number of cars
$x_3=$ age of condo, in years
$x_4=$ dummy variable taking the value 1 if the condo has a fireplace and 0 otherwise
$x_5=$ dummy variable taking the value 1 if the condo has hardwood floors and 0 if it has vinyl floors
a. Interpret the estimated coefficient of $x_4$.
b. Interpret the estimated coefficient of $x_5$.
c. Find a $95 \%$ confidence interval for the impact of a fireplace on selling price, all other things being equal.
d. Test the null hypothesis that type of flooring has no impact on selling price against the alternative that, all other things equal, condos with hardwood floors have a higher selling price than those with vinyl flooring.

Key Concepts

Coefficient Interpretation in Regression

In multiple linear regression, each estimated coefficient quantifies the expected change in the dependent variable for a one-unit change in the corresponding independent variable, holding all other variables constant. This concept is fundamental in understanding how different predictors contribute to explaining the outcome variable.

Confidence Intervals

A confidence interval provides a range of plausible values for an unknown population parameter and is constructed using the point estimate plus and minus a margin of error, which reflects sampling variability. It allows researchers to assess the precision and uncertainty associated with an estimated regression coefficient.

Hypothesis Testing in Regression

Hypothesis testing in regression involves evaluating whether the coefficients of certain predictors are statistically significantly different from zero (or another specified value). This is typically accomplished using t-tests, where the test statistic is compared against a critical value from the t-distribution to decide if the null hypothesis can be rejected.

Dummy Variables

Dummy variables are used in regression models to represent categorical data with binary outcomes (e.g., yes/no, presence/absence). Their coefficients indicate the effect of belonging to one category relative to a reference category, holding other factors constant.

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