Question 2: A real estate analyst is studying the relationship betwee…

1. A real estate analyst is studying the relationship between the size of a house ($x$, in square feet) and its selling price ( $y$, in thousands of dollars). The analyst collects data from $n$ houses in a city and wants to determine if there is a statistically significant relationship between the size of the house and its selling price.

The relationship between the variables is modeled as:

$y_i = \beta_0 + \beta_1 x_i + \epsilon_i \quad ; \quad 1 \leq i \leq n$

where:

• $y_i$ : Selling price of the $i^{th}$ house (in thousands of dollars).
• $x_i$ : Size of the $i^{th}$ house (in square feet).
• $\beta_0$ : The intercept, representing the average selling price of a house with size $x = 0$.
• $\beta_1$ : The slope, representing the change in selling price for every additional square foot of house size.
• $\epsilon_i$: Random error terms, assumed to follow a normal distribution $\epsilon_i \sim N(0, \sigma^2)$.

The following output has been obtained for a particular dataset :

1Call:
2lm(formula = y ~ x, data = house_data)
3
4Residuals:
5   Min     1Q Median     3Q    Max
6-6.458 -4.247 -2.036  2.631 12.253
7
8Coefficients:
9             Estimate Std. Error t value Pr(>|t|)
10(Intercept) 26.857367   7.262252   3.698  0.00606 **
11x            0.183111   0.002945  62.172 4.98e-12 ***
12---
13Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
14
15Residual standard error: 6.654 on 8 degrees of freedom
16Multiple R-squared:  0.9979,    Adjusted R-squared:  0.9977
17F-statistic:  3865 on 1 and 8 DF,  p-value: 4.98e-12

(a) Define the null and alternative hypotheses. [1 Mark]

(b) What are the estimates of $\beta_0$ and $\beta_1$. Interpret the meaning with respect to the given model. [2 Marks]

(c) Write the fitted linear regression model. [1 Mark]

(d) Based on the output obtained, justify if we can reject the null hypothesis. Also, what can you conclude about the relationship between $x$ (Size of house) and $y$ (Selling price)? [2 Marks]